297Source_NASA51_Rules_for_Exponents52__Addition_and

297

Source: NASA.

5.1 Rules for Exponents 5.2 Addition and Subtrac-

tion of Polynomials 5.3 Multiplication of

Polynomials 5.4 Special Products 5.5 Integer Exponents and

the Quotient Rule 5.6 Division of Polynomials

Digital images were first sent between New York and London by cable in the early 1920s. Unfortunately, the transmission time was 3 hours and the quality was poor. Digital photography was developed further by NASA in the 1960s because ordinary pic- tures were subject to interference when transmitted through space. Today, digital pic- tures remain crystal clear even if they travel millions of miles. The following digital picture shows the planet Mars.

Whether they are taken with a webcam, with a smartphone, or by the Mars rover, digital images comprise tiny units called pixels, which are represented by numbers. As a result, mathematics plays an important role in digital images. In this chapter we illustrate some of the ways mathematics is used to describe digital pictures (see Example 4 and Exercise 80 in Section 5.4). We also discuss how mathematics is used to model things such as heart rate, computer sales, motion of the planets, and interest on money.

5 Polynomials and Exponents

If you want to do something, do it!

—PLAUTUS

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When evaluating expressions, evaluate exponents before performing addition, subtraction, multiplication, division, or negation.

5.1 Rules for Exponents Review of Bases and Exponents ● Zero Exponents ● The Product Rule ● Power Rules

A LOOK INTO MATH N Electronic devices such as tablet computers and smartphones store information as bits. A bit is either a 0 or a 1, and a string of 8 bits is called a byte. In the 1970s, IBM devel- oped punch cards made out of paper that could hold up to 120 bits of information. Today, many computer hard drives can hold more than 1 terabyte of information; that’s more than 8,000,000,000,000 bits! In mathematics, we often use exponents to express such large numbers. In this section, we discuss the rules for exponents.

Review of Bases and Exponents The expression 53 is an exponential expression with base 5 and exponent 3. Its value is

5 # 5 # 5 = 125. In general, bn is an exponential expression with base b and exponent n. If n is a natural number, it indicates the number of times the base b is to be multiplied with itself.

Exponent T bn = b # b # b # g # b

Base c n times

STUDY TIP

Exponents occur throughout mathematics. Because expo- nents are so important, this section is essential for your success in mathematics. It takes practice, so set aside some extra time.

EVALUATING EXPRESSIONS

When evaluating expressions, use the following order of operations.

1. Evaluate exponents. 2. Perform negation. 3. Do multiplication and division from left to right. 4. Do addition and subtraction from left to right.

EXAMPLE 1 Evaluating exponential expressions

Evaluate each expression.

(a) 1 + 24

4 (b) 3a 1

3 b2 (c) - 24 (d) ( - 2)4

Solution (a) Evaluate the exponent first.

4 factors

1 + 24

4 = 1 +

2 # 2 # 2 # 2 4

= 1 + 16

4 = 1 + 4 = 5

∂

2 factors

(b) 3a 1 3 b2 = 3a 1

3 # 1

3 b = 3 # 1

9 =

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2995.1 RULES FOR EXPONENTS

4 factors

- 24 = - (2 # 2 # 2 # 2) = - 16.∂ 4 factors

(d) (�2)4 = (�2)(�2)(�2)(�2) = 16

Now Try Exercises 9, 11, 15, 17

NOTE: Parts (c) and (d) of Example 1 appear to be very similar. However, the negation sign is inside the parentheses in part (d), which means that the base for the exponential expression is - 2. In part (c), no parentheses are used, indicating that the base of the expo- nential expression is 2.

READING CHECK

• Explain how to tell the difference between a negative number raised to a power and the opposite of a positive number raised to a power.

TECHNOLOGY NOTE

Evaluating Exponents Exponents can often be evaluated on calculators by using the ^ key. The four expressions from Example 1 are evaluated with a calculator and the results are shown in the following two figures. When evaluating the last two expressions on your calculator, remember to use the negation key rather than the subtraction key.

1�2^4/4 5

3(1/3)2�Frac 1/3

�2^4 �16

(�2)^4 16

CALCULATOR HELP To evaluate exponents, see Appendix A (page AP-1).

Zero Exponents So far we have discussed natural number exponents. What if an exponent is 0? What does 20 equal? To answer these questions, consider Table 5.1, which shows values for decreas- ing powers of 2. Note that each time the power of 2 decreases by 1, the resulting value is divided by 2. For this pattern to continue, we need to define 20 to be 1 because dividing 2 by 2 results in 1.

This discussion suggests that 20 = 1, and is generalized as follows.

TABLE 5.1 Powers of 2

Power of 2 Value

23 8

22 4

21 2

20 ? ZERO EXPONENT

For any nonzero real number b,

b0 = 1.

The expression 00 is undefined.

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EXAMPLE 2 Evaluating zero exponents

Evaluate each expression. Assume that all variables represent nonzero numbers.

(a) 70 (b) 3a 4 9 b0 (c) a x2y5

3z b0

Solution (a) 70 = 1 (b) 314920 = 3(1) = 3. (Note that the exponent 0 does not apply to 3.) (c) All variables are nonzero, so the expression inside the parentheses is also nonzero.

Thus 1x2y53z 20 = 1. Now Try Exercises 13, 41, 67

THE PRODUCT RULE

For any real number a and natural numbers m and n,

am # an = am + n.

READING CHECK

• State the product rule in your own words.

The expression 43 # 42 has 3 + 2 = 5 factors of 4, so the result is 43 + 2 = 45. To multiply exponential expressions with the same base, add exponents and keep the base.

5 factors

The Product Rule We can use a special rule to calculate products of exponential expressions provided their bases are the same. For example,

43 # 42 = (4 # 4 # 4) # (4 # 4) = 45.v

NOTE: The product 24 # 35 cannot be simplified by using the product rule because the exponential expressions have different bases: 2 and 3.

EXAMPLE 3 Using the product rule

Multiply and simplify. (a) 23 # 22 (b) x4x5 (c) 2 x2 # 5x6 (d) x3(2 x + 3x2) Solution (a) 23 # 22 = 23 + 2 = 25 = 32 (b) x4x5 = x4 + 5 = x9 (c) Begin by applying the commutative property of multiplication to write the product in a

more convenient order.

2 x2 # 5x6 = 2 # 5 # x2 # x6 = 10x2 + 6 = 10x8 (d) To simplify this expression, first apply the distributive property.

3�1 T T 3�2

x3(2 x + 3x2) = x3 # 2 x + x3 # 3x2 = 2 x4 + 3x5 c Exponent is 1.

Now Try Exercises 21, 23, 27, 71

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3015.1 RULES FOR EXPONENTS

EXAMPLE 4 Applying the product rule

Multiply and simplify. (a) x # x3 (b) (a + b)(a + b)4

Solution (a) Begin by writing x as x1. Then x1 # x3 = x1 + 3 = x4. (b) First write (a + b) as (a + b)1. Then

(a � b)1 # (a � b)4 = (a � b)1 + 4 = (a � b)5. Now Try Exercises 19, 63

Power Rules How should (43)2 be evaluated? To answer this question, consider how the product rule can be used in evaluating

(43)2 = 43 # 43 = 43 + 3 = 46.

5 Product rule

5 3 + 3=3 # 2 Similarly,

5 + 5 + 5=5 # 3

$%& (a5)3 = a5 # a5 # a5 = a5 + 5 + 5 = a15.

Product rule

This discussion suggests that to raise a power to a power, we multiply the exponents.

RAISING A POWER TO A POWER

For any real number a and natural numbers m and n,

(am)n = amn.

EXAMPLE 5 Raising a power to a power

Simplify each expression. (a) (32)4 (b) (a3)2

Solution (a) (32)4 = 32 # 4 = 38 (b) (a3)2 = a3 # 2 = a6

Now Try Exercises 31, 33

To decide how to simplify the expression (2 x)3, consider

(2 x)3 = 2 x # 2 x # 2 x = (2 # 2 # 2) # (x # x # x) = 23x3. 3 factors 3 factors 3 factors

∂ ¶ ¶

To raise a product to a power, we raise each factor to the power.

NOTE: If an exponent does not appear in an expression, it is assumed to be 1. For example, x can be written as x1 and (x + y) can be written as (x + y)1.

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READING CHECK

• State the rule for raising a product to a power in your own words.

RAISING A PRODUCT TO A POWER

For any real numbers a and b and natural number n,

(ab)n = anbn.

EXAMPLE 6 Raising a product to a power

Simplify each expression. (a) (3z)2 (b) ( - 2 x2)3 (c) 4(x2y3)5 (d) ( - 22a5)3

Solution (a) (3z)2 = 32z2 = 9z2 (b) ( - 2 x2)3 = ( - 2)3(x2)3 = - 8x6 (c) 4(x2y3)5 = 4(x2)5( y3)5 = 4 x10y15 (d) ( - 22a5)3 = ( - 4a5)3 = ( - 4)3(a5)3 = - 64a15

Now Try Exercises 37, 39, 43, 45

The following equation illustrates another power rule.a 2 3 b4 = 2

3 # 2

3 =

2 # 2 # 2 # 2 3 # 3 # 3 # 3 =

4 factors

To raise a quotient to a power, raise both the numerator and the denominator to the power.

RAISING A QUOTIENT TO A POWER

For any real numbers a and b and natural number n,a a b bn = an

bn . b � 0

EXAMPLE 7 Raising a quotient to a power

Simplify each expression.

(a) a 2 3 b3 (b) a a

b b9 (c) a a + b

5 b2

Solution

(a) a 2 3 b3 = 23

33 =

27 (b) a a

b b9 = a9

(c) Because the numerator is an expression with more than one term, we must place paren- theses around it before raising it to the power 2.a a + b

5 b2 = (a + b)2

52 =

(a + b)2

Now Try Exercises 51, 53, 55

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The five rules for exponents discussed in this section are summarized as follows.

MAKING CONNECTIONS

Raising a Sum or Difference to a Power

Although there are power rules for products and quotients, there are not similar rules for sums and differences. In general, (a + b)n 3 an + bn and (a - b)n 3 an - bn. For example, (3 + 4)2 = 72 = 49 but 32 + 42 = 9 + 16 = 25. Similarly, (4 - 1)3 = 33 = 27 but 43 - 13 = 64 - 1 = 63.

RULES FOR EXPONENTS

The following rules hold for real numbers a and b, and natural numbers m and n.

Description Rule Example

Zero Exponent b0 = 1, for b � 0 ( - 13)0 = 1

The Product Rule am # an = am + n 54 # 53 = 54 + 3 = 57 Power to a Power (am)n = am # n ( y2)5 = y2 # 5 = y10

Product to a Power (ab)n = anbn ( pq)7 = p7q7

Quotient to a Power a a b bn = an

bn , for b � 0 a x

y b3 = x3

y3 , for y � 0

Simplification of some expressions may require the application of more than one rule of exponents. This is demonstrated in the next example.

EXAMPLE 8 Combining rules for exponents

Simplify each expression.

(a) (2a)2(3a)3 (b) a a2b3 c b4 (c) (2 x3y)2( - 4 x2y3)3

Solution (a) (2a)2(3a)3 = 22a2 # 33a3 Raising a product to a power = 4 # 27 # a2 # a3 Evaluate powers; commutative property = 108a5 Product rule

(b) a a2b3 c b4 = (a2)4(b3)4

c4 Raising a quotient to a power; raising

a product to a power

= a8b12

c4 Raising a power to a power

(c) (2 x3y)2( - 4 x2y3)3 = 22(x3)2y2( - 4)3(x2)3( y3)3 Raising a product to a power = 4 x6y2( - 64)x6y9 Raising a power to a power = 4( - 64)x6x6y2y9 Commutative property = - 256 x12y11 Product rule

Now Try Exercises 47, 49, 61

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N REAL-WORLD CONNECTION Exponents occur frequently in calculations involving yearly percent increases, such as the increase in property value illustrated in the next example.

EXAMPLE 9 Calculating growth in property value

If a parcel of property increases in value by about 11% each year for 20 years, then its value will double three times. (a) Write an exponential expression that represents “doubling three times.” (b) If the property is initially worth $25,000, how much will it be worth after it doubles

3 times?

Solution (a) Doubling three times is represented by 23. (b) 23(25,000) = 8(25,000) = $200,000

Now Try Exercise 85

5.1 Putting It All Together

Bases and Exponents In the expression bn, b is the base and n is the exponent. If n is a natural number, then

bn = b # b # g # b. 5 n times

23 has base 2 and exponent 3.

91 = 9, 32 = 3 # 3 = 9, 43 = 4 # 4 # 4 = 64, and - 62 = - (6 # 6) = - 36

Zero Exponents For any nonzero number b, b0 = 1. 50 = 1, x0 = 1, and (xy3)0 = 1

The Product Rule For any real number a and natural numbers m and n,

am # an = am + n.

24 # 23 = 24 + 3 = 27, x # x2 # x6 = x1 + 2 + 6 = x9, and (x + 1) # (x + 1)2 = (x + 1)3

CONCEPT EXPLANATION EXAMPLES

Raising a Power to a Power

For any real number a and natural numbers m and n,

(am)n = amn.

(24)2 = 24 # 2 = 28, (x2)5 = x2 # 5 = x10, and (a4)3 = a4 # 3 = a12

Raising a Product to a Power

For any real numbers a and b and natural number n,

(ab)n = anbn.

(3x)3 = 33x3 = 27x3, (x2y)4 = (x2)4y4 = x8y4, and ( - xy)6 = ( - x)6y6 = x6y6

Raising a Quotient to a Power

For any real numbers a and b and natural number n,a a

b bn = an

bn . b � 0

a x y b5 = x5

y5 and

a a2b d3 b4 = (a2)4b4

(d3)4 =

a8b4

d12

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3055.1 RULES FOR EXPONENTS

5.1 Exercises

CONCEPTS AND VOCABULARY

1. In the expression bn, b is the _____ and n is the _____.

2. The expression b0 = for any nonzero number b.

3. am # an = 4. (am)n = 5. (ab)n = 6. a a

b bn =

PROPERTIES OF EXPONENTS

Exercises 7–18: Evaluate the expression.

7. 82 8. 43

9. ( - 2)3 10. ( - 3)4

11. - 23 12. - 34

13. 60 14. ( - 0.5)0

15. 3 + 42

2 16. 6 - a - 4

2 b2

17. 4a 1 2 b3 18. 16a 1

4 b2

Exercises 19–74: Simplify the expression. Assume that all variables represent nonzero numbers.

19. 3 # 32 20. 53 # 53 21. 42 # 46 22. 104 # 103

23. x3x6 24. a5a2

25. x2x2x2 26. y7y3y0

27. 4 x2 # 5x5 28. - 2y6 # 5y2 29. 3( - xy3)(x2y) 30. (a2b3)( - ab2)

31. (23)2 32. (103)4

33. (n3)4 34. (z7)3

35. x(x3)2 36. (z3)2(5z5)

37. ( - 7b)2 38. ( - 4z)3

39. (ab)3 40. (xy)8

41. (2 x2)0 42. (3a2)4

43. ( - 4b2)3 44. ( - 3r4t3)2

45. (x2y3)7 46. (rt2)5

47. ( y3)2(x4y)3 48. (ab3)2(ab)3

49. (a2b)2(a2b2)3 50. (x3y)(x2y4)2

51. a 1 3 b3 52. a 5

2 b2

53. a a b b5 54. a x

2 b4

55. a x - y 3 b3 56. a 4

x + y b2

57. a 5 a + b

b2 58. a a - b 2 b3

59. a 2 x 5 b3 60. a 3y

2 b4

61. a 3x2 5y4 b3 62. a a2b3

3 b5

63. (x + y)(x + y)3 64. (a - b)2(a - b)

65. (a + b)2(a + b)3 66. (x - y)5(x - y)4

67. 6(x4y6)0 68. a xy z2 b0

69. a(a2 + 2b2) 70. x3(3x - 5y4) 71. 3a3(4a2 + 2b) 72. 2 x2(5 - 4y3)

73. (r + t)(rt)

74. (x - y)(x2y3)

75. Thinking Generally Students sometimes mistakenly apply the “rule” am # bn � (ab)m + n. In general, this equation is not true. Find values for a, b, m, and n with a � b and m � n that will make this equation true.

76. Thinking Generally Students sometimes mistakenly

apply the “rule” (a + b)n � an + bn. In general, this equation is not true. Find values for a, b, and n with a � b that will make this equation true.

APPLICATIONS

Exercises 77–80: Write a simplified expression for the area of the given figure.

77. 2x2

5x2

78. 2ab

2ab

79.

3x2

80.

4y2

7y3

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Exercises 81 and 82: Write a simplified expression for the volume of the given figure.

81.

82.

3a2

3a2 3a2

83. Compound Interest If P dollars are deposited in an account that pays 5% annual interest, then the amount of money in the account after 3 years is P(1 + 0.05)3. Find the amount when P = $1000.

84. Compound Interest If P dollars are deposited in an account that pays 9% annual interest, then the amount of money in the account after 4 years is P(1 + 0.09)4. Find the amount when P = $500.

85. Investment Growth If an investment increases in value by about 10% each year for 22 years, then its value will triple two times.

(a) Write an exponential expression that represents “tripling two times.”

(b) If the investment has an initial value of $8000, how much will it be worth if it triples two times?

86. Stock Value If a stock decreases in value by about 23% each year for 9 years, then its value will be halved three times. (a) Write an exponential expression that represents

“halved three times.” (b) If the stock is initially worth $88 per share, how

much will it be worth if it is halved three times?

WRITING ABOUT MATHEMATICS

87. Are the expressions (4 x)2 and 4 x2 equal in value? Explain your answer.

88. Are the expressions 33 # 23 and 66 equal in value? Explain your answer.

5.2 Addition and Subtraction of Polynomials Monomials and Polynomials ● Addition of Polynomials ● Subtraction of Polynomials ● Evaluating Polynomial Expressions

A LOOK INTO MATH N If you have ever exercised strenuously and then taken your pulse immediately afterward, you may have discovered that your pulse slowed quickly at first and then gradually leveled off. A typical scatterplot of this phenomenon is shown in Figure 5.1(a). These data points cannot be modeled accurately with a line, so a new expression, called a polynomial, is needed to model them. A graph of a polynomial that models these data is shown in Figure 5.1(b) and discussed in Exercise 71. (Source: V. Thomas, Science and Sport.)

NEW VOCABULARY

n Monomial n Degree of a monomial n Coefficient of a monomial n Polynomial n Polynomial in one variable n Binomial n Trinomial n Degree of a polynomial n Like terms

Figure 5.1 Heart Rate After Exercising

10 2 3 4 5 6 7

100

125

150

175

200

Time (minutes)

(a)

Pu ls

e (b

ea ts

p er

m in

ut e)

Heart Rate

10 2 3 4 5 6 7

100

125

150

175

200

Time (minutes)

(b)

Pu ls

e (b

ea ts

p er

m in

ut e)

Polynomial Model

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Monomials and Polynomials A monomial is a number, a variable, or a product of numbers and variables raised to natural number powers. Examples of monomials include

- 3, xy2, 5a2, - z 3, and - 1

2 xy3.

A monomial may contain more than one variable, but monomials do not contain division by variables. For example, the expression 3z is not a monomial. If an expression contains addi- tion or subtraction signs, it is not a monomial.

The degree of a monomial is the sum of the exponents of the variables. If the mono- mial has only one variable, its degree is the exponent of that variable. Remember, when a variable does not have a written exponent, the exponent is implied to be 1. A nonzero number has degree 0, and the number 0 has undefined degree. The number in a monomial is called the coefficient of the monomial. Table 5.2 contains the degree and coefficient of several monomials.

READING CHECK

• How do you determine the degree of a monomial?

A polynomial is a monomial or the sum of two or more monomials. Each monomial is called a term of the polynomial. Addition or subtraction signs separate terms. The expres- sion 2 x2 - 3x + 5 is a polynomial in one variable with three terms. Examples of polyno- mials in one variable include

- 2 x, 3x + 1, 4y2 - y + 7, and x5 - 3x3 + x - 7.

These polynomials have 1, 2, 3, and 4 terms, respectively. A polynomial with two terms is called a binomial, and a polynomial with three terms is called a trinomial.

A polynomial can have more than one variable, as in

x2y2, 2 xy2 + 5x2y - 1, and a2 + 2ab + b2.

Note that all variables in a polynomial are raised to natural number powers. The degree of a polynomial is the degree of the term (or monomial) with greatest degree.

READING CHECK

• How do you determine the degree of a polyomial?

TABLE 5.2 Properties of Monomials

Monomial - 5 6a3b - xy 7y3

Degree 0 4 2 3

Coe fficient - 5 6 - 1 7

EXAMPLE 1 Identifying properties of polynomials

Determine whether the expression is a polynomial. If it is, state how many terms and vari- ables the polynomial contains and give its degree.

(a) 7x2 - 3x + 1 (b) 5x3 - 3x2y3 + xy2 - 2y3 (c) 4 x2 + 5

x + 1 Solution (a) The expression 7x2 - 3x + 1 is a polynomial with three terms and one variable. The

first term 7x2 has degree 2 because the exponent on the variable is 2. The second term - 3x has degree 1 because the exponent on the variable is implied to be 1. The third term 1 has degree 0 because it is a nonzero number. The term with greatest degree is 7x2, so the polynomial has degree 2.

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(b) The expression 5x3 - 3x2y3 + xy2 - 2y3 is a polynomial with four terms and two variables, x and y. The first term has degree 3 because the exponent on the variable is 3. The second term has degree 5 because the sum of the exponents on the variables x and y is 5. Likewise, the third term has degree 3 and the fourth term has degree 3. The term with greatest degree is - 3x2y3, so the polynomial has degree 2 + 3 = 5.

Now Try Exercises 21, 23, 27

Figure 5.2 Adding lw + lw

l lw

+ l lw

= l 2lw

If two monomials contain the same variables raised to the same powers, we call them like terms. We can add or subtract (combine) like terms but cannot combine unlike terms. The terms lw and 2lw are like terms and can be combined geometrically, as shown in Figure 5.3. If we joined one of the small rectangles with area lw and a larger rect- angle with area 2 lw, then the total area is 3lw.

The distributive property justifies combining like terms.

1lw + 2lw = (1 + 2)lw = 3lw

The rectangles shown in Figure 5.4 have areas of ab and xy. Together, their area is the sum, ab + xy. However, because these monomials are unlike terms, they cannot be combined into one term.

Figure 5.3 Adding lw + 2lw

l lw 3lw

+ l 2lw = l

2w 3w

STUDY TIP

Do you want to know what material will be covered on your next exam? Often, the best place to look is on previ- ously completed assignments and quizzes. If a topic is not discussed in class, is not found on the syllabus, and is not part of your assignments, then your time may be better spent studying other topics.

Figure 5.4 Unlike terms: ab + xy

a ab xy

+ x

xy x

= a

Addition of Polynomials Suppose that we have 2 identical rectangles with length l and width w, as illustrated in Figure 5.2. Then the area of one rectangle is lw and the total area is

lw + lw.

This area is equivalent to 2 times lw, which can be expressed as 2lw, or

lw + lw = 2lw.

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EXAMPLE 2 Adding like terms

State whether each pair of expressions contains like terms or unlike terms. If they are like terms, add them.

(a) 5x2, - x2 (b) 7a2b, 10ab2 (c) 4rt2, 1

2 rt2

Solution (a) The terms 5x2 and - x2 have the same variable raised to the same power, so they are like

terms. To add like terms, add their coefficients. Note that the coefficient of - x2 is - 1.

5x2 + ( - x2) = (5 + ( - 1))x2 Distributive property = 4 x2 Add.

(b) The terms 7a2b and 10 ab2 have the same variables, but these variables are not raised to the same powers. They are unlike terms and cannot be added.

like terms. We add them as follows.

4rt2 + 1 2

rt2 = a4 + 1 2 brt2 Distributive property

= 9

2 rt2 Add.

Now Try Exercises 29, 31, 33

EXAMPLE 3 Adding polynomials

Add by combining like terms. (a) (3x + 4) + ( - 4 x + 2) (b) ( y2 - 2y + 1) + (3y2 + y + 11)

Solution (a) (3x + 4) + (�4 x + 2) = 3x + (�4 x) + 4 + 2 = (3 - 4)x + (4 + 2) = - x + 6

(b) ( y2 - 2y + 1) + (3y2 + y + 11) = y2 + 3y2 - 2y + y + 1 + 11 = (1 + 3)y2 + ( - 2 + 1)y + (1 + 11) = 4y2 - y + 12

Now Try Exercises 37, 39

Recall that the commutative and associative properties of addition allow us to rearrange a sum in any order. For example, if we write each subtraction in 2 x - 5 - 4 x + 10 as addition of the opposite, we have

2 x � 5 � 4 x + 10 = 2 x + (�5) + (�4 x) + 10,

and the terms can be rearranged as

2 x + (�4 x) + (�5) + 10 = 2 x � 4 x � 5 + 10.

If we pay attention to the sign in front of each term in a polynomial, the like terms can be combined without rearranging the terms, as demonstrated in the next example.

To add two polynomials, combine like terms, as illustrated in the next example.

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EXAMPLE 4 Adding polynomials

Add (x3 - 3x2 + 7x - 4) + (4 x3 - 5x + 9) by combining like terms.

Solution Remove parentheses and identify like terms with their signs as shown.

x3 � 3x2 � 7x � 4 � 4 x3 � 5x � 9

When like terms (of the same color) are added, the resulting sum is

5x3 � 3x2 � 2 x � 5.

Now Try Exercise 41

Polynomials can also be added vertically, as demonstrated in the next example.

EXAMPLE 5 Adding polynomials vertically

Simplify (3x2 - 3x + 5) + ( - x2 + x - 6).

Solution Write the polynomials in a vertical format and then add each column of like terms.

3x2 - 3x + 5 - x2 + x - 6 2 x2 - 2 x - 1 Add like terms in each column.

Regardless of the method used, the same answer should be obtained. However, adding ver- tically requires that like terms be placed in the same column.

Now Try Exercise 47

Subtraction of Polynomials To subtract one integer from another, add the first integer and the additive inverse or oppo- site of the second integer. For example, 3 - 5 is evaluated as follows.

3 - 5 = 3 + ( - 5) Add the opposite. = - 2 Simplify.

Similarly, to subtract one polynomial from another, add the first polynomial and the opposite of the second polynomial. To find the opposite of a polynomial, simply negate each term. Table 5.3 lists some polynomials and their opposites.

READING CHECK

• How do you subtract one polynomial from another?

CRITICAL THINKING

What is the result when a polynomial and its opposite are added?

TABLE 5.3 Opposites of Polynomials

Polynomial Opposite

2 x - 4 - 2 x + 4

- x2 - 2 x + 9 x2 + 2 x - 9

6 x3 - 12 - 6 x3 + 12

- 3x4 - 2 x2 - 8 x + 3 3x4 + 2 x2 + 8 x - 3

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EXAMPLE 6 Subtracting polynomials

Simplify each expression. (a) (3x - 4) - (5x + 1) (b) (5x2 + 2 x - 3) - (6 x2 - 7x + 9) (c) (6 x3 + x2) - ( - 3x3 - 9)

Solution (a) To subtract (5x + 1) from (3x - 4), we add the opposite of (5x + 1), or ( - 5x - 1).

(3x - 4) - (5x + 1) = (3x - 4) + (�5x - 1) = (3 � 5)x + (�4 � 1) = �2 x � 5

(b) The opposite of (6 x2 - 7x + 9) is ( - 6 x2 + 7x - 9).

(5x2 + 2 x - 3) - (6 x2 - 7x + 9) = (5x2 + 2 x - 3) + ( - 6 x2 + 7x - 9) = (5 - 6)x2 + (2 + 7)x + ( - 3 - 9) = - x2 + 9x - 12

(6 x3 + x2) - ( - 3x3 - 9) = (6 x3 + x2) + (3x3 + 9) = (6 + 3)x3 + x2 + 9 = 9x3 + x2 + 9

Now Try Exercises 57, 59, 61

NOTE: Some students prefer to subtract one polynomial from another by noting that a sub- traction sign in front of parentheses changes the signs of all of the terms within the paren- theses. For example, part (a) of the previous example could be worked as follows.

(3x - 4) - (5x + 1) = 3x - 4 - 5x - 1 = (3 - 5)x + ( - 4 - 1) = - 2 x - 5

EXAMPLE 7 Subtracting polynomials vertically

Simplify (5x2 - 2 x + 7) - ( - 3x2 + 3).

Solution To subtract one polynomial from another vertically, simply add the first polynomial and the opposite of the second polynomial. No x-term occurs in the second polynomial, so insert 0x.

5x2 - 2 x + 7 3x2 + 0x - 3 The opposite of - 3x2 + 3 is 3x2 - 3 or 3x2 + 0x - 3. 8 x2 - 2 x + 4 Add like terms in each column.

Now Try Exercise 69

Evaluating Polynomial Expressions Frequently, monomials and polynomials represent formulas that may be evaluated. We illustrate such applications in the next two examples.

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EXAMPLE 8 Writing and evaluating a monomial

Write the monomial that represents the volume of the box having a square bottom, as shown in Figure 5.5. Find the volume of the box if x = 3 feet and y = 2 feet.

Solution The volume of a box is found by multiplying the length, width, and height together. Because the length and width are both x and the height is y, the monomial xxy represents the volume of the box. This can be written x2y. To calculate the volume, let x = 3 and y = 2 in the monomial x2y.

x2y = 32 # 2 = 9 # 2 = 18 cubic feet Now Try Exercise 73

Figure 5.5

EXAMPLE 9 Modeling sales of personal computers

Worldwide sales of personal computers have increased dramatically in recent years, as illustrated in Figure 5.6. The polynomial

0.7868 x2 + 16.72 x + 122.58

approximates the number of computers sold in millions, where x = 0 corresponds to 2000, x = 1 to 2001, and so on. Estimate the number of personal computers sold in 2008 by using both the graph and the polynomial. (Source: International Data Corporation.)

Solution From the graph shown in Figure 5.7, it appears that personal computer sales were slightly more than 300 million, or about 310 million, in 2008.

Figure 5.6

4 6 8 10

100

200

300

400

Year (0 ↔ 2000)

C om

pu te

r Sa

le s

(m ill

io ns

)

Worldwide Computer Sales

Figure 5.7

4 6 8 10

100

200

300

400

Year (0 ↔ 2000)

C om

pu te

r Sa

le s

(m ill

io ns

)

Worldwide Computer Sales

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5.2 Putting It All Together

Monomial A number, variable, or product of numbers and variables raised to natu- ral number powers

The degree is the sum of the exponents.

The coefficient is the number in a monomial.

4 x2y Degree: 3; coefficient: 4

- 6 x2 Degree: 2; coefficient: - 6 - a4 Degree: 4; coefficient: - 1

x Degree: 1; coefficient: 1

- 8 Degree: 0; coefficient: - 8

Polynomial A monomial or the sum of two or more monomials

4 x2 + 8 xy2 + 3y2 Trinomial - 9x4 + 100 Binomial - 3x2y3 Monomial

Like Terms Monomials containing the same vari- ables raised to the same powers

10x and - 2 x, 4 x2 and 3x2

5ab2 and - ab2, 5z and 12 z

CONCEPT EXPLANATION EXAMPLES

Addition of Polynomials To add polynomials, combine like terms.

(x2 + 3x + 1) + (2 x2 - 2 x + 7) = (1 + 2)x2 + (3 - 2)x + (1 + 7) = 3x2 + x + 8

3xy + 5xy = (3 + 5)xy = 8 xy

Opposite of a Polynomial To obtain the opposite of a polyno- mial, negate each term.

Polynomial Opposite

- 2 x2 + x - 6 2 x2 - x + 6 a2 - b2 - a2 + b2

- 3x - 18 3x + 18

Subtraction of Polynomials

To subtract one polynomial from another, add the first polynomial and the opposite of the second polynomial.

(x2 + 3x) - (2 x2 - 5x) = (x2 + 3x) + ( - 2 x2 + 5x) = (1 - 2)x2 + (3 + 5)x = - x2 + 8 x

Evaluating a Polynomial To evaluate a polynomial in x, sub- stitute a value for x in the expression and simplify.

To evaluate the polynomial

3x2 - 2 x + 1 for x = 2,

substitute 2 for x and simplify.

3(2)2 - 2(2) + 1 = 9

The year 2008 corresponds to x = 8 in the given polynomial, so substitute 8 for x and evaluate the resulting expression.

0.7868 x2 + 16.72 x + 122.58 = 0.7868(8)2 + 16.72(8) + 122.58 � 307 million

The graph and the polynomial give similiar results.

Now Try Exercise 71

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5.2 Exercises

CONCEPTS AND VOCABULARY

1. A(n) _____ is a number, a variable, or a product of numbers and variables raised to a natural number power.

2. A(n) _____ is a monomial or a sum of monomials.

3. The _____ of a monomial is the sum of the exponents of the variables.

4. The _____ of a polynomial is the degree of the term with the greatest degree.

5. A polynomial with two terms is called a(n) _____. 6. A polynomial with three terms is called a(n) _____.

7. Two monomials with the same variables raised to the same powers are terms.

8. To add two polynomials, combine terms.

9. To subtract two polynomials, add the first polynomial to the _____ of the second polynomial.

10. Polynomials can be added horizontally or _____.

PROPERTIES OF POLYNOMIALS

Exercises 11–18: Identify the degree and coefficient of the monomial.

11. 3x2 12. y

13. - ab 14. - 2 xy

15. - 5rt 16. 8 x2y5

17. 6 18. - 12

Exercises 19–28: Determine whether the expression is a polynomial. If it is, state how many terms and variables the polynomial contains. Then state its degree.

19. - x 20. 7z

21. 4 x2 - 5x + 9 22. x3 - 9

23. x + 1 x

24. 5

xy + 1

25. 3x -2y-3 26. 52a3b4

27. - 23a4bc 28. - 7y-1z-3

Exercises 29–36: State whether the given pair of expres- sions are like terms. If they are like terms, add them.

29. 5x, - 4 x 30. x2, 8 x2

31. x3, - 6 x3 32. 4 xy, - 9xy

33. 9x, - xy 34. 5x2y, - 3xy2

35. ab, ba 36. rt2, - 2t2r

ADDITION OF POLYNOMIALS

Exercises 37–46: Add the polynomials.

37. (3x + 5) + ( - 4 x + 4)

38. ( - x + 5) + (2 x - 5)

39. (3x2 + 4 x + 1) + (x2 + 4 x)

40. ( - x2 - x) + (2 x2 + 3x - 1)

41. ( y3 + 3y2 - 5) + (3y3 + 4y - 4)

42. (4z4 + z2 - 10) + ( - z4 + 4z - 5)

43. ( - xy + 5) + (5xy - 4)

44. (2a2 + b2) + (3a2 - 5b2)

45. (a3b2 + a2b3) + (a2b3 - a3b2)

46. (a2 + ab + b2) + (a2 - ab + b2)

Exercises 47–50: Add the polynomials vertically.

47. 4 x2 - 2 x + 1 5x2 + 3x - 7

48. 8 x2 + 3x + 5 - x2 - 3x - 9

49. - x2 + x 2 x2 - 8 x - 1

50. a3 - 3a2b + 3ab2 - b3 a3 + 3a2b + 3ab2 + b3

SUBTRACTION OF POLYNOMIALS

Exercises 51–56: Write the opposite of the polynomial.

51. 5x2 52. 17x + 12

53. 3a2 - a + 4

54. - b3 + 3b

55. - 2t2 - 3t + 4 56. 7t2 + t - 10

Exercises 57–66: Subtract the polynomials.

57. (3x + 1) - ( - x + 3)

58. ( - 2 x + 5) - (x + 7)

59. ( - x2 + 6 x) - (2 x2 + x - 2)

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60. (2y2 + 3y - 2) - ( y2 - y)

61. (z3 - 2z2 - z) - (4z2 + 5z + 1)

62. (3z4 - z) - ( - z4 + 4z2 - 5)

63. (4 xy + x2y2) - (xy - x2y2)

64. (a2 + b2) - ( - a2 + b2)

65. (ab2) - (ab2 + a3b)

66. (x2 + 3xy + 4y2) - (x2 - xy + 4y2)

Exercises 67–70: Subtract the polynomials vertically.

67. (x2 + 2 x - 3) - (2 x2 + 7x + 1)

68. (5x2 - 9x - 1) - (x2 - x + 3)

69. (3x3 - 2 x) - (5x3 + 4 x + 2)

70. (a2 + 3ab + 2b2) - (a2 - 3ab + 2b2)

APPLICATIONS

71. Exercise and Heart Rate The polynomial given by 1.6t2 - 28t + 200 calculates the heart rate shown in Figure 5.1(b) in A Look Into Math for this section, where t represents the elapsed time in minutes since exercise stopped. (a) What is the heart rate when the athlete first stops

exercising? (b) What is the heart rate after 5 minutes? (c) Describe what happens to the heart rate after

exercise stops.

72. Cellular Phone Subscribers In the early years of cel- lular phone technology—from 1986 through 1991— the number of subscribers in millions could be mod- eled by the polynomial 0.163x2 - 0.146 x + 0.205, where x = 1 corresponds to 1986, x = 2 to 1987, and so on. The graph illustrates this growth.

1987 0

1989 1991

Year

C el

l P ho

ne S

ub sc

ri be

rs (m

ill io

ns )

(a) Use the graph to estimate the number of cellular phone subscribers in 1990.

(b) Use the polynomial to estimate the number of cel- lular phone subscribers in 1990.

73. Areas of Squares Write a monomial that equals the sum of the areas of the squares. Then calculate this sum for z = 10 inches.

74. Areas of Rectangles Find a monomial that equals the sum of the areas of the three rectangles. Find this sum for a = 5 yards and b = 3 yards.

75. Area of a Figure Find a polynomial that equals the area of the figure. Calculate its area for x = 6 feet.

76. Area of a Rectangle Write a polynomial that gives the area of the rectangle. Calculate its area for x = 3 feet.

77. Areas of Circles Write a polynomial that gives the sum of the areas of two circles, one with radius x and the other with radius y. Find this sum for x = 2 feet and y = 3 feet. Leave your answer in terms of p.

78. Squares and Circles Write a polynomial that gives

the sum of the areas of a square having sides of length x and a circle having diameter x. Approximate this sum to the nearest hundredth of a square foot for x = 6 feet.

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79. World Population The table lists actual and projected world population P in billions for selected years t.

(a) Find the slope of each line segment connecting consecutive data points in the table. Can these data be modeled with a line? Explain.

(b) Does the polynomial 0.077t - 148 give good estimates for the world population in year t? Explain how you decided.

Source: U.S. Census Bureau.

t 1974 1987 1999 2012

P 4 5 6 7

Source: U.S. Postal Service.

t 1963 1975 1987 2002 2007 2011

P 5¢ 13¢ 25¢ 37. 41¢ 44¢

(a) Does the polynomial 0.835t - 1635 model the data in the table exactly?

(b) Does it give approximations that are within 1.5¢ of the actual values?

WRITING ABOUT MATHEMATICS

81. Explain what the terms monomial, binomial, trino- mial, and polynomial mean. Give an example of each.

82. Explain how to determine the degree of a polynomial having one variable. Give an example.

83. Explain how to obtain the opposite of a polynomial. Give an example.

84. Explain how to subtract two polynomials. Give an example.

80. Price of a Stamp The table lists the price P of a first- class postage stamp for selected years t.

Checking Basic ConceptsSECTIONS5.1 and 5.2 1. Evaluate each expression. (a) - 52 (b) 32 - 23

2. Simplify each expression. (a) 103 # 105 (b) (3x2) ( - 4 x5) (c) (a3b)2 (d) a x

z3 b4

3. Simplify each expression. (a) (4y3)0 (b) (x3)2(3x4)2 (c) 2a2(5a3 - 7)

4. State the number of terms and variables in the polynomial 5x3y - 2 x2y + 5. What is its degree?

5. A box has a rectangular bottom twice as long as it is wide.

(a) If the bottom has width w and the box has height h, write a monomial that gives the vol- ume of the box.

(b) Find the volume of the box for w = 12 inches and h = 10 inches.

6. Simplify each expression. (a) (2a2 + 3a - 1) + (a2 - 3a + 7) (b) (4z 3 + 5z) - (2z 3 - 2z + 8) (c) (x2 + 2 xy + y2) - (x2 - 2 xy + y2)

5.3 Multiplication of Polynomials Multiplying Monomials ● Review of the Distributive Properties ● Multiplying Monomials and Polynomials ● Multiplying Polynomials

A LOOK INTO MATH N The study of polynomials dates back to Babylonian civilization in about 1800–1600 B.C. Many eighteenth-century mathematicians devoted their entire careers to the study of poly- nomials. Today, polynomials still play an important role in mathematics, often being used to approximate unknown quantities. In this section we discuss the basics of multiplying polynomials. (Source: Historical Topics for the Mathematics Classroom, Thirty-first Yearbook, NCTM.)

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Multiplying Monomials A monomial is a number, a variable, or a product of numbers and variables raised to natural number powers. To multiply monomials, we often use the product rule for exponents.

EXAMPLE 1 Multiplying monomials

Multiply. (a) - 5x2 # 4 x3 (b) (7xy4)(x3y2) Solution (a) - 5x2 # 4 x3 = ( - 5)(4)x2x3 Commutative property = - 20x2 + 3 The product rule = - 20x5 Simplify.

(b) (7xy4)(x3y2) = 7xx3y4y2 Commutative property = 7x1 + 3y4 + 2 The product rule = 7x4y6 Simplify.

Now Try Exercises 9, 13

READING CHECK

• Which rule for exponents is commonly used to multiply monomials?

Review of the Distributive Properties Distributive properties are used frequently for multiplying monomials and polynomials. For all real numbers a, b, and c,

a(b + c) = ab + ac and a(b - c) = ab - ac.

The first distributive property above can be visualized geometrically. For example,

3(x + 2) = 3x + 6

is illustrated in Figure 5.8. The dimensions of the large rectangle are 3 by x + 2, and its area is 3(x + 2). The areas of the two small rectangles, 3x and 6, equal the area of the large rectangle. Therefore 3(x + 2) = 3x + 6.

In the next example we use the distributive properties to multiply expressions.

EXAMPLE 2 Using distributive properties

Multiply. (a) 2(3x + 4) (b) (3x2 + 4)5 (c) - x(3x - 6)

Solution

(a) 2(3x + 4) = 2 # 3x + 2 # 4 = 6 x + 8 (b) (3x2 + 4)5 = 3x2 # 5 + 4 # 5 = 15x2 + 20 (c) - x(3x - 6) = - x # 3x + x # 6 = - 3x2 + 6 x Now Try Exercises 15, 19, 21

Figure 5.8 Area: 3x + 6

x + 2

3 3x 6

x 2

Distributive Property 3(x � 2) � 3x � 6

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Multiplying Monomials and Polynomials A monomial consists of one term, whereas a polynomial consists of one or more terms separated by + or - signs. To multiply a monomial by a polynomial, we apply the distribu- tive properties and the product rule.

READING CHECK

• What properties are com- monly used to multiply a monomial and a polynomial?

EXAMPLE 3 Multiplying monomials and polynomials

Multiply. (a) 9x(2 x2 - 3) (b) (5x - 8)x2 (c) - 7(2 x2 - 4 x + 6) (d) 4 x3(x4 + 9x2 - 8)

Solution

(a) 9x(2 x2 - 3) = 9x # 2 x2 - 9x # 3 Distributive property = 18 x3 - 27x The product rule

(b) (5x - 8)x2 = 5x # x2 - 8 # x2 Distributive property = 5x3 - 8 x2 The product rule

(d) 4 x3(x4 + 9x2 - 8) = 4 x3 # x4 + 4 x3 # 9x2 - 4 x3 # 8 Distributive property = 4 x7 + 36 x5 - 32 x3 The product rule

Now Try Exercises 23, 25, 27, 29

We can also multiply monomials and polynomials that contain more than one variable.

EXAMPLE 4 Multiplying monomials and polynomials

Multiply. (a) 2 xy(7x2y3 - 1) (b) - ab(a2 - b2)

Solution

(a) 2 xy(7x2y3 - 1) = 2 xy # 7x2y3 - 2 xy # 1 Distributive property = 14 xx2yy3 - 2 xy Commutative property = 14 x3y4 - 2 xy The product rule

(b) - ab(a2 - b2) = - ab # a2 + ab # b2 Distributive property = - aa2b + abb2 Commutative property = - a3b + ab3 The product rule

Now Try Exercises 31, 35

Multiplying Polynomials Monomials, binomials, and trinomials are examples of polynomials. Recall that a monomial has one term, a binomial has two terms, and a trinomial has three terms. In the next example we multiply two binomials, using both geometric and symbolic techniques.

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EXAMPLE 5 Multiplying binomials

Multiply (x + 4)(x + 2) (a) geometrically and (b) symbolically.

Solution (a) To multiply (x + 4)(x + 2) geometrically, draw a rectangle x + 4 long and x + 2

wide, as shown in Figure 5.9(a). The area of this rectangle equals length times width, or (x + 4)(x + 2). The large rectangle can be divided into four smaller rectangles, which have areas of x2, 4 x, 2 x, and 8, as shown in Figure 5.9(b). Thus

(x + 4)(x + 2) = x2 � 4 x � 2 x + 8 = x2 � 6 x + 8.

(b) To multiply (x + 4)(x + 2) symbolically, apply the distributive property two times.

(x � 4) (x + 2) = (x � 4) (x) + (x � 4) (2) = x # x + 4 # x + x # 2 + 4 # 2 = x2 � 4 x � 2 x + 8 = x2 � 6 x + 8

r Now Try Exercise 39

The distributive properties used in part (b) of the previous example show that if we want to multiply (x + 4) by (x + 2), we should multiply every term in x + 4 by every term in x + 2.

(x + 4)(x + 2) = x2 + 2 x + 4 x + 8 = x2 + 6 x + 8

NOTE: This process of multiplying binomials is sometimes called FOIL. This acro- nym may be used to remind us to multiply the first terms (F), outside terms (O), inside terms (I), and last terms (L). The FOIL process is a shortcut for the process used in Example 5(b).

Multiply the First terms to obtain x2. (x + 4)(x + 2)

Multiply the Outside terms to obtain 2 x. (x + 4)(x + 2)

Multiply the Inside terms to obtain 4 x. (x + 4)(x + 2)

Multiply the Last terms to obtain 8. (x + 4)(x + 2)

MULTIPLYING POLYNOMIALS

The product of two polynomials may be found by multiplying every term in the first polynomial by every term in the second polynomial and then combining like terms.

READING CHECK

• What kind of polynomials can be multiplied using the FOIL method?

Figure 5.9

x + 2

x + 4

(a) Area = (x + 4)(x + 2)

2 2x

x 4

4xx

(b) Area = x2 + 4x + 2x + 8

The following statement summarizes how to multiply two polynomials in general.

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EXAMPLE 6 Multiplying binomials

Multiply. Draw arrows to show how each term is found. (a) (3x + 2)(x + 1) (b) (1 - x)(1 + 2 x) (c) (4 x - 3)(x2 - 2 x)

Solution

(a) (3x + 2)(x + 1) = 3x # x + 3x # 1 + 2 # x + 2 # 1 = 3x2 � 3x � 2 x + 2 = 3x2 � 5x + 2

(b) (1 - x)(1 + 2 x) = 1 # 1 + 1 # 2 x - x # 1 - x # 2 x = 1 � 2 x � x - 2 x2

= 1 � x - 2 x2

Now Try Exercises 51, 53, 59

The FOIL process may be helpful for remembering how to multiply two binomials, but it cannot be used for every product of polynomials. In the next example, the general process for multiplying polynomials is used to find products of binomials and trinomials.

EXAMPLE 7 Multiplying polynomials

Multiply. (a) (2 x + 3)(x2 + x - 1) (b) (a - b)(a2 + ab + b2) (c) (x4 + 2 x2 - 5)(x2 + 1)

Solution (a) Multiply every term in (2 x + 3) by every term in (x2 + x - 1).

(2 x + 3)(x2 + x - 1) = 2 x # x2 + 2 x # x - 2 x # 1 + 3 # x2 + 3 # x - 3 # 1 = 2 x3 � 2 x2 � 2 x � 3x2 � 3x - 3 = 2 x3 � 5x2 � x - 3

(b) (a - b)(a2 + ab + b2) = a # a2 + a # ab + a # b2 - b # a2 - b # ab - b # b2 = a3 � a2b � ab2 � a2b � ab2 - b3

= a3 - b3

(c) (x4 + 2 x2 - 5)(x2 + 1) = x4 # x2 + x4 # 1 + 2 x2 # x2 + 2 x2 # 1 - 5 # x2 - 5 # 1 = x6 � x4 � 2 x4 � 2 x2 � 5x2 - 5 = x6 � 3x4 � 3x2 - 5

Now Try Exercises 63, 67, 69

STUDY TIP

Even if you know exactly how to do a math problem correctly, a simple computational error will often cause you to get an incorrect answer. Be sure to take your time on simple calculations.

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Polynomials can be multiplied vertically in a manner similar to multiplication of real numbers. For example, multiplication of 123 times 12 is performed as follows.

1 2 3

* 1 2 2 4 6

1 2 3

1 4 7 6

A similar method can be used to multiply polynomials vertically.

EXAMPLE 8 Multiplying polynomials vertically

Multiply 2 x2 - 4 x + 1 and x + 3 vertically.

Solution Write the polynomials vertically. Then multiply every term in the first polynomial by each term in the second polynomial. Arrange the results so that like terms are in the same column.

2 x2 - 4 x + 1 x + 3

6 x2 - 12 x + 3 2 x3 - 4 x2 + x 2 x3 + 2 x2 - 11x + 3

Multiply top polynomial by 3. Multiply top polynomial by x.

Add each column.

Now Try Exercise 71

MAKING CONNECTIONS

Vertical and Horizontal Formats

Whether you decide to add, subtract, or multiply polynomials vertically or horizontally, remember that the same answer is obtained either way.

EXAMPLE 9 Finding the volume of a box

A box has a width 3 inches less than its height and a length 4 inches more than its height. (a) If h represents the height of the box, write a polynomial that represents the volume of

the box. (b) Use this polynomial to calculate the volume of the box if h = 10 inches.

Solution (a) If h is the height, then h - 3 is the width and h + 4 is the length, as illustrated in

Figure 5.10. Its volume equals the product of these three expressions.

h(h - 3)(h + 4) = (h2 - 3h)(h + 4)

= h2 # h + h2 # 4 - 3h # h - 3h # 4 = h3 + 4h2 - 3h2 - 12h = h3 + h2 - 12h

(b) If h = 10, then the volume is

103 + 102 - 12(10) = 1000 + 100 - 120 = 980 cubic inches.

Now Try Exercise 79

Figure 5.10

h – 3

h + 4

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5.3 Putting It All Together

Distributive Properties For all real numbers a, b, and c,

a(b + c) = ab + ac and a(b - c) = ab - ac.

5(x + 3) = 5x + 15, 3(x - 6) = 3x - 18, and - 2 x(3 - 5x3) = - 6 x + 10x4

Multiplying Polynomials The product of two polynomials may be found by multiplying every term in the first polynomial by every term in the second polynomial and then com- bining like terms.

3x(5x2 + 2 x - 7) = 3x # 5x2 + 3x # 2 x - 3x # 7 = 15x3 + 6 x2 - 21x

(x + 2)(7x - 3) = x # 7x - x # 3 + 2 # 7x - 2 # 3 = 7x2 - 3x + 14 x - 6 = 7x2 + 11x - 6

CONCEPT EXPLANATION EXAMPLES

5.3 Exercises

CONCEPTS AND VOCABULARY

1. The equation x2 # x3 = x5 illustrates what rule of exponents?

2. The equation 3(x - 2) = 3x - 6 illustrates what property?

3. The product of two polynomials may be found by multiplying every in the first polynomial by every in the second polynomial and then com- bining like terms.

4. Polynomials can be multiplied horizontally or _____.

MULTIPLICATION OF MONOMIALS

Exercises 5–14: Multiply.

5. x2 # x5 6. - a # a5 7. - 3a # 4a 8. 7x # 5x 9. 4 x3 # 5x2 10. 6b6 # 3b5 11. xy2 # 4 xy 12. 3ab # ab2 13. ( - 3xy2)(4 x2y) 14. ( - r2t2)( - r3t)

MULTIPLICATION OF MONOMIALS AND POLYNOMIALS

Exercises 15–36: Multiply and simplify the expression.

15. 3(x + 4) 16. - 7(4 x - 1)

19. (4 - z)z 20. 3z(1 - 5z)

17. - 5(9x + 1)

18. 10(1 - 6 x)

21. - y(5 + 3y)

22. (2y - 8)2y

23. 3x(5x2 - 4) 24. - 6 x(2 x3 + 1)

27. - 8(4t2 + t + 1)

28. 7(3t2 - 2t - 5)

31. xy(x + y)

32. ab(2a - 3b)

25. (6 x - 6)x2

26. (1 - 2 x2)3x2

29. n2( - 5n2 + n - 2)

30. 6n3(2 - 4n + n2)

33. x2(x2y - xy2)

34. 2y2(xy - 5)

35. - ab(a3 - 2b3)

36. 5rt(r2 + 2rt + t2)

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MULTIPLICATION OF POLYNOMIALS

Exercises 37–42: (Refer to Example 5.) Multiply the given expression (a) geometrically and (b) symbolically.

37. x(x + 3)

38. 2 x(x + 5)

39. (x + 2)(x + 2)

40. (x + 1)(x + 3)

41. (x + 3)(x + 6)

42. (x + 5)(x + 2)

43. (x + 3)(x + 5)

44. (x - 4)(x - 7)

45. (x - 8)(x - 9)

46. (x + 10)(x + 10)

51. (10y + 7)( y - 1) 52. ( y + 6)(2y + 7)

57. (x - 1)(x2 + 1)

58. (x + 2)(x2 - x)

47. (3z - 2)(2z - 5) 48. (z + 6)(2z - 1)

53. (5 - 3a)(1 - 2a) 54. (4 - a)(5 + 3a)

59. (x2 + 4)(4 x - 3)

60. (3x2 - 1)(3x2 + 1)

49. (8b - 1)(8b + 1) 50. (3t + 2)(3t - 2)

55. (1 - 3x)(1 + 3x) 56. (10 - x)(5 - 2 x)

61. (2n + 1)(n2 + 3) 62. (2 - n2)(1 + n2)

Exercises 43–70: Multiply and simplify the expression.

63. (m + 1)(m2 + 3m + 1)

64. (m - 2)(m2 - m + 5)

65. (3x - 2)(2 x2 - x + 4)

66. (5x + 4)(x2 - 3x + 2)

67. (x + 1)(x2 - x + 1)

68. (x - 2)(x2 + 4 x + 4)

69. (4b2 + 3b + 7)(b2 + 3)

70. ( - 3a2 - 2a + 1)(3a2 - 3)

Exercises 71–76: Multiply the polynomials vertically.

71. (x + 2)(x2 - 3x + 1)

72. (2y - 3)(3y2 - 2y - 2)

73. (a - 2)(a2 + 2a + 4)

74. (b - 3)(b2 + 3b + 9)

75. (3x2 - x + 1)(2 x2 + 1)

76. (2 x2 - 3x - 5)(2 x2 + 3)

77. Thinking Generally If a polynomial with m terms and a polynomial with n terms are multiplied, how many terms are there in the product before like terms are combined?

78. Thinking Generally When a polynomial with m terms is multiplied by a second polynomial, the product contains k terms before like terms are combined. How many terms does the second polynomial contain?

APPLICATIONS

79. Volume of a Box (Refer to Example 9.) A box has a width 4 inches less than its height and a length 2 inches more than its height. (a) If h is the height of the box, write a polynomial

that represents the volume of the box. (b) Use this polynomial to calculate the volume for

h = 25 inches.

80. Surface Area of a Box Use the drawing of the box to write a polynomial that represents each of the following.

b + 1

2b + 1

(a) The area of its bottom (b) The area of its front (c) The area of its right side (d) The total area of its six sides

81. Perimeter of a Pen A rectangular pen for a pet has a perimeter of 100 feet. If one side of the pen has length x, then its area is given by x(50 - x). (a) Multiply this expression. (b) Evaluate the expression obtained in part (a) for

x = 25.

82. Rectangular Garden A rectangular garden has a perimeter of 500 feet. (a) If one side of the garden has length x, then write

a polynomial expression that gives its area. Mul- tiply this expression completely.

(b) Evaluate the expression for x = 50 and interpret your answer.

83. Surface Area of a Cube Write a polynomial that rep- resents the total area of the six sides of the cube hav- ing edges with length x + 1.

x + 1

x + 1 x + 1

84. Surface Area of a Sphere The surface area of a sphere with radius r is 4pr2. Write a polynomial that gives the surface area of a sphere with radius x + 2. Leave your answer in terms of p.

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85. Toy Rocket A toy rocket is shot straight up into the air. Its height h in feet above the ground after t sec- onds is represented by the expression t(64 - 16t). (a) Multiply this expression. (b) Evaluate the expression obtained in part (a) and

the given expression for t = 2. (c) Are your answers in part (b) the same? Should

they be the same?

86. Toy Rocket on the Moon (Refer to the preceding exercise.) If the same toy rocket were flown on the moon, then its height h in feet after t seconds would be t164 - 52 t2 . (a) Multiply this expression. (b) Evaluate the expression obtained in part (a) and

the given expression for t = 2. Did the rocket go higher on the moon?

WRITING ABOUT MATHEMATICS

87. Explain how the acronym FOIL relates to multiplying two binomials, such as x + 3 and 2 x + 1.

88. Does the FOIL method work for multiplying a bino- mial and a trinomial? Explain.

89. Explain in words how to multiply any two polynomi- als. Give an example.

90. Give two properties of real numbers that are used for multiplying 3x(5x2 - 3x + 2). Explain your answer.

Working with Real DataGroup Activity

Biology Some types of worms have a remarkable ability to live without moisture. The following table from one study shows the number of worms W surviv- ing after x days without moisture.

(a) Use the equation W = - 0.0014 x2 - 0.076 x + 50 to find W for each x-value in the table.

Directions: Form a group of 2 to 4 people. Select someone to record the group’s responses for this activity. All members of the group should work cooperatively to answer the questions. If your instructor asks for your results, each member of the group should be prepared to respond.

(b) Discuss how well this equation approximates the data.

Source: D. Brown and P. Rothery, Models in Biology.

x (days) 0 20 40 80 120 160

W (worms) 50 48 45 36 20 3

5.4 Special Products Product of a Sum and Difference ● Squaring Binomials ● Cubing Binomials

A LOOK INTO MATH N Polynomials are often used to approximate real-world phenomena in applications. Poly- nomials have played an important role in the development of everyday products such as tablet computers, cell phones, and automobiles. Even digital images in computers and interest calculations at a bank make use of polynomials. In this section we discuss how to multiply some special types of binomials.

Product of a Sum and Difference Products of the form (a + b)(a - b) occur frequently in mathematics. Other examples include

(x + y)(x - y) and (2r + 3t)(2r - 3t).

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These products can always be multiplied by using the techniques discussed in Section 5.3. However, there is a faster way to multiply these special products.

(a + b)(a - b) = a # a � a # b � b # a - b # b = a2 � ab � ba - b2

= a2 - b2

In words, the product of a sum of two numbers and their difference equals the difference of their squares. We generalize this method as follows.

STUDY TIP

In mathematics, there are often several correct ways to perform a particular process. If your instructor does not require you to use a specified method, choose the one that works best for you.

READING CHECK

• Explain in words how you can find the product of the sum of two numbers and their difference.

PRODUCT OF A SUM AND DIFFERENCE

For any real numbers a and b,

(a + b)(a - b) = a2 - b2.

EXAMPLE 1 Finding products of sums and differences

Multiply. (a) (x + y)(x - y) (b) (z - 2)(z + 2) (c) (2r + 3t)(2r - 3t) (d) (5m2 - 4n2)(5m2 + 4n2)

Solution (a) If we let a = x and b = y, then we can apply the rule

(a + b)(a - b) = a2 - b2.

Thus

(x + y)(x - y) = (x)2 - ( y)2

= x2 - y2.

(b) Because the expressions (z + 2)(z - 2) and (z - 2)(z + 2) are equal by the commu- tative property, we can apply the formula for the product of a sum and difference.

(z - 2)(z + 2) = (z)2 - (2)2

= z2 - 4

(2r + 3t)(2r - 3t) = (2r)2 - (3t)2

= 4r2 - 9t2

(d) (5m2 - 4n2)(5m2 + 4n2) = (5m2)2 - (4n2)2 = 25m4 - 16n4

Now Try Exercises 7, 13, 17

The next example demonstrates how the product of a sum and difference can be used to find some products of numbers mentally.

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EXAMPLE 2 Finding a product

Use the product of a sum and difference to find 22 # 18. Solution Because 22 = 20 + 2 and 18 = 20 - 2, rewrite and evaluate 22 # 18 as follows. 22 # 18 = (20 + 2)(20 - 2) Rewrite 22 as 20 + 2 and 18 as 20 - 2. = 202 - 22 Product of a sum and difference = 400 - 4 Evaluate exponents. = 396 Subtract.

Now Try Exercise 21

Squaring Binomials Because each side of the square shown in Figure 5.11 has length (a + b), its area equals

(a + b)(a + b),

which can be written as (a + b)2. We can multiply this expression as follows.

(a + b)2 = (a + b)(a + b) = a2 � ab � ba + b2

= a2 � 2ab + b2

This result is illustrated geometrically in Figure 5.12, where the area of the large square is (a + b)2. This area can also be found by adding the areas of the four small rectangles.

a2 � ab � ba + b2 = a2 � 2ab + b2

The geometric and symbolic results are the same. Note that to obtain the middle term, 2ab, we can multiply the two terms, a and b, in the binomial and double the result.

A similar product that is also the square of a binomial can be calculated as

(a - b)2 = (a - b)(a - b) = a2 � ab � ba + b2

= a2 � 2ab + b2.

These results are summarized as follows.

READING CHECK

• Explain in words how you can square a binomial.

SQUARING A BINOMIAL

For any real numbers a and b,

(a + b)2 = a2 + 2ab + b2 and (a - b)2 = a2 - 2ab + b2.

That is, the square of a binomial equals the square of the first term, plus (or minus) twice the product of the two terms, plus the square of the last term.

NOTE: (a + b)2 3 a2 + b2. Do not forget the middle term when squaring a binomial.

EXAMPLE 3 Squaring a binomial

Multiply. (a) (x + 3)2 (b) (2 x - 5)2 (c) (1 - 5y)2 (d) (7a2 + 3b)2

Figure 5.11

a + b

Area � (a � b)2

Figure 5.12

a b

(a � b)2 � a2 � 2ab � b2

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Solution (a) Let a = x and b = 3 in the formula (a + b)2 = a2 + 2ab + b2.

(x + 3)2 = (x)2 + 2(x)(3) + (3)2

= x2 + 6 x + 9

(b) Apply the formula (a - b)2 = a2 - 2ab + b2 with a = 2 x and b = 5.

(2 x - 5)2 = (2 x)2 - 2(2 x)(5) + (5)2

= 4 x2 - 20x + 25

(d) (7a2 + 3b)2 = (7a2)2 + 2(7a2)(3b) + (3b)2 = 49a4 + 42a2b + 9b2

Now Try Exercises 27, 29, 35, 39

MAKING CONNECTIONS

Multiplying Binomials and Special Products

If you forget these special products, you can still multiply polynomials by using earlier techniques. For example, the binomial in Example 3(b) can be multiplied as

(2 x - 5)2 = (2 x - 5)(2 x - 5)

= 2 x # 2 x - 2 x # 5 - 5 # 2 x + 5 # 5 = 4 x2 - 10x - 10x + 25 = 4 x2 - 20x + 25.

N REAL-WORLD CONNECTION NASA first developed digital pictures because they were easy to transmit through space and because they provided clear images. A digital image from outer space is shown in Figure 5.13.

Today, digital cameras are readily available, and the Internet uses digital images exclu- sively. The next example shows how polynomials relate to digital pictures.

EXAMPLE 4 Calculating the size of a digital picture

A digital picture comprises tiny square units called pixels. Shading individual pixels creates a picture. A simplified version of a digital picture of the letter T is shown in Figure 5.14. This picture includes an image of the letter T that measures 3 pixels by 3 pixels and a 1-pixel border. (a) If a square digital image measures x pixels by x pixels, then a picture that includes the

image and a 1-pixel border will measure x + 2 pixels by x + 2 pixels. Find a polyno- mial that gives the total number of pixels in the picture, including the border.

(b) Let x = 3 and evaluate the polynomial. Does it agree with Figure 5.14?

Solution (a) The total number of pixels equals (x + 2) times (x + 2), or (x + 2)2.

(x + 2)2 = x2 + 4 x + 4

(b) For x = 3, the polynomial evaluates to 32 + 4 # 3 + 4 = 25, the total number of pixels. This result agrees with Figure 5.14, which has a total of 5 # 5 = 25 pixels with a 3 pixel by 3 pixel image of the letter T inside.

Now Try Exercise 79

Figure 5.13 Digital Picture

Figure 5.14

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Cubing Binomials To calculate the volume of the cube shown in Figure 5.15, we find the product of its length, width, and height. Because all sides have the same measure, its volume is (x + 2)3. That is, the volume equals the cube of x + 2.

To multiply the expression (x + 2)3, we proceed as follows.

(x + 2)3 = (x + 2)(x � 2)2 a3 = a # a2

= (x + 2)(x2 � 4 x � 4) Square the binomial.

= x # x2 + x # 4 x + x # 4 + 2 # x2 + 2 # 4 x + 2 # 4 Multiply the polynomials. = x3 + 4 x2 + 4 x + 2 x2 + 8 x + 8 Simplify terms. = x3 + 6 x2 + 12 x + 8 Combine like terms.

Figure 5.15 Volume = (x + 2)3

x + 2

x + 2 x + 2

EXAMPLE 5 Cubing a binomial

Multiply (2z - 3)3.

Solution

(2z - 3)3 = (2z - 3)(2z � 3)2 a3 = a # a2

= (2z - 3)(4z2 � 12z � 9) Square the binomial.

= 8z3 - 24z2 + 18z - 12z2 + 36z - 27 Multiply the polynomials. = 8z3 - 36z2 + 54z - 27 Combine like terms.

NOTE: (2z - 3)3 3 (2z)3 - (3)3 = 8z3 - 27.

Now Try Exercise 47

CRITICAL THINKING

Suppose that a student is convinced that the expressions

(x + y)3 and x3 + y3

are equal. How could you convince the student otherwise?

EXAMPLE 6 Calculating interest

If a savings account pays x percent annual interest, where x is expressed as a decimal, then after 3 years a sum of money will grow by a factor of (1 + x)3. (a) Multiply this expression. (b) Evaluate the expression for x = 0.10 (or 10%), and interpret the result.

Solution (a) (1 + x)3 = (1 + x)(1 + x)2 a3 = a # a2 = (1 + x)(1 + 2 x + x2) Square the binomial. = 1 + 2 x + x2 + x + 2 x2 + x3 Multiply the polynomials. = 1 + 3x + 3x2 + x3 Combine like terms.

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(b) Let x = 0.1 in the expression 1 + 3x + 3x2 + x3.

1 + 3(0.1) + 3(0.1)2 + (0.1)3 = 1.331

The sum of money will increase by a factor of 1.331. For example, $1000 deposited in this account will grow to $1331 in 3 years.

Now Try Exercise 75

5.4 Putting It All Together

Product of a Sum and Difference

For any real numbers x and y,

(x + y)(x - y) = x2 - y2. (x + 6)(x - 6) = x2 - 36, (2 x - 3)(2 x + 3) = 4 x2 - 9, and (x2 + y2)(x2 - y2) = x4 - y4

Squaring a Binomial For all real numbers x and y,

(x + y)2 = x2 + 2 xy + y2 and (x - y)2 = x2 - 2 xy + y2.

(x + 4)2 = x2 + 8 x + 16, (5x - 2)2 = 25x2 - 20x + 4, and (x2 + y2)2 = x4 + 2 x2y2 + y4

Cubing a Binomial Multiply the binomial by its square. (x + 3)3

= (x + 3)(x + 3)2

= (x + 3)(x2 + 6 x + 9) = x3 + 6 x2 + 9x + 3x2 + 18 x + 27 = x3 + 9x2 + 27x + 27

CONCEPT EXPLANATION EXAMPLES

5.4 Exercises

CONCEPTS AND VOCABULARY

1. (a + b)(a - b) = _____

2. (a + b)2 = _____

3. (a - b)2 = _____

4. (a + b)3 = (a + b) # _____ 5. (True or False?) The two expressions (x + y)2 and

x2 + y2 are equal for all real numbers x and y.

6. (True or False?) The two expressions (r - t)2 and r2 - t2 are equal for all real numbers r and t.

PRODUCT OF A SUM AND DIFFERENCE

Exercises 7–20: Multiply.

7. (x - 3)(x + 3)

8. (x + 6)(x - 6)

9. (4 x - 1)(4 x + 1) 10. (10x + 3)(10x - 3)

11. (1 + 2a)(1 - 2a) 12. (4 - 9b)(4 + 9b)

13. (2 x + 3y)(2 x - 3y) 14. (5r - 6t)(5r + 6t)

15. (ab - 5)(ab + 5)

16. (2 xy + 7)(2 xy - 7)

17. (a2 - b2)(a2 + b2)

18. (3x2 + y2)(3x2 - y2)

19. (x3 - y3)(x3 + y3)

20. (2a4 + b4)(2a4 - b4)

Exercises 21–26: (Refer to Example 2.) Use the product of a sum and a difference to evaluate the expression.

21. 101 # 99 22. 52 # 48 23. 23 # 17 24. 29 # 31 25. 90 # 110 26. 38 # 42

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SQUARING BINOMIALS

Exercises 27–40: Multiply.

27. (a - 2)2

28. (x - 7)2

31. (3b + 5)2 32. (7t + 10)2

35. (1 - b)2 36. (1 - 4a)2

29. (2 x + 3)2

30. (7x - 2)2

33. 134 a - 422

37. (5 + y3)2 38. (9 - 5x2)2

34. 115 a + 122

39. (a2 + b)2

40. (x3 - y3)2

CUBING BINOMIALS

Exercises 41–48: Multiply.

41. (a + 1)3

42. (b + 4)3

43. (x - 2)3

44. ( y - 7)3

45. (2 x + 1)3

46. (4z + 3)3

47. (6u - 1)3 48. (5v + 3)3

MULTIPLICATION OF POLYNOMIALS

Exercises 49–66: Multiply, using any appropriate method.

(b) Find the sum of the areas of the smaller rectan- gles inside the large square.

49. 4(5x + 9) 50. (2 x + 1)(3x - 5)

51. (x - 5)(x + 7)

52. (x + 10)(x + 10)

53. (3x - 5)2 54. (x - 3)(x + 9)

55. (5x + 3)(5x + 4) 56. - x3(x2 - x + 1)

57. (4b - 5)(4b + 5) 58. (x + 5)3

59. - 5x(4 x2 - 7x + 2)

60. (4 x2 - 5)(4 x2 + 5)

61. (4 - a)3 62. 2 x(x - 3)3

63. x(x + 3)2

64. (x - 1)2(x + 1)

65. (x + 2)(x - 2)(x + 1)(x - 1)

66. (x - y)(x + y)(x2 + y2)

67. Thinking Generally Multiply (an + bn)(an - bn). 68. Thinking Generally Multiply (an + bn)2.

APPLICATIONS

Exercises 69–72: Do each part and verify that your answers are the same.

(a) Find the area of the large square by multiplying its length and width.

69. 2

x 2

70. 4

a 4

71.

2x 3

72. x

3y x

Exercises 73 and 74: Find a polynomial that represents the following.

(a) The outside surface area given by the six sides of the cube

(b) The volume of the cube

73.

x + 5

x + 5 x + 5

74.

2x + 1

2x + 1 2x + 1

75. Compound Interest (Refer to Example 6.) If a sum of money is deposited in a savings account that is paying x percent annual interest (expressed as a deci- mal), then this sum of money increases by a factor of (1 + x)2 after 2 years. (a) Multiply this expression. (b) Evaluate the polynomial expression found in

part (a) for an annual interest rate of 10%, or x = 0.10, and interpret the answer.

76. Compound Interest If a sum of money is deposited in a savings account that is paying x percent annual interest, then this sum of money increases by a factor of 11 + 1100x23 after 3 years. (a) Multiply this expression.

(b) Evaluate the polynomial expression in part (a) for an annual interest rate of 8%, or x = 8, and interpret the answer.

77. Probability If there is an x percent chance of rain on each of two consecutive days, then the expres- sion (1 - x)2 gives the percent chance that neither day will have rain. Assume that all percentages are expressed as decimals.

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(a) Multiply this expression. (b) Evaluate the polynomial expression in part (a)

for a 50% chance of rain, or x = 0.50, and inter- pret the answer.

78. Probability If there is an x percent chance of rolling a 6 with one die, then the expression (1 - x)3 gives the percent chance of not rolling a 6 with three dice. Assume that all percentages are expressed as deci- mals or fractions. (a) Multiply this expression. (b) Evaluate the polynomial expression found in part

(a) for a 16.6% chance of rolling a 6, or x = 16, and interpret the answer.

79. Swimming Pool A square swimming pool has an 8-foot-wide sidewalk around it. (a) If the sides of the pool have length z, as shown in

the accompanying figure, find a polynomial that gives the area of the sidewalk.

(b) Evaluate the polynomial in part (a) for z = 60 and interpret the answer.

80. Digital Picture (Refer to Example 4.) Suppose that a digital picture, including its border, is x + 2 pixels by x + 2 pixels and that the actual image inside the border is x - 2 pixels by x - 2 pixels, as shown in the following figure.

x + 2

x + 2 x – 2

x – 2

(a) Find a polynomial that gives the number of pixels in the border.

(b) Evaluate the polynomial in part (a) for x = 5. (c) Sketch a digital picture of the letter H with

x = 5. Does the picture agree with the answer in part (b)?

(d) Digital pictures typically have large values for x. If a picture has x = 500, find the total number of pixels in its border.

WRITING ABOUT MATHEMATICS

81. Explain why (a + b)2 does not equal a2 + b2 in general for real numbers a and b.

82. Explain how to find the cube of a binomial.

Checking Basic ConceptsSECTIONS5.3 and 5.4 1. Multiply each expression. (a) ( - 3xy4)(5x2y) (b) - x(6 - 4 x) (c) 3ab(a2 - 2ab + b2)

2. Multiply each expression. (a) (x + 3)(4 x - 3) (b) (x2 - 1)(2 x2 + 2) (c) (x + y)(x2 - xy + y2)

3. Multiply each expression. (a) (5x + 2)(5x - 2) (b) (x + 3)2

4. Complete each part and verify that your answers are the same.

(a) Find the area of the large square by squaring the length of one of its sides.

(b) Find the sum of the areas of the smaller rect- angles inside the large square.

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NEW VOCABULARY

n Scientific notation

5.5 Integer Exponents and the Quotient Rule Negative Integers as Exponents ● The Quotient Rule ● Other Rules for Exponents ● Scientific Notation

A LOOK INTO MATH N In 2009, astronomers discovered a large planet that orbits a distant star. The planet, named WASP-17b, is about 5,880,000,000,000,000 miles from Earth. Also in 2009, the H1N1 virus was identified in a worldwide influenza pandemic. A typical flu virus measures about 0.00000468 inch across. In this section, we will discuss how integer exponents can be used to write such numbers in scientific notation. (Source: Scientific American.)

READING CHECK

• How is a negative integer power on a base related to the corresponding posi- tive integer power on that base?

TABLE 5.4 Powers of 2

Power of 2 Value

21 2

20 1

2-1 12 = 1 21

2-2 14 = 1 22

Decrease exponent by 1

Divide by 2

Table 5.4 shows that 2-1 = 1 21

and 2-2 = 1 22

. In other words, if the exponent on 2 is nega- tive, then the expression is equal to the reciprocal of the corresponding expression with a positive exponent on 2. This discussion suggests the following definition for negative inte- ger exponents.

NEGATIVE INTEGER EXPONENTS

Let a be a nonzero real number and n be a positive integer. Then

a-n = 1

an .

That is, a-n is the reciprocal of an.

STUDY TIP

Mathematics often builds on concepts that have already been studied. Try to get in the regular habit of reviewing topics from earlier parts of the text.

Negative Integers as Exponents So far we have defined exponents that are whole numbers. For example,

50 = 1 and 23 = 2 # 2 # 2 = 8. What if an exponent is a negative integer? To answer this question, consider Table 5.4, which shows values for decreasing powers of 2. Note that each time the exponent on 2 decreases by 1, the resulting value is divided by 2.

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EXAMPLE 1 Evaluating negative exponents

Simplify each expression. (a) 2-3 (b) 7-1 (c) x -2 (d) (x + y)-8

Solution

(a) Because a�n = 1

an , 2�3 =

23 =

2 # 2 # 2 = 1

8 .

(b) 7�1 = 1

71 =

(d) (x + y)�8 = 1

(x + y)8

Now Try Exercises 7, 19, 25(b)

TECHNOLOGY NOTE

Negative Exponents Calculators can be used to evaluate negative exponents. The figure shows how a graphing cal- culator evaluates the expressions in parts (a) and (b) of Example 1.

2^( �3)�Frac 1/8

7^( �1)�Frac 1/7

CALCULATOR HELP To use the fraction feature (Frac), see Appendix A (pages AP-1 and AP-2).

The rules for exponents discussed in this chapter so far also apply to expressions hav- ing negative exponents. For example, we can apply the product rule, am # an = am + n, as follows.

T Add

2-3 # 22 = 2-3 + 2 = 2-1 = 1 2

We can check this result by evaluating the expression without using the product rule.

2-3 # 22 = 1 23

# 22 = 1 8

# 4 = 4 8 =

EXAMPLE 2 Using the product rule with negative exponents

Evaluate each expression. (a) 52 # 5-4 (b) 3-2 # 3-1

Solution T

Add

(a) 52 # 5�4 = 52 + (�4) = 5�2 = 1 52

= 1

(b) 3-2 # 3-1 = 3-2 + (-1) = 3-3 = 1 33

= 1

27 Now Try Exercise 9

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EXAMPLE 3 Using the rules of exponents

Simplify the expression. Write the answer using positive exponents. (a) x2 # x -5 (b) ( y3)-4 (c) (rt)-5 (d) (ab)-3(a-2b)3

Solution (a) Using the product rule, am # an = am�n, gives

x2 # x�5 = x2 + (�5) = x�3 = 1 x3

(b) Using the power rule, (am)n = amn, gives

( y3)�4 = y3(�4) = y�12 = 1

y12 .

(rt)�5 = r�5t�5 = 1

r5 # 1

t5 =

r5t5 .

This expression could also be simplified as follows.

(rt)-5 = 1

(rt)5 =

r5t5

(d) (ab)-3(a-2b)3 = a-3b -3a-6b3

= a-3 + (-6)b -3 + 3

= a-9b0

= 1

a9 # 1

= 1

Now Try Exercises 21, 27, 29(a)

The Quotient Rule Consider the division problem

32 =

3 # 3 # 3 # 3 3 # 3 =

3 3

# 3 3

# 3 # 3 = 1 # 1 # 32 = 32. Because there are two more 3s in the numerator than in the denominator, the result is

T Subtract

34 - 2 = 32.

That is, to divide exponential expressions having the same base, subtract the exponent of the denominator from the exponent of the numerator and keep the same base. This rule is called the quotient rule, which we express in symbols as follows.

THE QUOTIENT RULE

For any nonzero number a and integers m and n,

an = am - n.

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EXAMPLE 4 Using the quotient rule

Simplify each expression. Write the answer using positive exponents.

(a) 43

45 (b)

6a7

3a4 (c)

xy7

x2y5

Solution

T Subtract

(a) 43

45 = 43 - 5 = 4-2 =

42 =

(b) 6a7

3a4 =

3 # a7

a4 = 2a7 - 4 = 2a3

x2y5 =

x2 # y7

y5 = x1 - 2y7 - 5 = x -1y2 =

Now Try Exercises 13(b), 31(b), 33(a)

MAKING CONNECTIONS

The Quotient Rule and Simplifying Quotients

Some quotients can be simplified mentally. Because

x3 =

x # x # x # x # x x # x # x ,

the quotient x 5

x3 has five factors of x in the numerator and three factors of x in the denominator.

There are two more factors of x in the numerator than in the denominator, 5 - 3 = 2, so this expression simplifies to x2. Similarly,

x5 =

x # x # x x # x # x # x # x

has two more factors of x in the denominator than in the numerator. This quotient x 3

x5 sim-

plifies to 1 x2

. Use this technique to simplify the expressions

z4 ,

a8 , and

x6y2

x3y7 .

Other Rules for Exponents Other rules can be used to simplify expressions with negative exponents.

QUOTIENTS AND NEGATIVE EXPONENTS

The following three rules hold for any nonzero real numbers a and b and positive integers m and n.

1. 1

a-n = an 2.

a-n

b -m =

an 3. a a

b b-n = a b

a bn

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We demonstrate the validity of these rules as follows.

1. 1

a-n =

= 1 # a n

1 = an

2. a-n

b -m =

= 1

an # bm

1 =

3. a a b b-n = a-n

b -n =

= 1

an # bn

1 =

an = a b

a bn

EXAMPLE 5 Working with quotients and negative exponents

Simplify each expression. Write the answer using positive exponents.

(a) 1

2-5 (b)

3-3

4-2 (c)

5x -4y2

10x2y-4 (d) a 2

z2 b-4

Solution

(a) 1

2�5 = 25 = 2 # 2 # 2 # 2 # 2 = 32 (b) 3

�3

4�2 =

33 =

10x2y�4 =

y2y4

2 x2x4 =

2 x6 (d) a 2

z2 b�4 = a z2

2 b4 = z8

24 =

Now Try Exercises 15(b), 17, 37, 47

The rules for natural number exponents that are summarized in Section 5.1 on page 303 also hold for integer exponents. Additional rules for integer exponents are summarized as follows.

RULES FOR INTEGER EXPONENTS

The following rules hold for nonzero real numbers a and b, and positive integers m and n.

Description Rule Example

Negative Exponents (1) a-n = 1

an 9-2 =

92 =

The Quotient Rule am

an = am - n

2-2 = 23 - (-2) = 25

Negative Exponents (2) 1

a-n = an

7-5 = 75

Negative Exponents (3) a-n

b -m =

4-3

2-5 =

Negative Exponents (4) a a b b - n = a b

a bn a 1

5 b - 2 = a 5

1 b2 = 25

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Scientific Notation Powers of 10 are important because they are used in science to express numbers that are either very small or very large in absolute value. Table 5.5 lists some powers of 10. Note that if the power of 10 decreases by 1, the result decreases by a factor of 110, or equivalently, the decimal point is moved one place to the left. Table 5.6 shows the names of some important powers of 10.

READING CHECK

• What kinds of numbers are often expressed in scientific notation?

TABLE 5.5 Powers of 10

Power of 10 Value

103 1000

102 100

101 10

100 1

10-1 110 = 0.1

10-2 1100 = 0.01

10-3 11000 = 0.001

TABLE 5.6 Important Powers of 10

Power of 10 Name

1012 Trillion

109 Billion

106 Million

103 Thousand

10-1 Tenth

10-2 Hundredth

10-3 Thousandth

10-6 Millionth

Recall that numbers written in decimal notation are sometimes said to be in standard form. Decimal numbers that are either very large or very small in absolute value can be expressed in scientific notation.

TECHNOLOGY NOTE

Powers of 10 Calculators make use of sci- entific notation, as illustrated in the accompanying figure. The letter E denotes a power of 10. That is,

2.5E13 = 2.5 : 1013 and 5E�6 = 5 : 10�6.

25000000000000 2.5E13

.000005 5E �6

Note: The calculator has been set in scientific mode.

REAL-WORLD CONNECTION As mentioned in A Look Into Math for this section, the distance to the planet WASP-17b is about 5,880,000,000,000,000 miles. This distance can be written in scientific notation as 5.88 * 1015 because

5,880,000,000,000,000 = 5.88 * 1015. 8 15 decimal places

The 1015 indicates that the decimal point in 5.88 should be moved 15 places to the right. A typical virus is about 0.00000468 inch in diameter, which can be written in scientific

notation as 4.68 * 10-6 because

0.00000468 = 4.68 * 10-6. (1)1* 6 decimal places

The 10�6 indicates that the decimal point in 4.68 should be moved 6 places to the left. The following definition provides a more complete explanation of scientific notation.

SCIENTIFIC NOTATION

A real number a is in scientific notation when a is written in the form b * 10n, where 1 … 0 b 0 6 10 and n is an integer.

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EXAMPLE 6 Converting scientific notation to standard form

Write each number in standard form. (a) 5.23 * 104 (b) 8.1 * 10-3 (c) 6 * 10-2

Solution (a) The positive exponent 4 indicates that the decimal point in 5.23 is to be moved 4 places

to the right.

5.23 * 104 = 5. 2 3 0 0. = 52,300 1 2 3 4

(b) The negative exponent - 3 indicates that the decimal point in 8.1 is to be moved 3 places to the left.

8.1 * 10�3 = 0. 0 0 8. 1 = 0.0081 1 2 3

Now Try Exercises 61, 63

The following steps can be used for writing a positive number a in scientific notation.

NOTE: The scientific notation for a negative number a is the opposite of the scientific nota- tion of 0 a 0 . For example, 450 = 4.5 * 102 and - 450 = - 4.5 * 102.

WRITING A POSITIVE NUMBER IN SCIENTIFIC NOTATION

For a positive, rational number a expressed as a decimal, if 1 … a 6 10, then a = a * 100. Otherwise, use the following process to write a in scientific notation.

1. Move the decimal point in a until it becomes a number b such that 1 … b 6 10. 2. Let the positive integer n be the number of places the decimal point was moved. 3. Write a in scientific notation as follows. • If a Ú 10, then a = b * 10n. • If a 6 1, then a = b * 10-n.

EXAMPLE 7 Writing a number in scientific notation

Write each number in scientific notation. (a) 308,000,000 (U.S. population in 2010) (b) 0.001 (Approximate time in seconds for sound to travel one foot)

Solution (a) Move the assumed decimal point in 308,000,000 eight places to obtain 3.08.

3. 0 8 0 0 0 0 0 0.

1 2 3 4 5 6 7 8

Since 308,000,000 Ú 10, the scientific notation is 3.08 * 108.

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Numbers in scientific notation can be multiplied by applying properties of real numbers and properties of exponents.

(6 * 104) # (3 * 103) = (6 # 3) * (104 # 103) Properties of real numbers = 18 * 107 Product rule = 1.8 * 108 Scientific notation

Division can also be performed with scientific notation.

6 * 104

3 * 103 =

6 3

* 104

103 Property of fractions

= 2 * 101 Quotient rule

These results are supported in Figure 5.16, where the calculator is in scientific mode.

In the next example we show how to use scientific notation in an application.

CALCULATOR HELP To display numbers in scientific notation, see Appendix A (page AP-2).

Figure 5.16

(6�10^4)(3�10^3)

1.8E8 (6�10^4)/(3 �10^3 )

2E1

EXAMPLE 8 Analyzing the cost of Internet advertising

In 2009, a total of $2.38 * 1010 was spent on Internet advertising in the United States. At that time the population of the United States was 3.05 * 108. Determine how much was spent per person on Internet advertising. (Source: New York Times.)

Solution To determine the amount spent per person, divide $2.38 * 1010 by 3.05 * 108.

2.38 * 1010

3.05 * 108 =

2.38

3.05 * 1010 - 8 � 0.78 * 102 = 78

In 2009, about $78 per person was spent on Internet advertising.

Now Try Exercise 97

CRITICAL THINKING

Estimate the number of seconds that you have been alive. Write your answer in scientific notation.

(b) Move the decimal point in 0.001 three places to obtain 1.

0. 0 0 1.

1 2 3

Since 0.001 6 1, the scientific notation is 1 * 10�3.

Now Try Exercises 75, 79

5.5 Putting It All Together

For the rules for integer exponents in this table, assume that a and b are nonzero real num- bers and that m and n are integers.

CONCEPT EXPLANATION EXAMPLES

Negative Integer Exponents a

-n = 1

an 2-4 =

24 =

16 , a-8 =

a8 , and

(xy)-2 = 1

(xy)2 =

x2y2

continued on next page

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CONCEPT EXPLANATION EXAMPLES

Quotient Rule am

an = am - n 7

74 = 72 - 4 = 7-2 =

72 =

49 and

x3 = x6 - 3 = x3

Quotients and Negative Integer Exponents 1.

a-n = an

2. a-n

b -m =

3. a a b b-n = a b

a bn

1. 1

5-2 = 52 = 25

2. x -4

y-2 =

3. a 2 3 b-3 = a 3

2 b3 = 33

23 =

Scientific Notation Write a as b * 10n, where 1 … 0 b 0 6 10 and n is an integer. 23,500 = 2.35 * 104,0.0056 = 5.6 * 10-3, and

1000 = 1 * 103

5.5 Exercises

CONCEPTS AND VOCABULARY

Exercises 1–5: Complete the given rule for integer expo- nents m and n, where a and b are nonzero real numbers.

1. a-n = 2. 1

a-n =

3. am

an = 4.

a-n

b -m =

5. a a b b-n =

6. To write a positive number a in scientific notation as b * 10n, the number b must satisfy _____.

NEGATIVE EXPONENTS

Exercises 7–18: Simplify the expression.

7. (a) 4-1 (b) a 1 3 b-2

8. (a) 6-2 (b) 2.5-1

9. (a) 23 # 2-2 (b) 10-1 # 10-2

10. (a) 3-4 # 32 (b) 104 # 10-2 11. (a) 3-2 # 3-1 # 3-1 (b) (23)-1 12. (a) 2-3 # 25 # 2-4 (b) (3-2)-2

13. (a) (3243)-1 (b) 45

14. (a) (2-232)-2 (b) 55

15. (a) 19

17 (b)

4-3

16. (a) - 64

6 (b)

6-2

17. (a) 5-2

5-4 (b) a 2

7 b-2

18. (a) 7-3

7-1 (b) a 3

4 b-3

continued from previous page

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Exercises 19–50: Simplify the expression. Write the answer using positive exponents.

19. (a) x -1 (b) a-4

20. (a) y-2 (b) z-7

21. (a) x -2 # x -1 # x (b) a-5 # a-2 # a-1 22. (a) y-3 # y4 # y-5 (b) b5 # b -3 # b -6 23. (a) x2y-3x -5y6 (b) (xy)-3

24. (a) a-2b -6b3a-1 (b) (ab)-1

25. (a) (2t)-4 (b) (x + 1)-7

26. (a) (8c)-2 (b) (a + b)-9

27. (a) (a-2)-4 (b) (rt3)-2

28. (a) (4 x3)-3 (b) (xy-3)-2

29. (a) (ab)2(a2)-3 (b) x4

30. (a) (x3)-2(xy)4y-5 (b) y9

31. (a) a10

a-3 (b)

2z4

32. (a) b5

b -2 (b)

12 x2

24 x7

33. (a) - 4 xy5

6 x3y2 (b)

x -4

x -1

34. (a) 12a6b2

8ab3 (b)

y-2

y-7

35. (a) 10b -4

5b -5 (b) a a

b b3

36. (a) 8a-2

2a-3 (b) a 2 x

y b5

37. (a) 6 x2y-4

18 x -5y4 (b)

16a-3b -5

4a-8b

38. (a) m2n4

3m-5n4 (b)

7x -3y-5

x -3y-2

39. (a) 1

y-5 (b)

2t -3

40. (a) 1

z-6 (b)

10b -5

41. (a) 3a4

(2a-2)3 (b)

(2b5)-3

4b -6

42. (a) (2 x4)-2

5x -2 (b)

2y5

(3y-4)-2

43. (a) 1

(xy)-2 (b)

(a2b)-3

44. (a) 1

(ab)-1 (b)

(rt4)-2

45. (a) (3m4n)-2

(2mn-2)3 (b)

( - 4 x4y)2

(xy-5)-3

46. (a) (x4y2)2

( - 2 x2y-2)3 (b)

(m2n-6)-2

(4m2n-4)-3

47. (a) a a b b-2 (b) a u

4v b-1

48. (a) a 2 x y b-3 (b) a 5u

3v b-2

49. (a) a 3a4b 2ab -2

b-2 (b) a 4m4n 5m-3n2

b2 50. (a) a 2 x4y2

3x3y-3 b3 (b) a a-5b2

2ab -2 b-2

51. Thinking Generally For positive integers m and n

show that an

am =

am - n .

52. Thinking Generally For positive integers m and n

show that a-n

a-m = am - n.

SCIENTIFIC NOTATION

Exercises 53–58: (Refer to Table 5.6.) Write the value of the power of 10 in words.

53. 103 54. 106

55. 109 56. 10-1

57. 10-2 58. 10-6

Exercises 59–70: Write the expression in standard form.

59. 2 * 103 60. 5 * 102

61. 4.5 * 104 62. 7.1 * 106

63. 8 * 10-3 64. 9 * 10-1

65. 4.56 * 10-4 66. 9.4 * 10-2

67. 3.9 * 107 68. 5.27 * 106

69. - 5 * 105 70. - 9.5 * 103

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Exercises 71–82: Write the number in scientific notation.

71. 2000 72. 11,000

73. 567 74. 9300

75. 12,000,000 76. 600,000

77. 0.004 78. 0.0008

79. 0.000895 80. 0.0123

81. - 0.05 82. - 0.934

Exercises 83–90: Evaluate the expression. Write the answer in standard form.

83. (5 * 103)(3 * 102)

84. (2.1 * 102)(2 * 104)

85. ( - 3 * 10-3)(5 * 102)

86. (4 * 102)(1 * 103)(5 * 10-4)

87. 4 * 105

2 * 102 88.

9 * 102

3 * 106

89. 8 * 10-6

4 * 10-3 90.

6.3 * 102

2 * 10-3

APPLICATIONS

91. Light-year The distance that light travels in 1 year is called a light-year. Light travels at 1.86 * 105 miles per second, and there are about 3.15 * 107 seconds in 1 year. (a) Estimate the number of miles in 1 light-year. (b) Except for the sun, Alpha Centauri is the near-

est star, and its distance is 4.27 light-years from Earth. Estimate its distance in miles. Write your answer in scientific notation.

92. Milky Way It takes 2 * 108 years for the sun to make one orbit around the Milky Way galaxy. Write this number in standard form.

93. Speed of the Sun (Refer to the two previous exer- cises.) Assume that the sun’s orbit in the Milky Way galaxy is circular with a diameter of 105 light-years. Estimate how many miles the sun travels in 1 year.

94. Distance to the Moon The moon is about 240,000 miles from Earth.

(a) Write this number in scientific notation. (b) If a rocket traveled at 4 * 104 miles per hour,

how long would it take for it to reach the moon? 95. Online Exploration In 1997, the creators of the

Internet search engine BackRub renamed it Google. This new name is a play on the word googol, which is a very large number. Look up a googol and write it in scientific notation.

96. Online Exploration An astronomical unit (AU) is based on the distance from Earth to the sun. Look up the distance in kilometers from Earth to the sun.

(a) Write an astronomical unit in standard form to the nearest million kilometers.

(b) Convert your rounded answer from part (a) to scientific notation.

97. Gross Domestic Product The gross domestic prod- uct (GDP) is the total national output of goods and services valued at market prices within the United States. The GDP of the United States in 2005 was $12,460,000,000,000. (Source: Bureau of Economic Analysis.)

(a) Write this number in scientific notation. (b) In 2005, the U.S. population was 2.98 * 108. On

average, how many dollars of goods and services were produced by each individual?

98. Average Family Net Worth A family refers to a group of two or more people related by birth, mar- riage, or adoption who reside together. In 2000, the average family net worth was $280,000, and there were about 7.2 * 107 families. Calculate the total family net worth in the United States in 2000. (Source: U.S. Census Bureau.)

WRITING ABOUT MATHEMATICS

99. Explain what a negative exponent is and how it is different from a positive exponent. Give an example.

100. Explain why scientific notation is helpful for writ- ing some numbers.

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Working with Real DataGroup Activity

Water in a Lake East Battle Lake in Minnesota covers an area of about 1950 acres, or 8.5 * 107 square feet, and its average depth is about 3.2 * 101 feet. (a) Estimate the cubic feet of water in the lake. (Hint:

volume = area * average depth.) (b) One cubic foot of water equals about 7.5 gallons.

How many gallons of water are in this lake?

(c) The population of the United States is about 3.1 * 108, and the average American uses about 1.5 * 102 gallons of water per day. Could this lake supply the American population with water for 1 day?

Directions: Form a group of 2 to 4 people. Select a person to record the group’s responses for this activity. All members of the group should work cooperatively to answer the questions. If your instructor asks for your results, each member of the group should be prepared to respond.

5.6 Division of Polynomials Division by a Monomial ● Division by a Polynomial

A LOOK INTO MATH N The study of polynomials has occupied the minds of mathematicians for centuries. During the sixteenth century, Girolamo Cardano and other Italian mathematicians discovered how to solve higher degree polynomial equations. In this section we demonstrate how to divide polynomials. Division is often needed to factor polynomials and to solve polynomial equa- tions. (Source: H. Eves, An Introduction to the History of Mathematics.)

Division by a Monomial To add two fractions with like denominators, we use the property

d +

d =

a + b d

For example, 17 + 3 7 =

1 + 3 7 =

4 7.

To divide a polynomial by a monomial, we use the same property, only in reverse. That is,

a + b d

= a

d +

d .

Note that each term in the numerator is divided by the monomial in the denominator. The next example shows how to divide a polynomial by a monomial.

Girolamo Cardano (1501–1576)

EXAMPLE 1 Dividing a polynomial by a monomial

Divide.

(a) a5 + a3

a2 (b)

5x4 + 10x 10x

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Solution

(a) a5 � a3

a2 =

a2 �

a2 = a5 - 2 + a3 - 2 = a3 + a

(b) 5x4 � 10x

10x =

5x4

10x �

10x

10x =

2 + 1

6y =

3y2

6y �

6y =

2 +

3 -

2 y

Now Try Exercises 17, 19, 21

MAKING CONNECTIONS

Division and Simplification

A common mistake made when dividing expressions is to “cancel” incorrectly. Note in Example 1(b) that

5x4 + 10x 10x

3 5x4 + 10x

10x .

The monomial must be divided into every term in the numerator.

When dividing two natural numbers, we can check our work by multiplying. For exam- ple, 105 = 2, and we can check this result by finding the product 5 # 2 = 10. Similarly, to check

a5 � a3

a2 = a3 � a

in Example 1(a) we can multiply a2 and a3 � a.

a2(a3 � a) = a2 # a3 + a2 # a Distributive property = a5 � a3 ✓ It checks.

EXAMPLE 2 Dividing and checking

Divide the expression 8 x 3 - 4 x2 + 6 x

2 x2 and then check the result.

Solution Be sure to divide 2 x2 into every term in the numerator.

8 x3 � 4 x2 � 6 x

2 x2 =

8 x3

2 x2 �

4 x2

2 x2 �

6 x

2 x2 = 4 x - 2 +

3 x

Check:

2 x2a4 x - 2 + 3 x b = 2 x2 # 4 x - 2 x2 # 2 + 2 x2 # 3

= 8 x3 - 4 x2 + 6 x ✓

Now Try Exercise 23

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EXAMPLE 3 Finding the length of a rectangle

The rectangle in Figure 5.17 has an area A = x2 + 2 x and width x. Write an expression for its length l in terms of x.

STUDY TIP

Do you have enough time to study your notes and complete your assign- ments? One way to man- age your time is to make a list of your time com- mitments and determine the amount of time that each activity requires. Remember to include time for eating, sleeping, and relaxing!

Figure 5.17

x A = x2 + 2x

Solution The area A of a rectangle equals length l times width w, or A = lw. Solving for l gives

l = A w

Thus to find the length of the given rectangle, divide the area by the width.

l = x2 + 2 x

x =

x +

2 x x

= x + 2

The length of the rectangle is x + 2. The answer checks because x(x + 2) = x2 + 2 x.

Now Try Exercise 49

To check this result, we find the product of the quotient and divisor and then add the remain- der. Because 67 # 4 + 3 = 271, the answer checks. The quotient and remainder can also be expressed as 6734. Division of polynomials is similar to long division of natural numbers.

Division by a Polynomial To understand division by a polynomial better, we first need to review some terminology related to long division of natural numbers. To compute 271 , 4, we complete long divi- sion as follows.

67 R 3

4�271 24

Quotient h v Remainder Divisorh v Dividend

EXAMPLE 4 Dividing polynomials

Divide 6 x 2 + 13x + 3

3x + 2 and check.

Solution Begin by dividing the first term of 3x + 2 into the first term of 6 x2 + 13x + 3. That is, divide 3x into 6 x2 to obtain 2 x. Then find the product of 2 x and 3x + 2, or 6 x2 + 4 x, place it below 6 x2 + 13x, and subtract. Bring down the 3.

2x 3x + 2�6 x2 + 13x + 3

6 x2 + 4 x 9x + 3

6 x2

3x = 2 x

2 x(3x + 2) = 6 x2 + 4 x Subtract: 13x - 4 x = 9x. Bring down the 3.

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346 CHAPTER 5 POLYNOMIALS AND EXPONENTS

In the next step, divide 3x into the first term of 9x + 3 to obtain 3. Then find the product of 3 and 3x + 2, or 9x + 6, place it below 9x + 3, and subtract.

2 x + 3 3x + 2�6 x2 + 13x + 3

6 x2 + 4 x 9x + 3 9x + 6

- 3

9x 3x

= 3

3(3x + 2) = 9x + 6 Subtract: 3 - 6 = - 3.

The quotient is 2 x + 3 with remainder - 3. This result can also be written as

2 x + 3 + - 3

3x + 2 , Quotient +

Remainder

Divisor

in the same manner that 67 R 3 was written as 6734. Check polynomial division by adding the remainder to the product of the divisor and

the quotient. That is,

(Divisor)(Quotient) + Remainder = Dividend.

For this example, the equation becomes

(3x � 2)(2 x � 3) + (�3) = 3x # 2 x + 3x # 3 + 2 # 2 x + 2 # 3 � 3 = 6 x2 + 9x + 4 x + 6 - 3 = 6 x2 � 13x � 3. ✓ It checks.

Now Try Exercise 27

READING CHECK

• How can you check a polynomial division problem?

EXAMPLE 5 Dividing polynomials having a missing term

Simplify (3x3 + 2 x - 4) , (x - 2).

Solution Because the dividend does not have an x2-term, insert 0x2 as a “place holder.” Then begin by dividing x into 3x3 to obtain 3x2.

3x2

x - 2�3x3 + 0 x2 + 2 x - 4 3x3 - 6 x2

6 x2 + 2 x

3x3

x = 3x2

3x2(x - 2) = 3x3 - 6 x2

Subtract 0x2 - ( - 6 x2) = 6 x2. Bring down 2x.

In the next step, divide x into 6 x2.

3x2 + 6 x x - 2�3x3 + 0 x2 + 2 x - 4

3x3 - 6 x2

6 x2 + 2 x 6 x2 - 12 x

14 x - 4

6 x2

x = 6 x

6 x(x - 2) = 6 x2 - 12 x Subtract: 2 x - ( - 12 x) = 14 x. Bring down - 4.

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Now divide x into 14 x.

3x2 + 6 x + 14 x - 2�3x3 + 0 x2 + 2 x - 4

3x3 - 6 x2

6 x2 + 2 x 6 x2 - 12 x

14 x - 4 14 x - 28

The quotient is 3x2 + 6 x + 14 with remainder 24. This result can also be written as

3x2 + 6 x + 14 + 24

x - 2 .

Now Try Exercise 37

EXAMPLE 6 Dividing when the divisor is not linear

Divide x3 - 3x2 + 3x + 2 by x2 + 1.

Solution Begin by writing x2 + 1 as x2 + 0x + 1.

x - 3 x2 + 0x + 1�x3 - 3x2 + 3x + 2

x3 + 0x2 + x - 3x2 + 2 x + 2 - 3x2 + 0x - 3

2 x + 5

The quotient is x - 3 with remainder 2 x + 5. This result can also be written as

x - 3 + 2 x + 5 x2 + 1

Now Try Exercise 41

5.6 Putting It All Together

Division by a Monomial Use the property

a + b d

= a

d +

d .

Be sure to divide the denominator into every term in the numerator.

2 x3 + 4 x 2 x2

= 2 x3

2 x2 +

4 x

2 x2 = x +

2 x

and

a2 - 2a 4a

= a2

4a -

4a =

4 -

CONCEPT EXPLANATION EXAMPLES

continued on next page

14 x x

= 14

14(x - 2) = 14 x - 28 Subtract: - 4 - ( - 28) = 24.

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CONCEPT EXPLANATION EXAMPLES

Division by a Polynomial

Is done similarly to the way long divi- sion of natural numbers is performed

If either the divisor or the dividend is missing a term, be sure to insert as a “place holder” the missing term with coefficient 0.

Divide x2 + 3x + 3 by x + 1.

x + 2 x + 1�x2 + 3x + 3

x2 + x 2 x + 3 2 x + 2

The quotient is x + 2 with remainder 1, which can be expressed as

x + 2 + 1

x + 1 .

Checking a Result Dividend = (Divisor)(Quotient) + Remainder

When x2 + 3x + 3 is divided by x + 1, the quotient is x + 2 with remainder 1. Thus

(x + 1)(x + 2) + 1 = x2 + 3x + 3,

and the answer checks.

continued from previous page

CHAPTER 5 POLYNOMIALS AND EXPONENTS348

CONCEPTS AND VOCABULARY

1. a + bd = 2. a + b - c

d =

3. When dividing a polynomial by a monomial, the mono- mial must be divided into every of the polynomial.

4. (True or False?) The expressions 5x 2 + 2 x

2 x and 5x 2 + 1

are equal.

5. (True or False?) The expressions 5x 2 + 2 x

2 x and 5x2 2 x are

equal.

6. Because 379 = 4 with remainder 1, it follows that 37 = # + .

7. Because 2 x3 - x + 5 divided by x + 1 equals 2 x2 - 2 x + 1 with remainder 4, it follows that 2 x3 - x + 5 = _____ # _____ + _____.

8. When dividing 2 x3 + 3x - 1 by x - 1, insert

into the dividend as a “place holder” for the missing x2-term.

DIVISION BY A MONOMIAL

Exercises 9–16: Divide and check.

9. 6 x2

3x 10.

- 5x2

10x4

11. z4 + z3

z 12.

t3 - t t

13. a5 - 6a3

2a3 14.

b4 - 4b 4b2

15. y + 6y2

3y3 16.

8z2 - z 4z2

Exercises 17–26: Divide.

17. 4 x - 7x4

x2 18.

1 + 6 x4

3x3

19. 6y2 + 3y

3y3 20.

5z2 - 10z3

5z4

5.6 Exercises

21. 9x4 - 3x + 6

3x 22.

y3 - 4y + 6 y

23. 12y4 - 3y2 + 6y

3y2 24.

2 x2 - 6 x + 9 12 x

25. 15m4 - 10m3 + 20m2

5m2 26.

n8 - 8n6 + 4n4

2n5

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47. Thinking Generally If the quotient in a polynomial division problem is an integer, what must be true about the degrees of the dividend and divisor?

48. Thinking Generally If the quotient in a polynomial division problem is a polynomial of degree 1, what must be true about the degrees of the dividend and divisor?

APPLICATIONS

Exercises 49 and 50: Area of a Rectangle The area of a rectangle and its width are given. Find an expression for the length l.

49. 50.

51. Volume of a Box The volume V of a box is 2 x3 + 4 x2, and the area of its bottom is 2 x2. Find the height of the box in terms of x. Make a possible sketch of the box, and label the length of each side.

52. Area of a Triangle A triangle has height h and area A = 2h2 - 4h. Find its base b in terms of h. Make a possible sketch of the triangle, and label the height and base. (Hint: A = 12 bh.)

WRITING ABOUT MATHEMATICS

53. Suppose that one polynomial is divided into another polynomial and the remainder is 0. What does the product of the divisor and quotient equal? Explain.

54. A student simplifies the expression 4 x 3 - 1 4 x2

to x - 1. Explain the student’s error.

2x A = 8x2

l l

x – 1 A = x2 – 1

27. 2 x2 - 3x + 1

x - 2 28.

4 x2 - x + 3 x + 2

Exercises 27–34: Divide and check.

29. x2 + 2 x + 1

x + 1 30.

4 x2 - 4 x + 1 2 x - 1

31. x3 - x2 + x - 2

x - 1 32.

2 x3 + 3x2 + 3x - 1 2 x + 1

33. x3 + x2 - 7x + 2

x - 2

34. x3 + x2 - 2 x + 12

x + 3

Exercises 35–46: Divide.

35. 4 x3 - 3x2 + 7x + 3

4 x + 1

36. 10x3 - x2 - 17x - 7

5x + 2

37. x3 - x + 2

x - 2

38. 6 x3 + 8 x2 + 4

3x + 4

39. (3x3 + 2) , (x - 1)

40. ( - 3x3 + 8 x2 + x) , (3x + 4)

41. (x3 + 3x2 + 1) , (x2 + 1)

42. (x4 - x3 + x2 - x + 1) , (x2 - 1)

43. x3 + 1

x2 - x + 1 44.

4 x3 + 3x + 2 2 x2 - x + 1

45. x3 + 8 x + 2

46. x4 - 16 x - 2

Checking Basic ConceptsSECTIONS5.5 and 5.6

1. Simplify each expression. Write the result with positive exponents.

(a) 9-2 (b) 3x -3

6 x4 (c) (4ab -4)-2

2. Simplify each expression. Write the result with positive exponents.

(a) 1

z-5 (b)

x -3

y-6 (c) a 3

x2 b-3

3. Write each number in scientific notation. (a) 45,000 (b) 0.000234 (c) 0.01

4. Write each expression in standard form. (a) 4.71 * 104 (b) 6 * 10-3

5. Simplify 25a 4 - 15a3 5a3

6. Divide 3x2 - x - 4 by x - 1. State the quotient and remainder.

7. Divide x4 + 2 x3 - 2 x2 - 5x - 2 by x2 - 3. State the quotient and remainder.

8. Distance to the Sun The distance to the sun is

approximately 93 million miles. (a) Write this distance in scientific notation. (b) Light travels at 1.86 * 105 miles per second.

How long does it take for the sun’s light to reach Earth?

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CHAPTER 5 Summary SECTION 5.1 . RULES FOR EXPONENTS

Bases and Exponents The expression bn has base b and exponent n and equals the expression b # b # b # g # b, when n is a natural number.u

n times

Example: 23 has base 2 and exponent 3 and equals 2 # 2 # 2 = 8. Evaluating Expressions When evaluating expressions, evaluate exponents before performing addition, subtraction, multiplication, division, or negation. In general, operations within paren- theses should be evaluated before using the order of operations.

1. Evaluate exponents. 2. Perform negation. 3. Do multiplication and division from left to right. 4. Do addition and subtraction from left to right.

Example: - 32 + 3 # 4 = - 9 + 3 # 4 = - 9 + 12 = 3 Zero Exponents For any nonzero number b, b0 = 1. Note that 00 is undefined.

Examples: 50 = 1 and a x y b0 = 1, where x and y are nonzero.

Product Rule For any real number a and natural numbers m and n,

am # an = am + n. Examples: 34 # 32 = 36 and x3x2x4 = x9 Power Rules For any real numbers a and b and natural numbers m and n,

(am)n = amn, (ab)n = anbn, and a a b bn = an

bn , b � 0.

Examples: (x2)3 = x6, (3x)4 = 34x4 = 81x4, and a 2 y b3 = 23

y3 =

SECTION 5.2 . ADDITION AND SUBTRACTION OF POLYNOMIALS

Terms Related to Polynomials Monomial A number, variable, or product of numbers and variables

raised to natural number powers

Degree of a Monomial Sum of the exponents of the variables

Coefficient of a Monomial The number in a monomial

Example: The monomial - 3x2y3 has degree 5 and coefficient - 3.

Polynomial A monomial or the sum of two or more monomials

Term of a Polynomial Each monomial is a term of the polynomial.

Binomial A polynomial with two terms

Trinomial A polynomial with three terms

Degree of a Polynomial The degree of the term with highest degree

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Opposite of a Polynomial The opposite is found by negating each term.

Example: 2 x3 - 4 x + 5 is a trinomial with degree 3. Its opposite is - 2 x3 + 4 x - 5.

Like Terms Two monomials with the same variables raised to the same powers

Examples: 3xy2 and - xy2 are like terms.

5x3 and 3x3 are like terms.

5x2 and 5x are unlike terms.

Addition of Polynomials Combine like terms, using the distributive property.

Example: (2 x2 - 4 x) + ( - x2 - x) = (2 - 1)x2 + ( - 4 - 1)x = x2 - 5x

Subtraction of Polynomials Add the first polynomial to the opposite of the second polynomial.

Example: (4 x4 - 5x) - (7x4 + 6 x) = (4 x4 - 5x) + ( - 7x4 - 6 x) = (4 - 7)x4 + ( - 5 - 6)x = - 3x4 - 11x

SECTION 5.3 . MULTIPLICATION OF POLYNOMIALS

Multiplication of Monomials Use the commutative property and the product rule.

Examples: - 2 x3 # 3x2 = - 2 # 3 # x3 # x2 = - 6 x5 (2 xy2)(3x2y3) = 2 # 3 # x # x2 # y2 # y3 = 6 x3y5 c Assumed exponent of 1

Distributive Properties

a(b + c) = ab + ac and a(b - c) = ab - ac

Examples: 4 x(3x + 6) = 4 x # 3x + 4 x # 6 = 12 x2 + 24 x ab(a2 - b2) = ab # a2 - ab # b2 = a3b - ab3 Multiplication of Monomials and Polynomials Apply the distributive properties. Be sure to multiply every term in the polynomial by the monomial.

Example: - 2 x2(4 x2 - 5x - 3) = - 8 x4 + 10x3 + 6 x2

Multiplication of Polynomials The product of two polynomials may be found by multiplying every term in the first polynomial by every term in the second polynomial. Be sure to combine like terms.

Examples: (x + 3)(2 x - 5) = 2 x2 - 5x + 6 x - 15

= 2 x2 + x - 15

(2 x + 1)(x2 - 5x + 2) = 2 x3 - 10x2 + 4 x + x2 - 5x + 2 = 2 x3 - 9x2 - x + 2

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SECTION 5.4 . SPECIAL PRODUCTS

Product of a Sum and Difference

(a + b)(a - b) = a2 - b2

Examples: (x + 4)(x - 4) = x2 - 16 (2r - 3t)(2r + 3t) = (2r)2 - (3t)2 = 4r2 - 9t2

Squaring Binomials

(a + b)2 = a2 + 2ab + b2 and (a - b)2 = a2 - 2ab + b2

Examples: (2 x + 1)2 = (2 x)2 + 2(2 x)1 + 12 = 4 x2 + 4 x + 1 (z2 - 2)2 = (z2)2 - 2z2(2) + 22 = z4 - 4z2 + 4

Cubing Binomials To multiply (a + b)3, write it as (a + b)(a + b)2.

Example: (x + 4)3 = (x + 4)(x + 4)2

= (x + 4)(x2 + 8 x + 16) Square the binomial. = x3 + 8 x2 + 16 x + 4 x2 + 32 x + 64 Distributive property = x3 + 12 x2 + 48 x + 64 Combine like terms.

SECTION 5.5 . INTEGER EXPONENTS AND THE QUOTIENT RULE

Negative Integers as Exponents For any nonzero real number a and positive integer n,

a-n = 1

an .

Examples: 5-2 = 1

52 and x -4 =

The Quotient Rule For any nonzero real number a and integers m and n,

an = am - n.

Examples: 64

62 = 64 - 2 = 62 = 36 and

xy3

x4y2 = x1 - 4y3 - 2 = x -3y1 =

Other Rules For any nonzero real numbers a and b and positive integers m and n,

a-n = an,

a-n

b -m =

an , and a a

b b-n = a b

a bn.

Examples: 1

4-3 = 43,

x -3

y-2 =

x3 , and a 4

5 b-2 = a 5

4 b2

Scientific Notation A real number a written as b * 10n, where 1 … 0 b 0 6 10 and n is an integer Examples: 2.34 * 103 = 2340 Move the decimal point 3 places to the right. 2.34 * 10-3 = 0.00234 Move the decimal point 3 places to the left.

SECTION 5.6 . DIVISION OF POLYNOMIALS

Division of a Polynomial by a Monomial Divide the monomial into every term of the polynomial.

Example: 5x3 - 10x2 + 15x

5x =

5x3

5x -

10x2

5x +

15x

5x = x2 - 2 x + 3

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Division of a Polynomial by a Polynomial Division of polynomials is performed similarly to long division of natural numbers.

Example: Divide 2 x3 + 4 x2 - 3x + 1 by x + 1.

2 x2 + 2 x - 5 x + 1�2 x3 + 4 x2 - 3x + 1

2 x3 + 2 x2

2 x2 - 3x 2 x2 + 2 x

- 5x + 1 - 5x - 5

The quotient is 2 x2 + 2 x - 5 with remainder 6, which can be written as

2 x2 + 2 x - 5 + 6

x + 1 .

CHAPTER 5 Review Exercises SECTION 5.1

Exercises 1–6: Evaluate the expression.

1. 53 2. - 34

3. 4( - 2)0 4. 3 + 32 - 30

5. - 52

5 6. a - 5

5 b2

Exercises 7–24: Simplify the expression.

7. 62 # 63 8. 105 # 107 9. z4 # z5 10. y2 # y # y3 11. 5x2 # 6 x7 12. (ab3)(a3b) 13. (25)2 14. (m4)5

15. (ab)3 16. (x2y3)4

17. (xy)3(x2y4)2 18. (a2b9)0

19. (r - t)4(r - t)5 20. (a + b)2(a + b)4

21. a 3 x - y b2 22. a x + y2 b3

23. 2 x2(3x - 5) 24. 3x(4 x + x3)

SECTION 5.2

Exercises 25 and 26: Identify the degree and coefficient of the monomial.

25. 6 x7 26. - x2y3

Exercises 27–30: Determine whether the expression is a polynomial. If it is, state how many terms and variables the polynomial contains. Then state its degree.

27. 8y 28. 8 x3 - 3x2 + x - 5

29. a2 + 2ab + b2 30. 1 xy

31. Add the polynomials vertically.

3x2 + 4 x + 8 2 x2 - 5x - 5

32. Write the opposite of 6 x2 - 3x - 7.

Exercises 33–40: Simplify.

33. (4 x - 3) + ( - x + 7)

34. (3x2 - 1) - (5x2 + 12)

35. (x2 + 5x + 6) - (3x2 - 4 x + 1)

36. (x2 + 3x - 5) + (2 x2 - 5x - 1)

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37. (a3 + 4a2) + (a3 - 5a2 + 7a)

38. (4 x3 - 2 x + 6) - (4x3 - 6)

39. (xy + y2) + (4y2 - 4 xy)

40. (7x2 + 2 xy + y2) - (7x2 - 2 xy + y2)

SECTION 5.3

Exercises 41–54: Multiply and simplify.

41. - x2 # x3 42. - (r2t3)(rt) 43. - 3(2t - 5) 44. 2y(1 - 6y)

45. 6 x3(3x2 + 5x)

46. - x(x2 - 2 x + 9)

47. - ab(a2 - 2ab + b2)

48. (a - 2)(a + 5)

49. (8 x - 3)(x + 2) 50. (2 x - 1)(1 - x)

51. ( y2 + 1)(2y + 1) 52. ( y2 - 1)(2y2 + 1)

53. (z + 1)(z2 - z + 1)

54. (4z - 3)(z2 - 3z + 1)

Exercises 55 and 56: Multiply the expression (a) geometrically and (b) symbolically.

55. z(z + 1) 56. 2 x(x + 2)

SECTION 5.4

Exercises 57–72: Multiply.

57. (z + 2)(z - 2) 58. (5z - 9)(5z + 9) 59. (1 - 3y)(1 + 3y) 60. (5x + 4y)(5x - 4y)

61. (rt + 1)(rt - 1)

62. (2m2 - n2)(2m2 + n2)

63. (x + 1)2 64. (4 x + 3)2

65. ( y - 3)2 66. (2y - 5)2

67. (4 + a)2 68. (4 - a)2

69. (x2 + y2)2

70. (xy - 2)2

71. (z + 5)3

72. (2z - 1)3

Exercises 73 and 74: Use the product of a sum and a dif- ference to evaluate the expression.

73. 59 # 61 74. 22 # 18 SECTION 5.5

Exercises 75–82: Simplify the expression.

75. 9-1 76. 3-2

77. 43 # 4-2 78. 10-6 # 103

79. 1

6-2 80.

81. (3-1 22)-2 82. (2-4 53)0 1

Exercises 83–98: Simplify the expression. Write the answer using positive exponents.

83. z-2 84. y-4

85. a-4 # a2 86. x2 # x -5 # x 87. (2t)-2 88. (ab2)-3

89. (xy)-2(x -2y)-1 90. x6

91. 4 x

2 x4 92.

20x5y3

30xy6

93. a a b b5 94. 4

t -4

95. (3m3n)-2

(2m2n-3)3 96. a x -4y2

3xy-3 b-2

97. a x y b-2 98. a 3u

2v b-1

Exercises 99–102: Write the expression in standard form.

99. 6 * 102 100. 5.24 * 104

101. 3.7 * 10-3 102. 6.234 * 10-2

Exercises 103–106: Write the number in scientific notation.

103. 10,000 104. 56,100,000

105. 0.000054 106. 0.001

Exercises 107 and 108: Evaluate the expression. Write the result in standard form.

107. (4 * 102)(6 * 104) 108. 8 * 103

4 * 104

SECTION 5.6

Exercises 109–116: Divide and check.

109. 5x2 + 3x

3x 110.

6b4 - 4b2 + 2 2b2

111. 3x2 - x + 2

x - 1 112.

9x2 - 6 x - 2 3x + 2

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114. 2 x3 - x2 - 1

2 x - 1 115.

x3 - x2 - x + 1 x2 + 1

116. x4 + 3x3 + 8 x2 + 7x + 5

x2 + x + 1

113. 4 x3 - 11x2 - 7x - 1

4 x + 1

APPLICATIONS

117. Heart Rate An athlete starts running and continues for 10 seconds. The polynomial 12 t

2 + 60 calculates the heart rate of the athlete in beats per minute t sec- onds after beginning the run, where t … 10. (a) What is the athlete’s heart rate when the athlete

first starts to run? (b) What is the athlete’s heart rate after 10 seconds? (c) What happens to the athlete’s heart rate while

the athlete is running?

118. Areas of Rectangles Find a monomial equal to the sum of the areas of the rectangles. Calculate this sum for x = 3 feet and y = 4 feet.

119. Area of a Rectangle Write a polynomial that gives the area of the rectangle. Calculate its area for z = 6 inches.

120. Area of a Square Find the area of the square whose sides have length x2y.

x2y

121. Compound Interest If P dollars are deposited in an account that pays 6% annual interest, then the amount of money after 3 years is given by P(1 + 0.06)3. Find this amount when P = $700.

122. Volume of a Sphere The expression for the volume of a sphere with radius r is 43 pr

3. Find a polynomial that gives the volume of a sphere with radius x + 2. Leave your answer in terms of p.

x + 2

123. Height Reached by a Baseball A baseball is hit straight up. Its height h in feet above the ground after t seconds is given by t(96 - 16t). (a) Multiply this expression. (b) Evaluate both the expression in part (a) and the

given expression for t = 2. Interpret the result.

124. Rectangular Building A rectangular building has a perimeter of 1200 feet. (a) If one side of the building has length L, write

a polynomial expression that gives its area. (Be sure to multiply your expression.)

(b) Evaluate the expression in part (a) for L = 50 and interpret the answer.

125. Geometry Complete each part and verify that your answers are equal. (a) Find the area of the large square by multiplying

its length and width. (b) Find the sum of the areas of the smaller rect-

angles inside the large square.

x 5

126. Digital Picture A digital picture, including its border, is x + 4 pixels by x + 4 pixels, and the actual picture inside the border is x - 4 pixels by x - 4 pixels.

x + 4 x – 4

x + 4

x – 4

(a) Find a polynomial that gives the number of pix- els in the border.

(b) Let x = 100 and evaluate the polynomial.

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127. Federal Debt In 1990, the federal debt held by the public was $2.19 trillion, and the population of the United States was 249 million. Use scientific nota- tion to approximate the national debt per person. (Source: U.S. Department of the Treasury.)

128. Alcohol Consumption In 2007, about 239 million people in the United States were age 14 or older. They consumed, on average, 2.31 gallons of alcohol per person. Use scientific notation to estimate the total number of gallons of alcohol consumed by this age group. (Source: Department of Health and Human Services.)

1. Simplify each expression. (a) - 50 (b) - 92

2. Evaluate each expression by hand. (a) - 42 + 10 (b) 8-2

2-3 (d) - 3x0

3. State how many terms and variables the polynomial 5x2 - 3xy - 7y3 contains. Then state its degree.

4. Write the opposite of - x3 + 4 x - 8.

Exercises 5–8: Simplify.

5. ( - 3x + 4) + (7x + 2)

6. ( y3 - 2y + 6) - (4y3 + 5)

7. (5x2 - x + 3) - (4 x2 - 2 x + 10)

8. (a3 + 5ab) + (3a3 - 3ab)

Exercises 9–16: Write the given expression with positive exponents.

9. 6y4 # 4y7 10. (a2b3)2(ab2) 11. x7 # x -3 12. (a-1b2)-3

13. ab(a2 - b2) 14. a 3a2 2b -3

b-2 15.

12 xy4

6 x2y 16. a 2

a + b b4

Exercises 17–22: Multiply and simplify.

24. Write 6.1 * 10-3 in standard form.

25. Write 5410 in scientific notation.

Exercises 26 and 27: Divide.

CHAPTER 5 Test Step-by-step test solutions are found on the Chapter Test Prep Videos available via the Video Resources on DVD, in , and on (search “RockswoldBeginAlg” and click on “Channels”).

17. 3x2(4 x3 - 6 x + 1)

19. (7y2 - 3)(7y2 + 3) 20. (3x - 2)2

18. (z - 3)(2z + 4)

21. (m + 3)3

22. ( y + 2)( y2 - 2y + 3)

23. Evaluate 78 # 82 using the product of a sum and a difference.

26. 9x3 - 6 x2 + 3x

3x2 27.

x3 + x2 - x + 1 x + 2

28. Concert Tickets Tickets for a concert are sold for

$20 each. (a) Write a polynomial that gives the revenue from

selling t tickets. (b) Putting on the concert costs management $2000

to hire the band plus $2 for each ticket sold. What is the total cost of the concert if t tickets are sold?

(c) Subtract the polynomial that you found in part (b) from the polynomial that you found in part (a). What does this polynomial represent?

29. Areas of Rectangles Find a polynomial representing

the sum of the areas of two identical rectangles that have width 2 x and length 3x. Calculate this sum for x = 10 feet.

30. Volume of a Box Write a polynomial that repre- sents the volume of the box. Be sure to multiply your answer completely.

x + 6

x + 3 3x

31. Height Reached by a Golf Ball When a golf ball is hit into the air, its height in feet above the ground after t seconds is given by t(88 - 16t). (a) Multiply this expression. (b) Evaluate the expression in part (a) for t = 3.

Interpret the result.

IS B

N 1-256-49082-2

357CHAPTER 5 EXTENDED AND DISCOVERY EXERCISES

Exercises 1–6: Arithmetic and Scientific Notation The product (4 * 103) * (2 * 102) can be evaluated as

(4 * 2) * (103 * 102) = 8 * 105,

and the quotient (4 * 103) , (2 * 102) can be evalu- ated as

4 * 103

2 * 102 =

2 *

103

102 = 2 * 101.

How would you evaluate (4 * 103) + (2 * 102)? How would you evaluate (4 * 103) - (2 * 102)? Make a con- jecture as to how numbers in scientific notation should be added and subtracted. Try your method on these prob- lems and then check your answers with a calculator set in scientific mode. Does your method work?

1. (4 * 103) + (3 * 103)

2. (5 * 10-2) - (2 * 10-2)

3. (1.2 * 104) - (3 * 103)

4. (2 * 102) + (6 * 101)

5. (2 * 10-1) + (4 * 10-2)

6. (2 * 10-3) - (5 * 10-2)

Exercises 7 and 8: Constructing a Box A box is con- structed from a rectangular piece of metal by cutting squares from the corners and folding up the sides. The square, cutout corners are x inches by x inches.

x x x

7. Suppose that the dimensions of the metal piece are 20 inches by 30 inches. (a) Write a polynomial that gives the volume of the

box. (b) Find the volume of the box for x = 4 inches.

8. Suppose that the metal piece is square with sides of length 25 inches. (a) Write a polynomial expression that gives the out-

side surface area of the box. (Assume that the box does not have a top.)

(b) Find this area for x = 3 inches.

Exercises 9–12: Calculators and Polynomials A graph- ing calculator can be used to help determine whether two polynomial expressions in one variable are equal. For example, suppose that a student believes that (x + 2)2 and x2 + 4 are equal. Then the first two calculator tables shown demonstrate that the two expressions are not equal except for x = 0.

Y1�(X�2)2

X Y1 �3 1 �2 0 �1 1 0 4 1 9 2 16 3 25

Y1�X2�4

X Y1 �3 13 �2 8 �1 5 0 4 1 5 2 8 3 13

The next two calculator tables support the fact that (x + 1)2 and x2 + 2 x + 1 are equal for all x.

Y1�(X�1)2

X Y1 �3 4 �2 1 �1 0 0 1 1 4 2 9 3 16

Y1�X2�2X�1

X Y1 �3 4 �2 1 �1 0 0 1 1 4 2 9 3 16

Use a graphing calculator to determine whether the first expression is equal to the second expression. If the expressions are not equal, multiply the first expression and simplify it.

9. 3x(4 - 5x), 12 x - 5x

10. (x - 1)2, x2 - 1

11. (x - 1)(x2 + x + 1), x3 - 1

12. (x - 2)3, x3 - 8

CHAPTER 5 Extended and Discovery Exercises

IS B

N 1-

25 6-

49 08

2- 2

358 CHAPTER 5 POLYNOMIALS AND EXPONENTS

CHAPTERS 1–5 Cumulative Review Exercises

Exercises 1 and 2: Evaluate each expression by hand.

1. (a) 18 - 2 # 5 (b) 42 , 7 + 2 2. (a) 21 - ( - 8) (b) - 73 , 1 - 149 2 Exercises 3 and 4: Solve the equation. Note that these equations may have no solutions, one solution, or infi- nitely many solutions.

3. (a) (x - 3) + x = 4 + x (b) 2(5x - 4) = 1 + 10x

4. (a) 2 + 6 x = 2(3x + 1) (b) 11x - 9 = - 31

5. Find the average speed of a car that travels 306 miles in 4 hours 30 minutes.

6. Write each value as a fraction in lowest terms. (a) 42% (b) 0.076

7. Graph the equation 4 x - 5y = 20.

8. Sketch a line with slope - 23 that passes through the point (1, 1).

9. Write the slope–intercept form for the line shown.

–2–3 –1 2 3

–2

–3

–1

10. Find the x-and y-intercepts for the graph of the equa- tion 2y = 3x - 6.

Exercises 11 and 12: Write the slope–intercept form of a line that satisfies the given information.

11. Parallel to 3x - 6y = 7, passing through (2, - 3) 12. Passing through ( - 2, - 5) and (1, 4)

Exercises 13–16: Solve the system of equations. Note that these systems may have no solutions, one solution, or infinitely many solutions.

Exercises 17 and 18: Shade the solution set for the system of inequalities.

13. 4 x + 3y = - 6 8 x + 6y = 12

14. x - 3y = 5 3x + y = 5

15. x + 4y = - 8 - 3x - 12y = 24

16. x - 5y = 30 2 x + y = - 6

17. x + y 6 3 y Ú x + 2

18. x - 2y 7 4 3x + y 6 6

19. Simplify the expression. (a) 3x2 # 5x3 (b) (x3y)2(x4y5)

20. Simplify. (a) (5x2 - 3x + 4) - (3x2 - 2 x + 1) (b) (7a3 - 4a2 - 5) + (5a3 + 4a2 + a)

21. Multiply and simplify. (a) (2 x + 3)(x - 7) (b) (y + 3)(y2 - 3y - 1)

22. Simplify the expression. Write the answer using posi- tive exponents. (a) x -5 # x3 # x (b) a 2

x3 b-3

6 x -2y (d) (xy-2)3(x -2y)-2

23. Write 24,000,000,000 in scientific notation.

24. Write 4.71 * 10-7 in standard form.

25. Divide.

(a) 8 x3 - 2 x

2 x (b)

2 x2 + x - 14 x + 3

26. Price Decrease If the price of a computer is reduced from $1200 to $900, find the percent change.

27. Mixing an Acid Solution How many milliliters of a 3% acid solution should be added to 400 milliliters of a 6% acid solution to dilute it to a 5% acid solution?

28. Surface Area of a Box Use the drawing of the box to write a polynomial that represents the area of each of the following.

x + 5

x + 2 2x

(a) The bottom (b) The front (c) The right side (d) All six sides

IS B

N 1-256-49082-2

333

C h e c k l i s t s A A P P E N D I X

1. Is this budget static (not adjusted for volume) or flexible (adjusted for volume dur- ing the year)?

2. Are the figures designated as fixed or variable?

3. Is the budget for a defined unit of authority?

4. Are the line items within the budget all expenses (and revenues, if applicable) that are controllable by the manager?

5. Is the format of the budget comparable with that of previous periods so that sev- eral reports over time can be compared if so desired?

6. Are actual and budget for the same period?

7. Are the figures annualized?

8. Test one line-item calculation. Is the math for the dollar difference computed cor- rectly? Is the percentage properly computed based on a percentage of the budget figure?

Checklist A-1 Reviewing a Budget

334 APPENDIX A Checklists

1. What is the proposed volume for the new budget period?

2. What is the appropriate inflow (revenues) and outflow (cost of services delivered) relationship?

3. What will the appropriate dollar cost be?

(Note: this question requires a series of assumptions about the nature of the oper- ation for the new budget period.)

3a. Forecast service-related workload.

3b. Forecast non–service-related workload.

3c. Forecast special project workload if applicable.

3d. Coordinate assumptions for proportionate share of interdepartmental proj- ects.

4. Will additional resources be available?

5. Will this budget accomplish the appropriate managerial objectives for the organi- zation?

Checklist A-2 Building a Budget

1. What is the date on the balance sheet?

2. Are there large discrepancies in balances between the prior year and the current year?

3. Did total assets increase over the prior year?

4. Did current assets increase, decrease, or stay about the same?

5. Did current liabilities increase, decrease, or stay about the same?

6. Did land, plant, and equipment increase or decrease significantly over the prior year?

7. Did long-term debt increase or decrease significantly over the prior year?

Courtesy of Baker and Baker, Dallas, Texas.

Checklist A-3 Balance Sheet Review

Checklists 335

1. What is the period reported on the statement of revenue and expense?

2. Is it one year or a shorter period? If it is a shorter period, why is that?

3. Are there large discrepancies in balances between the prior year operations and the current year operations?

4. Did total operating revenue increase over the prior year?

5. Did total operating expenses increase, decrease, or stay about the same? Is any par- ticular line item unusually large or small?

6. Did income from operations increase, decrease, or stay about the same?

7. Are there unusual nonoperating gains or losses?

8. Did the current year result in an excess of revenue over expense? Is it as much as that of the prior year?

9. Did long-term debt increase or decrease significantly over the prior year?

Checklist A-4 Review of the Statement of Revenue and Expense

• Only one location?

• Equipment—single purpose or multi-purpose?

• Technology—new, middle-aged, old (obsolete vs. untested)

• Equipment compatibility

• Medical supply cost

• High or low capital investment?

• Buy new or used (refurbished)?

• Buy or lease?

• Lease for a number of years or lease on a pay-per-procedure deal?

• How much staff training is required?

• Certification required?

• Square footage required for equipment

• Is the required square footage available?

• Cleaning methods and equipment (and staff level required)

• Repairs and maintenance expense (high, medium, low?)

Checklist A-5 Considerations for Forecasting Equipment Acquisition

W eb-Based and Software

Learning To o l s B A P P E N D I X

HOMEPAGE FOR HEALTHCARE FINANCE

Health Care Finance: Basic Tools for Nonfinancial Managers, 3rd edition, has its own page on Jones and Bartlett Publishers’ Web site. The homepage contains resources for both in- structors and students. The site can be accessed using the following URL: http://www.jbpub.com/catalog/0763726605/.

SOFTWARE TOOLS

Microsoft software and its web-based applications are universally available across the United States, and Microsoft Office Excel offers an array of computation tools. Relevant informa- tion is available at www.microsoft.com/office.

For example, click on “function reference/financial” for a listing of Excel’s financial com- putations. And of course the formulas within Excel spreadsheets provide calculator capa- bility for every-day addition, subtraction, multiplication, and division. The Web site also offers supporting resources such as online tutorials and a “help and how-to” feature, along with tips about using the various Excel features.

OTHER WEB-BASED TOOLS

A user who prefers to use a business analyst calculator (as opposed to computer spread- sheets) can search the Web for a calculator distributor who posts an operating guidebook.

337

N o t e s

Chapter 1

1. C. S. George, Jr., The Histor y of Management Thought, 2nd ed. (Englewood Cliffs, NJ: Prentice Hall, 1972), 1–27.

2. Ibid., 87. 3. S. Williamson et al., Fundamentals of Strategic Planning for Healthcare Organizations (New York: The Ha-

worth Press, 1997).

Chapter 2

1. J. J. Baker, Activity-Based Costing and Activity-Based Management for Health Care (Gaithersburg, MD: Aspen Publishers, Inc., 1998).

2. L. V. Seawell, Chart of Accounts for Hospitals (Chicago: Probus Publishing Company, 1994).

Chapter 4

1. Texas Medical Association, American Medical Association, Texas Medical Foundation, and Texas Os- teopathic Medical Association, A Guide to Forming Physician-Directed Managed Care Networks (Austin, TX: Texas Medical Association, 1994), 3.

2. Health Care Financing Administration, Health Care Financing Review: Medicare and Medicaid Statistical Supplement (Baltimore, MD: U.S. Department of Health and Human Services, 1997), 8.

3. Ibid., 9. 4. D. I. Samuels, Capitation: New Opportunities in Healthcare Delivery (Chicago: Irwin Professional Publish-

ing, 1996), 20–21. 5. D. E. Goldstein, Alliances: Strategies for Building Integrated Delivery Systems (Gaithersburg, MD: Aspen Pub-

lishers, Inc., 1995), 283; and Texas Medical Association, American Medical Association, Texas Medical Foundation, and Texas Osteopathic Medical Association, A Guide to Forming Physician-Directed Managed Care Networks (Austin, TX: Texas Medical Association, 1994), 4–6.

6. C. Horngren et al., Cost Accounting: A Managerial Emphasis, 9th ed. (Englewood Cliffs, NJ: Prentice Hall, 1998), 116.

7. When ICD-10 is fully implemented, it is possible that the term “major diagnostic categories” (MDCs) may have to be replaced with some other universal designation. Whether these hospitals will change the names of their service line designations to match the new titles is unknown at this point. We do know it will take time to decide upon such a change and then additional time to implement the change.

8. A. Sharpe and G. Jaffe, “Columbia/HCA Plans for More Big Changes in Health-Care World,” Wall Street Journal, 28 May, 1997, A8.

339

340 NOTES

Chapter 5

1. S. A. Finkler, Essentials of Cost Accounting for Health Care Organizations, 2nd ed. (Gaithersburg, MD: Aspen Publishers, Inc., 1999).

2. At the time of this writing, 23 major diagnostic categories (MDCs) serve as the basic classification sys- tem for diagnosis-related groups (DRGs). When ICD-10 is fully implemented, it is probable that the number of MDCs will be increased. It is also possible that the terminology itself (MDCs) may be changed to some other designation.

3. G. F. Longshore, “Service-line Mmanagement/Bottom-line Management for Health Care,” Journal of Health Care Finance, 24, no. 4 (1998): 72–79.

Chapter 6

1. C. Horngren et al., Cost Accounting: A Managerial Emphasis, 9th ed. (Englewood Cliffs, NJ: Prentice Hall, 1998), 70.

2. J. J. Baker, Activity-Based Costing and Activity-Based Management for Health Care (Gaithersburg, MD: Aspen Publishers, Inc., 1998).

3. D. A. West, T. D. West, and P. J. Malone, “Managing Capital and Administrative (indirect) Costs to Achieve Strategic Objectives: The Dialysis Clinic versus the Outpatient Clinic,” Journal of Health Care Fi- nance, 25, no. 2 (1998): 20–24.

Chapter 7

1. C. Horngren et al., Cost Accounting: A Managerial Emphasis, 9th ed. (Englewood Cliffs, NJ: Prentice Hall, 1998).

2. J. J. Baker, Activity-Based Costing and Activity-Based Management for Health Care (Gaithersburg, MD: Aspen Publishers, Inc., 1998).

3. It is possible that the term “diagnosis-related groups” (DRGs) may be changed to some new terminol- ogy as a consequence of ICD-10 implementation.

Chapter 8

1. Department of the Treasury, Internal Revenue Service. How to Depreciate Property. Publication 946 (Washington, D.C.: U.S. Government, 2007): 31–32.

2. Ibid., Table 4-1, 38. 3. Ibid., “Required Use of ADS,” 31. 4. Ibid., “Election of ADS,” 38.

Chapter 9

1. J. J. Baker, Prospective Payment for Long-Term Care: An Annual Guide (Gaithersburg, MD: Aspen Publish- ers, Inc., 1999).

Chapter 10

1. S. A. Finkler, Essentials of Cost Accounting for Health Care Organizations, 2nd ed. (Gaithersburg, MD: Aspen Publishers, Inc., 1999).

Chapter 12

1. S. Williamson et al., Fundamentals of Strategic Planning for Healthcare Organizations (New York: The Ha- worth Press, 1997).

Chapter 13

1. Merriam Webster’s Collegiate Dictionary, 10th ed., s.v. “Forecast.” 2. B. A. Brotman, M. Bumgarner, and P. Prime, “Client Flow through the Women, Infants, and Children

Public Health Program,” Journal of Health Care Finance, 25, no. 1 (1998): 72–77.

Chapter 14

1. Merriam Webster’s Collegiate Dictionary, 10th ed., s.v. “Inflation.”

Chapter 15

1. W. O. Cleverly, Essentials of Health Care Finance, 4th ed. (Gaithersburg, MD: Aspen Publishers, Inc., 1997).

2. C. Horngren et al., Cost Accounting: A Managerial Emphasis, 9th ed. (Englewood Cliffs, NJ: Prentice Hall, 1998), 227.

3. Ibid., 228. 4. J. R. Pearson et al., “The Flexible Budget Process—A Tool for Cost Containment,” A. J. C. P., 84, no. 2

(1985): 202–208.

Chapter 17

1. S. A. Finkler, “Flexible Budget Variance Analysis Extended to Patient Acuity and DRGs,” Health Care Management Review, 10, no. 4 (1985): 21–34.

Chapter 18

1. Merriam Webster’s Collegiate Dictionary, 10th ed., s.v. “Estimate.”

Chapter 19

1. American Recovery and Reinvestment Act of 2009 (ARRA) Division A Title XIII Sec. 3000. 2. Ibid. 3. A. K. Jha et. al., “Use of Electronic Health Records in U.S. Hospitals,” New England Journal of Medicine

360, no. 16 (2009). http://content.nejm.org/full/3601/16/1628. 4. Ibid. 5. ARRA Division B. Title IV Sec. 4101. 6. 74 Federal Register (FR) 3328 (January 16, 2009). 7. 73 FR 69847 (November 19, 2008). 8. ARRA Division A. Title XIII “Health Information Technology” (HITECH) and Division B. Title IV

“Medicare and Medicaid Health Information Technology.” 9. ARRA Division A. Title XIII Sec. 3001. The Office of the National Coordinator for Health Informa-

tion Coordinator is located within the Department of Health and Human Services. 10. ARRA Division B. Title IV Sec. 4102. 11. Ibid. 12. ARRA Division B. Title IV Sec. 4101. 13. Ibid. 14. Ibid. 15. ARRA Division B. Title IV Sec. 4102. 16. National Center for Health Statistics, International Classification of Diseases, Tenth Revision (ICD-10).

www.cdc.gov/nchs/about/major/dvs/icd10des.htm. 17. World Health Organization, Classifications. www.who.int/classifications/icd/en/.

Notes 341

342 NOTES

18. CMS: ICD-10 Clinical Modification/Procedure Coding System Fact Sheet. www.cms.hhs.gov/MLN Products/downloads/ICD-10factsheet2008.pdf.

19. National Center for Health Statistics (NCHS), About the International Classification of Diseases, Tenth Revision, Clinical Modification (ICD-10-CM). www.cdc.gov/nchs/about/otheract/icd9/abticd10 .htm.

20. CMS: ICD-10-CM-PCS Fact Sheet. 21. Ibid. 22. 73 FR 49743 (August 22, 2008). 23. Ibid. 24. 74 FR 3328 (January 16, 2009). 25. 73 FR 49745 (August 22, 2008). 26. 74 FR 3357 (January 16, 2009). 27. 73 FR 49821 (August 22, 2008). 28. Ibid., 49769. 29. Ibid., 49811. 30. Ibid., 49813. 31. Ibid., 49818. 32. Ibid., 49829. 33. Ibid., 49769. 34. CMS: ICD-10-CM-PCS Fact Sheet. 35. Ibid. 36. Ibid. 37. Ibid.

Chapter 20

1. 74 FR 3348-9 (January 16, 2009). 2. CMS: ICD-10-CM-PCS Fact Sheet. 3. J. Zhang, “Why We Need 1,170 Angioplasty Codes,” Wall Street Journal, November 11, 2008. 4. 73 FR 49814-5 (August 22, 2008). 5. Ibid., 49815-6. 6. Ibid., 49815. 7. Ibid., 49816. 8. Ibid., 49815. 9. Ibid., 49816 and 74 FR 3346-7 (January 16, 2009).

10. Ibid., 49816. 11. M. Libicki and I. Brahmakulam, The Costs and Benefits of Moving to the ICD-10 Code Sets (Santa Monica,

CA: RAND Corporation, 2004), 10. http://www.rand.org/pubs/technical_reports/2004/RAND_ TR132.pdf.

12. 73 FR 49816 (August 22, 2008) and 74 FR 3346-7 (January 16, 2009). 13. 73 FR 49817 (August 22, 2008). 14. Ibid., 49816-7. 15. 70 FR 6265 (February 4, 2005). 16. Ibid. 17. 70 FR 67572 (November 7, 2005). 18. D. S. Bell and M. A. Friedman, “E-Prescribing and the Medicare Modernization Act of 2008,” Health

Affairs 24 (2005): 1159–1169. 19. 70 FR 67588 (November 7, 2005). 20. Ibid., 67569. 21. Ibid., 67592. 22. Ibid., 67588. 23. A. Zimmerman, “Sam’s Club to Provide Digital Medical Tools,” Wall Street Journal, March 12, 2009. The

name of the closely held software maker is eClinicalWorks.

24. Ibid. 25. Adapted from 70 FR 67569 (November 7, 2005). 26. 70 FR 6270 (February 4, 2005). 27. 70 FR 67589 (November 7, 2005). 28. CMS, Transmittal 459, CR 6394 (2009): C. 29. Medicare Learning Network, MLN Matters MM6394 (2009): 9. 30. Ibid., 3. 31. Ibid., 4. 32. CMS, Transmittal 459: C.1. 33. Medicare Learning Network, “2009 Electronic Prescribing (E-Prescribing) Incentive Program Made

Simple.” www.cms.hhs.gov/ERxIncentive. 34. Medicare Learning Network: 11. 35. Ibid., 10. 36. 73 FR 69847-8 (November 19, 2008). 37. CMS, Transmittal 459: C.4. 38. 73 FR 69847-8 (November 19, 2008). 39. Medicare Learning Network: 9. 40. Medicare Learning Network: adapted from Step 2, p. 4. 41. Ibid.

Chapter 21

1. Federal Deposit Insurance Corporation, “Who Is the FDIC?” Federal Deposit Insurance Corporation. www.fdic.gov/about/learn.

2. Ibid. 3. U.S. Securities and Exchange Commission, “The Investor’s Advocate: How the SEC Protects Investors,

Maintains Market Integrity, and Facilitates Capital Formation.” www.sec.gov/about/whatwedo.shtml. 4. Ibid. 5. Merriam Webster’s Collegiate Dictionary, 10th ed., s.v. “Inflation.” 6. Ibid., 303. 7. U.S. Department of Commerce, Bureau of Economic Analysis (BEA). News Release: Gross Domestic

Product: Fourth Quarter 2008 (Final). www.bea.gov/newsreleases/national/gdp. 8. Ibid.

Notes 343

345

G l o s s a r y

Accounting Rate of Return: See Unadjusted Rate of Return.

Accounting System: Records the evidence that some event has occurred in the healthcare financial system.

Accrual Basis Accounting: Revenue is recorded when it is earned, not when payment is re- ceived. Expenses are recorded when they are incurred, not when they are paid. The opposite of accrual basis is cash basis accounting.

Annualize: To convert data to an annual (12-month) period.

Assets: The net value of what an organization owns.

Balance Sheet: One of the four basic financial statements. Generally speaking, the balance sheet records what an organization owns, what it owes, and what it is worth at a partic- ular point in time.

Benchmarking: The continuous process of measuring products, services, and activities against the best levels of performance. Best levels may be found inside or outside of the organization.

Book Value: The book value (also known as net book value) of a fixed asset is a balance sheet figure that represents the remaining undepreciated portion of the fixed asset cost.

Break-Even Point: The point when the contribution margin (i.e., net revenues less variable costs) equals the fixed costs.

Budget: The organization-wide instrument through which activities are quantified in fi- nancial terms.

Business Plan: A document that is typically prepared in order to obtain funding and/or financing.

Capital: Represents the financial resources of the organization. Generally considered to be a combination of debt and equity.

346 GLOSSARY

Capital Expenditure Budget: A budget usually intended to plan, monitor, and control long-term financial issues.

Capital Structure: Means the proportion of debt versus equity within the organization. The phrase “capital structure” actually refers to the debt–equity relationship.

Case Mix Adjusted: A performance measure that has been adjusted for the acuity level of the patient and, presumably, the resource level required to provide care.

Cash Basis Accounting: A transaction does not enter the books until cash is either received or paid out. The opposite of cash basis is accrual basis accounting.

Cash Flow Analysis: This type of analysis illustrates how the project’s cash is expected to move over a period of time.

Chart of Accounts: Maps out account titles in a uniform manner through a method of nu- meric coding.

Code Users: Any individual who needs to have some level of understanding of the coding system, but does not actually assign codes.

Common Sizing: A process of converting dollar amounts to percentages to put information on the same relative basis. Also known as vertical analysis.

Common Stock: Stocks represent equity, or net worth, in a company. Common stock typi- cally pays a proportionate share of net income out as a dividend to its investors.

Contribution Income Statement: Specifically identifies the contribution margin within the income statement format.

Contribution Margin: Called this because it contributes to fixed costs and to profits. Com- puted as net revenues less variable costs.

Controllable Expenses: Subject to a manager’s own decision making and thus “controllable.”

Controlling: Making sure that each area of the organization is following the plans that have been established.

Cost: The amount of cash expended (or property transferred, services performed, or lia- bility incurred) in consideration of goods or services received or to be received.

Cost-Profit-Volume: A method of illustrating the break-even point, whereby the three elements of cost, profit, and volume are accounted for within the computation.

Cost Object: Any unit for which a separate cost measurement is desired.

Cumulative Cash Flow: The accumulated effect of cash inflows and cash outflows are added and/or subtracted to show the overall net accumulated result.

Current Ratio: A liquidity ratio considered to be a measure of short-term debt-paying abil- ity. Computed by dividing current assets by current liabilities.

Days Cash on Hand Ratio: A liquidity ratio that indicates the number of days of operating expenses represented in the amount of unrestricted cash on hand. Computed by di-

viding unrestricted cash and cash equivalents by the cash operating expenses divided by number of days in the period.

Days Receivables Ratio: A liquidity ratio that represents the number of days in receivables. Computed by dividing net receivables by net credit revenues divided by number of days in the period.

Debentures: Bonds that are unsecured. Debentures are backed by revenues that the issu- ing organization can earn.

Debt Service Coverage Ratio: A solvency ratio universally used in credit analysis to measure ability to pay debt service. Computed by dividing change in unrestricted net assets (net income) plus interest, depreciation, and amortization by maximum annual debt service.

Decision Making: Making choices among available alternatives.

Deflation: A contraction in the volume of available money and credit that results in a gen- eral decline in prices.

Depreciation: Depreciation expense spreads, or allocates, the cost of a fixed asset over the useful life of that asset.

Diagnoses: A common method of grouping healthcare expenses for purposes of planning and control. Such a grouping may be by major diagnostic categories or by diagnosis- related groups.

Direct Costs: These costs are incurred for the sole benefit of a particular operating unit. They can therefore be specifically associated with a particular unit or department or patient. Laboratory tests are an example of a direct cost.

Discounted Fee-for-Service: The provider of services is paid according to an agreed-upon contracted discount and after the service is delivered.

Dispenser: Either a person or other legal entity who provides drug products for human use on prescription in the course of professional practice, and who is licensed, registered, or otherwise permitted by the jurisdiction in which the person practices, or the entity is located, to do so.

Electronic Health Record (EHR): A health-related electronic record of an individual that includes patient demographic and clinical information and that has the ca- pacity to provide clinical decision support, support physician order entr y, capture and quer y quality information, and exchange and integrate electronic health information.

Electronic Prescribing (E-Prescribing): Transmitting a prescription or prescription-related information, using electronic media, between a prescriber, dispenser, PBM, or health plan. The transmission may be either direct or through an intermediary, including an e-prescribing network.

Eligible Professional: In this context, physicians, practitioners, and therapists who are eli- gible for payment in the e-prescribing incentives program.

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348 GLOSSARY

Equity: Claims held by the owners of the business because they have invested in the busi- ness; what the business is worth on paper, net of liabilities.

Estimates: A judgment that takes the place of actual measurement.

Expenses: Actual or expected cash outflows incurred in the course of doing business. Ex- penses are the costs that relate to the earning of revenue. An example is salary ex- pense for labor performed.

Expired Costs: Costs that are used up in the current period and are matched against cur- rent revenues.

Fee-for-Service: The provider of services is paid according to the service performed and after the service is delivered.

FIFO: The First-In, First-Out (FIFO) inventory costing method recognizes the first costs placed into inventory as the first costs moved out into cost of goods sold when a sale occurs.

Financial Accounting: Is generally for outside, or third-party, use and thus emphasizes ex- ternal reporting.

Financial Lease: A formal agreement that may be called a lease but is actually a contract to purchase. This type of lease must meet certain criteria.

Fixed Costs: Those costs that do not vary in total when activity levels or volume of opera- tions change. Rent expense is an example of fixed cost.

Flexible Budget: A budget based on a range of activity or volume. The flexible budget is ad- justed, or flexed (thus “flexible”) to the actual level of output achieved or expected to be achieved during the budget period.

Forecasts: Information used for purposes of planning for the future. Forecasts can be short, intermediate, or long range.

For-Profit Organization: A proprietary organization that is generally subject to income tax.

Full-Time Equivalent: A measure to express the equivalent of an employee (annualized) or a position (staffed) for the full time required (thus, “full-time equivalent” or FTE).

Fund Balance: The difference between net assets and net liabilities; a term generally used by not-for-profit organizations.

General Ledger: A document in which all transactions for the period reside.

General Services Expenses: This type of expense provides services necessary to maintain the patient, but the service is not directly related to patient care. Examples of general services expenses are laundry and dietary.

Gross Domestic Product (GDP): A measure of the output of goods and services produced by labor and property located in the United States. The Bureau of Economic Analysis (BEA) is responsible for releasing quarterly estimates of the GDP.

Health Information Technology (HIT): Technology that is designed for, or support use by, healthcare entities or patients. Includes hardware, software, integrated technologies or related licenses, intellectual property, upgrades, or packaged solutions.

Horizontal Analysis: The process of comparing and analyzing figures over several time pe- riods. Also known as trend analysis.

Indirect Cost: These costs are incurred on behalf of the overall operation and therefore cannot be associated with the provision of specific health services. The finance office is an example of an indirect cost. Also known as joint costs.

Inflation: An increase in the volume of money and credit relative to available goods and services resulting in a continuing rise in the general price level.

Information System: Gathers the evidence that some event has occurred in the healthcare financial system.

Internal Rate of Return: A return on investment method, defined as the rate of interest that dis- counts future net inflows (from the proposed investment) down to the amount invested.

Inventory: All the items (“goods”) that an organization has for sale in the normal course of its business.

Inventory Turnover: A ratio that shows how fast inventory is sold, or “turns over.”

Joint Costs: These costs are incurred on behalf of the overall operation and therefore can- not be associated with the provision of specific health services. The finance office is a typical example of a joint cost. Also known as indirect cost.

Liabilities: What the organization owes.

Liabilities to Fund Balance Ratio: A solvency ratio used as a quick indicator of debt load. Computed by dividing total liabilities by unrestricted net assets. Also known as Debt to Net Worth Ratio.

LIFO: The Last-In, First-Out or LIFO inventory costing method recognizes the latest, or last, costs placed into inventory as the first costs moved out into cost of goods sold when a sale occurs.

Liquidity Ratios: Ratios that reflect the ability of the organization to meet its current obli- gations. Liquidity ratios are measures of short-term sufficiency.

Loan Costs: Those costs necessary to close a loan.

Managed Care: A means of providing healthcare services within a network of healthcare providers. The central concept is coordination of all healthcare services for an individual.

Managerial Accounting: Is generally for inside, or internal, use and thus emphasizes infor- mation useful for managerial employees.

Medicaid Program: A federal and state matching entitlement program intended to provide medical assistance to eligible needy individuals and families. The program was estab- lished under Title XIX of the Social Security Act.

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350 GLOSSARY

Medicare Program: A federal health insurance program for the aged (and, in certain in- stances, for the disabled) intended to complement other federal benefits. The pro- gram was established under Title XVIII of the Social Security Act.

Mixed Cost: Those costs that contain an element of variable cost and an element of fixed cost.

Monetary Unit: A measure of units of currency, such as the dollar. Monetary units should be comparable when reporting financial results.

Mortgage Bonds: Bonds that are backed, or secured, by certain real property.

Municipal Bonds: Long-term obligations that are typically used to finance capital projects.

Net Worth: See Equity.

Noncontrollable Expenses: Outside the manager’s power to make decisions, and thus “noncontrollable.”

Nonproductive Time: Paid-for time when the employee is not on duty—that is, not pro- ducing. Paid-for vacation days and holidays are examples of nonproductive time.

Nonprofit Organization: Indicates the taxable status of the organization. A nonprofit (or voluntary) organization is exempt from paying income taxes.

Not-for-Profit Organization: See Nonprofit Organization.

Operating Budget: A budget that generally deals with actual short-term revenues and ex- penses necessary to operate the facility.

Operating Lease: A lease that is considered an operating expense and thus is treated as an expense of current operations. This type of lease does not meet the criteria to be treated as a financial lease.

Operating Margin: A profitability ratio generally expressed as a percentage, the operating margin is a multipurpose measure. It is used for a number of managerial purposes and also sometimes enters into credit analysis. Computed by dividing operating in- come (loss) by total operating revenues.

Operations Expenses: This type of expense provides service directly related to patient care. Examples of operations expenses are radiology expense and drug expense.

Organization Chart: Indicates the formal lines of communication and reporting and how responsibility is assigned to managers.

Organizing: Deciding how to use the resources of the organization to most effectively carry out the plans that have been established.

Original Records: Provide evidence that some event has occurred in the healthcare finan- cial system.

Overhead: Refers to the remaining expenses of operation that are necessary to produce the service but that are not directly attributable to that service.

Pareto Analysis: An analytical tool employing the Pareto principle, also known as the 80/20 rule. For example, the Pareto principle states that 80% of an organization’s problems are caused by 20% of the possible causes.

Patient Mix: A term indicating the mix of payers; thus, whether the individual is a Medicare patient, a Medicaid patient, a patient covered by private insurance, or a pri- vate pay patient varies the patient mix proportions. Patient mix information allows es- timated payment levels to be associated with the service utilization assumptions.

Payback Period: The length of time required for the cash coming in from an investment to equal the amount of cash originally spent when the investment was acquired.

Payer Mix: The proportion of revenues realized from different types of payers. A measure often included in the profile of a healthcare organization.

Performance Measures: Measures that compare and quantify performance. Performance measures may be financial, non-financial, or a combination of both types.

Period Cost: For purposes of healthcare businesses, period cost is necessary to support the existence of the organization itself, rather than actual delivery of a service. Period costs are matched with revenue on the basis of the period during which the cost is in- curred. The term originated with the manufacturing industry.

Planning: Identifying objectives of the organization and identifying the steps required to accomplish the objectives.

Preferred Stock: Stock that has preference over common stock in certain issues such as payment of dividends.

Prescriber: A physician, dentist, or other person who issues prescriptions for drugs for human use, and who is licensed, registered, or otherwise permitted by the United States or the jurisdiction in which he or she practices to do so.

Present Value Analysis: A concept based on the time value of money. The value of a dollar today is more than the value of a dollar in the future.

Procedures: A common method of grouping healthcare expenses for purposes of plan- ning and control. Such a grouping will generally be by Current Procedural Terminol- ogy (or CPT) codes, which list descriptive terms and identifying codes for medical services and procedures performed.

Product Cost: For the purposes of healthcare businesses, product cost is necessary to actu- ally deliver the service. The term originated with the manufacturing industry.

Productive Time: Equates to the employee’s net hours on duty when performing the func- tions in his or her job description.

Profit Center: Makes a manager responsible for both the revenue/volume (inflow) side and the expense (outflow) side of a department, division, unit, or program. Also known as a responsibility center.

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352 GLOSSARY

Profitability Ratios: Ratios that reflect the ability of the organization to operate with an ex- cess of operating revenue over operating expense.

Profit-Oriented Organization: Indicates the taxable status of the organization. A profit- oriented (or proprietary) organization is responsible for paying income taxes.

Profit-Volume (PV) Ratio: The contribution margin (i.e., net revenues less variable costs) expressed as a percentage of net revenue.

Proprietary Organization: Indicates the taxable status of the organization. A proprietary (or profit-oriented) organization is responsible for paying income taxes.

Quartiles: A distribution into four classes, each of which contains one quarter of the whole; any one of the four classes is a quartile.

Quick Ratio: A liquidity ratio considered the most severe test of short-term debt-paying ability (even more severe than the current ratio). Computed by dividing cash and cash equivalents plus net receivables by current liabilities. Also known as the acid-test ratio.

Reporting System: Produces reports of an event’s effect in the healthcare financial system.

Responsibility Centers: Makes a manager responsible for both the revenue/volume (in- flow) side and the expense (outflow) side of a department, division, unit, or program. Also known as a profit center.

Return on Total Assets: A profitability ratio generally expressed as a percentage, this is a broad measure of profitability in common use. Computed by dividing earnings before interest and taxes, or EBIT, by total assets. This ratio is known by its acronym, EBIT, in credit analysis circles.

Revenue: Actual or expected cash inflows due to the organization’s major business. Rev- enues are amounts earned in the course of doing business. In the case of health care, revenues are mostly earned by rendering services to patients.

Revenue Amount: Refers to how much each payer is expected to pay for the service and/or drug or device.

Revenue Sources: Refers to how many payers will pay for the service and/or drug and de- vice, and in what proportion.

Revenue Type: A designation as to whether, for example, revenue is derived entirely from services or whether part of the revenue is derived from drugs and devices.

Salvage Value: Salvage value, also known as residual value or scrap value, represents any ex- pected cash value of the asset at the end of its useful life.

Semifixed Costs: Those costs that stay fixed for a time when activity levels or volume of op- erations change; rises will occur, but not in direct proportion.

Semivariable Costs: Those costs that vary when activity levels or volume of operations change, but not in direct proportion. A supervisor’s salary is an example of a semi- variable cost.

Situational Analysis: Management tool that reviews, assesses, and analyzes the organiza- tion’s internal operations for strengths and weaknesses and the organization’s exter- nal environment for opportunities and threats.

Solvency Ratios: Ratios that reflect the ability of the organization to pay the annual interest and principal obligations on its long-term debt. These ratios determine ability to “be solvent.”

Space Occupancy: Within the context of a forecast or projection, refers to the overall cost of occupying the space required for the service or procedure. Considered to be an in- direct cost.

Staffing: A term that means the assigning of staff to fill scheduled positions.

Statement of Cash Flows: One of the four basic financial statements, this statement reports the current period cash flow by taking the accrual basis statements and converting them to an effective cash flow. This is accomplished by a series of reconciling adjust- ments that account for the noncash amounts.

Statement of Fund Balance/Net Worth: One of the four basic financial statements, this statement reports the excess of revenue over expenses (or vice versa) for the period as the excess flows into equity (or reduces equity, in the case of a loss for the period).

Statement of Revenue and Expense: One of the four basic financial statements, this state- ment reports the inflow of revenue and the outflow of expense over a stated period of time. The net result is also reported, either as excess of revenue over expense or, in the case of a loss for the period, excess of expense over revenue.

Static Budget: A budget based on a single level of operations, or volume. After it is ap- proved and finalized, the single level of operations (volume) is never adjusted; thus, the budget is “static” or unchanging.

Stock Warrants: Warrants allow the owner of the warrant to purchase additional shares of stock in the company, generally at a particular price and prior to an expiration date.

Subsidiary Journals: Documents that contain specific sets of transactions and that support the general ledger.

Subsidiary Reports: Reports that support, and thus are subsidiary to, the four major financial statements.

Supplies: Within the context of a forecast or projection, refers to the necessary supplies that are required to perform a procedure or service. Considered to be a direct expense.

Support Services Expenses: This type of expense provides support to both general services expenses and to operations expenses. It is necessary for support, but it is neither di- rectly related to patient care nor is it a service necessary to maintain the patient. Ex- amples of support services are insurance and payroll taxes.

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354 GLOSSARY

SWOT Analysis: Acronym for a method of situational analysis assessing an organization’s strengths-weaknesses-opportunities-and-threats; thus “SWOT.”

Target Operating Income: Allows the manager to determine, or target, how many units must be sold in order to yield a particular operating income.

Three-Variance Method: A method of variance analysis that compares volume variance to use (or quantity) variance and to spending (or price) variance.

Time Value of Money: The present value concept, which is that the value of a dollar today is more than the value of a dollar in the future.

Trend Analysis: The process of comparing and analyzing figures over several time periods. Also known as horizontal analysis.

Trial Balance: A document used to balance the general ledger accounts and to produce fi- nancial statements.

Two-Variance Method: A method of variance analysis that compares volume variance to budgeted costs (defined as standard hours for actual production)—thus the “two- variance” method.

Unadjusted Rate of Return: An unsophisticated return on investment method, the answer for which is an estimate containing no precision.

Unexpired Costs: Costs that are not yet used up and will be matched against future revenues.

Useful Life: The useful life of a fixed asset determines the period over which the fixed asset’s cost will be spread.

Variable Costs: Those costs that vary in direct proportion to changes in activity levels of vol- ume of operations. Food for meal preparation is an example of variable cost.

Variance Analysis: A variance is the difference between standard and actual prices and quantities. Variance analysis analyzes these differences.

Vertical Analysis: A process of converting dollar amounts to percentages to put informa- tion on the same relative basis. Also known as common sizing.

Voluntary Organization: Indicates the taxable status of the organization. A voluntary (or nonprofit) organization is exempt from paying income taxes.

The following examples and exercises include examples, practice exercises, and assign- ment exercises. Solutions to the practice exercises are found at the end of this section. Ex- ercises are designated by chapter number.

EXAMPLES AND EXERCISES

CHAPTER 1

Assignment Exercise 1–1

Review the chapter text about types of organizations and examine the list in Exhibit 1-1.

Require d

1. Obtain listings of healthcare organizations from the yellow pages of a telephone book.

2. Set up a worksheet listing the classifications of organizations found in Exhibit 1-1. 3. Enter the organizations you found in the yellow pages onto the worksheet. 4. For each organization indicate the type of organization. 5. If some cannot be identified by type, comment on what you would expect them to be;

that is, proprietary, voluntary, or government owned.

Assignment Exercise 1–2

Review the chapter text about organization charts. Also examine the organization charts appearing in Figures 1-2 and 1-3.

Require d

1. Refer to the Metropolis Health System (MHS) case study appearing in Chapter 25. Read about the various types of services offered by MHS.

2. The MHS organization chart has seven major areas of responsibility, each headed by a senior vice president. Select one of the seven areas and design additional levels of

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Examples and Exercises,

Supplemental Materials, and

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356 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

detail that indicate the managers. If you have considerable detail you may choose one department (such as ambulatory operations) instead of the entire area of responsi- bility for that senior vice president.

3. Do you believe your design of the detailed organization chart indicates centralized or decentralized lines of authority for decision making? Can you explain your approach in one to two sentences?

CHAPTER 2

Assignment Exercise 2–1: Health System Flowsheets

Review the chapter text about information flow and Figures 2-2 and 2-3.

Require d

1. Find an information flowsheet from a healthcare organization. It can be from a pub- lished source or from an actual organization.

2. Based on this flowsheet, comment on what the structure of the organization’s infor- mation system appears to be.

3. If you were a manager (at this organization), would you want to change the structure? If so, why? If not, why not?

Assignment Exercise 2–2: Chart of Accounts

Review the chapter text about the chart of accounts and how it is a map of the company el- ements. Also review Exhibits 2-1, 2-2, and 2-3.

Require d

1. Find an excerpt from a healthcare organization’s chart of accounts. It can be from a published source or from an actual organization.

2. Based on this chart of accounts excerpt, comment on what the structure of the orga- nization’s reporting system appears to be.

3. If you were a manager (at this organization), would you want to change the system? If so, why? If not, why not?

CHAPTER 3

Example 3A: Assets and Liabilities

Study the chapter text concerning examples of assets and liabilities. Is the difference be- tween short-term and long-term assets and liabilities clear to you?

Practice Exercise 3–I

Place an “X” in the appropriate classification for each balance sheet item listed below.

Assignment Exercise 3–1: Balance Sheet

Locate a healthcare-related balance sheet. The source of the balance sheet can be internal (within a healthcare facility of some type) or external (from a published article or from a company’s annual report, for example). Write your impressions and/or comments about the assets, liabilities, and net worth found on your balance sheet. Would you have preferred more detail in this statement? If so, why?

Assignment Exercise 3–2: Balance Sheet

Locate a second healthcare-related balance sheet. Again, the source of the balance sheet can be either internal or external. Compare the balance sheet you acquired for Assignment Exercise 3-1 with the second balance sheet you have now obtained. What is the same? What is different? Which one do you find more informative? Why?

CHAPTER 4

Example 4A: Contractual Allowances

Contractual allowances represent the difference between the full established rate and the agreed-upon contractual rate that will be paid. An example was given in the text of Chapter 4 by which the hospital’s full established rate for a certain procedure is $100, but Giant Health Plan has negotiated a managed care contract whereby the plan pays only $90 for that procedure. The contractual allowance is $10 ($100 less $90 � $10). Assume instead that Near-By Health Plan has negotiated its own managed care contract whereby this plan pays $95 for that procedure. In this case the contractual allowance is $5 ($100 less $95 � $5).

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Short-Term Long-Term Short-Term Long-Term Asset Asset Liability Liability

Payroll taxes due

Accounts receivable

Land

Mortgage payable (non-current)

Buildings

Note payable (due in 24 months)

Inventory

Accounts payable

Cash on hand

358 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

Assignment Exercise 4–1: Contractual Allowances

Physician office revenue for visit code 99214 has a full established rate of $72.00. Of ten dif- ferent payers, there are nine different contracted rates, as follows:

Contracted Payer Rate

FHP $35.70 HPHP 58.85 MC 54.90 UND 60.40 CCN 70.20 MO 70.75 CGN 10.00 PRU 54.90 PHCS 50.00 ANA 45.00

Rates for illustration only.

Require d

1. Set up a worksheet with four columns: Payer, Full Rate, Contracted Rate, and Con- tractual Allowance.

2. For each payer, enter the full rate and the contracted rate. 3. For each payer, compute the contractual allowance.

The first payer has been computed below:

Full Contracted Contractual Payer Rate (less) Rate (equals) Allowance FHP $72.00 $35.70 $36.30

Example 4B: Revenue Sources and Grouping Revenue

Sources of healthcare revenue are often grouped by payer. Thus, services might be grouped as follows:

Revenue from the Medicare Program (payer � Medicare) Revenue from the Medicaid Program (payer � Medicaid) Revenue from Blue Cross Blue Shield (payer � Commercial Insurance) or Revenue from Blue Cross Blue Shield (payer � Managed Care Contract)

Assignment Exercise 4–2: Revenue Sources and Grouping Revenue

The Metropolis Health System has revenue sources from operations, donations, and inter- est income. The revenue from operations is primarily received for services. MHS groups its revenue first by cost center. Within each cost center the services revenue is then grouped by payer.

Require d

1. Set up a worksheet with individual columns across the top for six revenue sources (payers): Medicare, Medicaid, Other Public Programs, Patients, Commercial Insur- ance, and Managed Care Contracts.

2. Certain situations concerning the Intensive Care Unit and the Laboratory are de- scribed below.

Set up six vertical line items on your worksheet, numbered (1) through (6). Six situations are described below. For each of the six situations, indicate its number (1 through 6) and enter the appropriate cost center (either Intensive Care Unit or Laboratory). Then place an X in the column(s) that represents the correct revenue source(s) for the item. The six situations are as follows:

(1) ICU stay billed to employee’s insurance program. (2) Lab test paid for by an individual. (3) Pathology work performed for the state. (4) ICU stay billed to member’s health plan. (5) ICU stay billed for Medicare beneficiary. (6) Series of allergy tests run for eligible Medicaid beneficiary.

Headings for your worksheet:

Other Public Commercial Managed Medicare Medicaid Programs Patients Insurance Care Contracts

(1)

(2)

(3)

(4)

(5)

(6)

CHAPTER 5

Example 5A: Grouping Expenses by Cost Center

Cost centers are one method of grouping expenses. For example, a nursing home may con- sider the Admitting department as a cost center. In that case the expenses grouped under the Admitting department cost center may include:

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360 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

• Administrative and Clerical Salaries • Admitting Supplies • Dues • Periodicals and Books • Employee Education • Purchased Maintenance

Practice Exercise 5–I: Grouping Expenses by Cost Center

The Metropolis Health System groups expenses for the Intensive Care Unit into its own cost center. Laboratory expenses and Laundry expenses are likewise grouped into their own cost centers.

Require d

1. Set up a worksheet with individual columns across the top for the three cost centers: Intensive Care Unit, Laboratory, and Laundry.

2. Indicate the appropriate cost center for each of the following expenses: • Drugs Requisitioned • Pathology Supplies • Detergents and Bleach • Nursing Salaries • Clerical Salaries • Uniforms (for Laundry Aides) • Repairs (parts for microscopes)

(Hint: One of the expenses will apply to more than one cost center.)

Headings for your worksheet:

Intensive Care Unit Laboratory Laundry

Assignment Exercise 5–1: Grouping Expenses by Cost Center

The Metropolis Health System’s Rehabilitation and Wellness Center offers outpatient ther- apy and return-to-work services plus cardiac and pulmonary rehabilitation to get people back to a normal way of living. The Rehabilitation and Wellness Center expenses include the following:

• Nursing Salaries • Physical Therapist Salaries • Occupational Therapist Salaries • Cardiac Rehab Salaries • Pulmonary Rehab Salaries • Patient Education Coordinator Salary • Nursing Supplies • Physical Therapist Supplies

• Occupational Therapist Supplies • Cardiac Rehab Supplies • Pulmonary Rehab Supplies • Training Supplies • Clerical Office Supplies • Employee Education

Require d

1. Decide how many cost centers should be used for the above expenses at the Center. 2. Set up a worksheet with individual columns across the top for the cost centers you

have chosen. 3. For each of the expenses listed above, indicate to which of your cost centers it should

be assigned.

Example 5B

Study the chapter text concerning grouping expenses by diagnoses and procedures. Refer to Exhibits 5-3 and 5-4 (about major diagnostic categories), Exhibit 5-5 (about DRGs and MDCs), and Table 5-1 (about procedure codes) for examples of different ways to group ex- penses by diagnoses and procedures.

Assignment Exercise 5–2

Require d

Find a listing of expenses by diagnosis or by procedure. The source of the list can be inter- nal (within a healthcare facility of some type) or external (such as a published article, re- port, or survey). Comment upon whether you believe the expense grouping used is appropriate. Would you have grouped the expenses in another way?

CHAPTER 6

Example 6A: Direct and Indirect Costs

Review the chapter text regarding direct and indirect costs. In particular, review the exam- ple of freestanding dialysis center direct costs (Exhibit 6-1) and indirect costs (Exhibit 6-2). Remember that indirect costs are shared and are sometimes called joint costs or common costs. Because such costs are shared they must be allocated. Also, remember that one test of a direct cost is to ask: “If the operating unit (such as a department) did not exist, would this cost not be in existence?”

Practice Exercise 6–I: Identifying Direct and Indirect Costs

Make a worksheet with two columns labeled: Direct Cost and Indirect Cost. Place each of the following items in the appropriate column:

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362 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

• Managed care marketing expense • Real estate taxes • Liability insurance • Clinic telephone expense • Utilities (for the entire facility) • Emergency room medical supplies

Assignment Exercise 6–1: Allocating Indirect Costs

Study Table 6-1, “Example of Radiology Departments Direct and Indirect Cost Totals,” and Table 6-2, “Example of Indirect Costs Allocated to Radiology Departments,” and review the chapter text describing how the indirect cost is allocated. This assignment will change the allocation bases: A) Volumes, B) Direct Costs, and C) Number of Films.

Required

1. Compute the costs allocated to cost centers #557, 558, 559, 560, and 561 using the new allocation bases shown below. Use a worksheet replicating the setup in Table 6-2. Total the new results.

The new allocation bases are:

A) Volumes 120,000 130,000 70,000 110,000 70,000 500,000 B) Direct costs $1,100,000 $700,000 $1,300,000 $1,600,000 $1,300,000 $6,000,000 C) No. of films 400,000 20,000 55,000 25,000 20,000 520,000

2. Using a worksheet replicating the setup in Table 6-1, enter the new direct cost and the new totals for indirect costs resulting from your work. Total the new results.

Practice Exercise 6–II: Responsibility Centers

The Metropolis Health System has one director who supervises the areas of Security, Com- munications, and Ambulance Services. This director also supervises the medical records relevant to Ambulance Services, the educational training for Security and Ambulance Ser- vices personnel, and the human resources for Security, Communications, and Ambulance Services personnel.

Require d

Of the duties and services described above, all of which are supervised by one director, which areas should be responsibility centers and which areas should be support centers? Draw them in a visual and indicate the reporting requirements.

Assignment Exercise 6–2: Responsibility Centers

Choose between the physician’s practice in Mini-Case Study 2 or the clinic in Mini-Case Study 3. Designate the responsibility centers and the support centers for the organization selected. Prepare a rationale for the structure you have designed.

CHAPTER 7

Example 7A: Fixed, Variable, and Semivariable Distinction

Review the chapter text for the distinction between fixed, variable, and semivariable costs. Pay particular attention to the accompanying Figures 7-1, 7-2, 7-3, 7-4, and 7-5.

Practice Exercise 7–I: Analyzing Mixed Costs

The Metropolis Health System has a system-wide training course for nurse aides. The course requires a packet of materials that MHS calls the training pack. Due to turnover and because the course is system-wide, there is a monthly demand for new packs. In addition, the local community college also obtains the training packs used in their credit courses from MHS.

The education coordinator needs to know how much of the cost is fixed and how much of the cost is variable for these training packs. She decides to use the high-low method of computation.

Require d

Using the monthly utilization information presented below, find the fixed and variable por- tion of costs through the high-low method.

Number of Month Training Packs Cost

January 1,000 $6,200 February 200 1,820 March 250 2,350 April 400 3,440 May 700 4,900 June 300 2,730 July 150 1,470 August 100 1,010 September 1,100 7,150 October 300 2,850 November 250 2,300 December 100 1,010

Assignment Exercise 7–1: Analyzing Mixed Costs

The education coordinator decides that the community college packs may be unduly in- fluencing the high-low computation. She decides to re-run the results, omitting the com- munity college volume.

Require d

1. Using the monthly utilization information presented below, and omitting the com- munity college training packs, find the fixed and variable portion of costs through

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364 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

the high-low method. Note that the college only acquires packs in three months of the year: January, May, and September. These dates coincide with the start dates of their semesters and summer school.

2. The reason the education coordinator needs to know how much of the cost is fixed is because she is supposed to collect the appropriate variable cost from the commu- nity college for their packs. For her purposes, which computation do you believe is better? Why?

Total Number of Total Community College Community College Month Training Packs Cost Number Packs Cost

January 1,000 $6,200 200 $1,240 February 200 1,820 March 250 2,350 April 400 3,440 May 700 4,900 300 2,100 June 300 2,730 July 150 1,470 August 100 1,010 September 1,100 7,150 300 1,950 October 300 2,850 November 250 2,300 December 100 1,010

Example 7B: Contribution Margin

Computation of a contribution margin is simplified if the fixed and variable expense has al- ready been determined. Examine Table 7-1, which contains Operating Room fixed and variable costs. We can see that the total costs are $1,217,756. Of this amount, $600,822 is designated as variable cost and $616,934 is designated as fixed ($529,556 + $87,378 = $616,934). For purposes of our example, assume the Operating Room revenue amounts to $1,260,000. The contribution margin is computed as follows:

Amount

Revenue $1,260,000 Less Variable Cost (600,822) Contribution Margin $659,178

Thus, $659,178 is available to contribute to fixed costs and to profit. (In this example fixed costs amount to $616,934, so there is an amount left to contribute toward profit.)

Practice Exercise 7–II: Calculating the Contribution Margin

Greenside Clinic has revenue totaling $3,500,000. The clinic has costs totaling $3,450,000. Of this amount, 40% is variable cost and 60% is fixed cost.

Require d

Compute the contribution margin for Greenside Clinic.

Assignment Exercise 7–2: Calculating the Contribution Margin

The Mental Health program for the Community Center has just completed its fiscal year end. The program director determines that his program has revenue for the year of $1,210,000. He believes his variable expense amounts to $205,000 and he knows his fixed expense amounts to $1,100,000.

Require d

1. Compute the contribution margin for the Community Center Mental Health Pro- gram.

2. What does the result tell you about the program?

Example 7C: Cost-Volume-Profit (CVP) Ratio and Profit-Volume (PV) Ratio

Closely review the examples of ratio calculations in the chapter text. Also note that exam- ples are presented in visuals as well as text.

Practice Exercise 7–III: Calculating the PV Ratio

The profit-volume (PV) ratio is also known as the contribution margin (CM) ratio. Use the same assumptions for the Community Center Mental Health Program. In addition to the contribution margin figures already computed, now compute the PV ratio (also known as the CM ratio).

Assignment Exercise 7–3: Calculating the PV Ratio and the CVP Ratio

Use the same assumptions for the Greenside Clinic. One more assumption will be added: the clinic had 35,000 visits.

Required

1. In addition to the contribution margin figures already computed, now compute the PV ratio (also known as the CM ratio).

2. Add another column to your worksheet and compute the clinic’s per-visit revenue and costs.

3. Create a Cost-Volume-Profit chart. Refer to the chapter text along with Figure 7-6.

CHAPTER 8

Assignment Exercise 8–1: FIFO and LIFO Inventory

Study the FIFO and LIFO explanations in the chapter.

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366 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

Require d

a.1. Use the format in Exhibit 8-1 “FIFO Inventory Effect” to compute the ending FIFO inventory and the cost of goods sold, assuming $90,000 in sales.

a.2. Also compute the cost of goods sold percentage of sales. b.1. Use the format in Exhibit 8-2 “LIFO Inventory Effect” to compute the ending

LIFO inventory and the cost of goods sold, assuming $90,000 in sales. b.2. Also compute the cost of goods sold percentage of sales.

c. Comment on the difference in outcomes.

Assignment Exercise 8–2: Inventory Turnover

Study the “Calculating Inventory Turnover” portion of the chapter closely, whereby the cost of goods sold divided by the average inventory equals the inventory turnover.

Require d

Compute two inventory turnover calculations as follows: 1. Use the LIFO information in the previous assignment to first compute the average

inventory and then to compute the inventory turnover. 2. Use the FIFO information in the previous assignment to first compute the average

inventory and then to compute the inventory turnover.

Example 8A: Depreciation Concept

Assume that MHS purchased equipment for $200,000 cash on April 1st (the first day of its fiscal year). This equipment has an expected life of 10 years. The salvage value is 10 percent of cost. No equipment was traded in on this purchase.

Straight-line depreciation is a method that charges an equal amount of depreciation for each year the asset is in service. In the case of this purchase, straight-line depreciation would amount to $18,000 per year for 10 years. This amount is computed as follows:

Step 1. Compute the cost net of salvage or trade-in value: 200,000 less 10 percent salvage value or 20,000 equals 180,000.

Step 2. Divide the resulting figure by the expected life (also known as estimated useful life): 180,000 divided by 10 equals 18,000 depreciation per year for 10 years.

Accelerated depreciation represents methods that are speeded up, or accelerated. In other words a greater amount of depreciation is taken earlier in the life of the asset. One example of accelerated depreciation is the double declining balance method. Unlike straight-line depreciation, trade-in or salvage value is not taken into account until the end of the depreciation schedule. This method uses book value, which is the net amount re- maining when cumulative previous depreciation is deducted from the asset’s cost. The com- putation is as follows:

Step 1. Compute the straight-line rate: 1 divided by 10 equals 10 percent. Step 2. Now double the rate (as in double declining method): 10 percent times 2 equals

20 percent.

Step 3. Compute the first year’s depreciation expense: 200,000 times 20 percent equals 40,000.

Step 4. Compute the carry-forward book value at the beginning of the second year: 200,000 book value beginning Year 1 less Year 1 depreciation of 40,000 equals book value at the beginning of the second year of 160,000.

Step 5. Compute the second year’s depreciation expense: 160,000 times 20 percent equals 32,000.

Step 6. Compute the carry-forward book value at the beginning of the third year: 160,000 book value beginning Year 2 less Year 2 depreciation of 32,000 equals book value at the beginning of the third year of 128,000. —Continue until the asset’s salvage or trade-in value has been reached. —Do not depreciate beyond the salvage or trade-in value.

Practice Exercise 8–I: Depreciation Concept

Assume that MHS purchased equipment for $600,000 cash on April 1st (the first day of its fiscal year). This equipment has an expected life of 10 years. The salvage value is 10 percent of cost. No equipment was traded in on this purchase.

Required

1. Compute the straight-line depreciation for this purchase. 2. Compute the double declining balance depreciation for this purchase.

Assignment Exercise 8–3: Depreciation Concept

Assume that MHS purchased two additional pieces of equipment on April 1st (the first day of its fiscal year), as follows:

(1) The laboratory equipment cost $300,000 and has an expected life of 5 years. The sal- vage value is 5 percent of cost. No equipment was traded in on this purchase.

(2) The radiology equipment cost $800,000 and has an expected life of 7 years. The sal- vage value is 10 percent of cost. No equipment was traded in on this purchase.

Require d

For both pieces of equipment:

1. Compute the straight-line depreciation. 2. Compute the double declining balance depreciation.

Example 8B: Depreciation

This example shows straight-line depreciation computed at a five-year useful life with no sal- vage value. Straight-line depreciation is the method commonly used for financing projec- tions and funding proposals.

Depreciation Expense Computation: Straight Line

5-year useful life; no salvage value

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Annual Remaining Year # Depreciation Balance

Beginning Balance � 60,000 1 12,000 48,000 2 12,000 36,000 3 12,000 24,000 4 12,000 12,000 5 12,000 -0-

Example 8C: Depreciation

This example shows straight-line depreciation computed at a five-year useful life with a remain- ing salvage value of $10,000. Note the difference in annual depreciation between 16B and 16C.

Depreciation Expense Computation: Straight Line

5-year useful life; $10,000 salvage value

Annual Remaining Year # Depreciation Balance

Beginning Balance � 60,000 1 10,000 50,000 2 10,000 40,000 3 10,000 30,000 4 10,000 20,000 5 10,000 10,000

Example 8D: Depreciation

This example shows double declining depreciation computed at a five-year useful life with no salvage value. As is often the case with a five-year life, the double declining method is used for the first three years and the straight-line method is used for the remaining two years. The double declining method first computes what the straight-line percentage would be. In this case 100 percent divided by five years equals 20 percent. The 20 percent is then doubled. In this case 20 percent times 2 equals 40 percent. Then the 40 percent is multi- plied by the remaining balance to be depreciated. Thus 60,000 times 40 percent for year one equals 24,000 depreciation, with a remaining balance of 36,000. Then 36,000 times 40 percent for year two equals 14,400 depreciation, and 36,000 minus 14,400 equals 21,600 re- maining balance, and so on.

Now note the difference in annual depreciation between 8B, using straight-line for all five years, and 8D, using the combined double declining and straight-line methods.

Depreciation Expense Computation: Double Declining Balance

5-year useful life; $10,000 salvage value

Annual Remaining Year # Depreciation Balance

Beginning Balance � 60,000 1 24,000* 36,000 2 14,400* 21,600 3 8,640* 12,960 4 6,480** 6,480 5 6,480** 6,480

* � double declining balance depreciation ** � straight-line depreciation for remaining two years (12,960 divided by 2 � 6,480/yr)

Practice Exercise 8–II: Depreciation

Compute the straight-line depreciation for each year for equipment with a cost of $50,000, a 5-year useful life, and a $5,000 salvage value.

Assignment Exercise 8–4: Depreciation

Set up a purchase scenario of your own and compute the depreciation with and without sal- vage value.

Assignment Exercise 8–5: Depreciation Computation: Units-of-Service

Study the “Units-of-Service” portion of the chapter closely.

Require d

1. Using the format in Table 8-7, compute units-of-service depreciation using the fol- lowing assumptions:

Cost to be depreciated � $50,000 Salvage value � zero Total units of service � 10,000 Units of service per year: Year 1 � 2,200; Year 2 � 2,100;

Year 3 � 2,300; Year 4 � 2,200; Year 5 � 200

2. Using the same format, compute units-of-service depreciation using adjusted as- sumptions as follows:

Cost to be depreciated � $50,000 Salvage value � $5,000 Total units of service � 10,000 Units of service per year: Year 1 � 2,200; Year 2 � 2,100;

Year 3 � 2,300; Year 4 � 2,200; Year 5 � 200

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CHAPTER 9

Example 9A

Review the chapter text about annualizing positions. In particular review Exhibit 9-2, which contains the annualizing calculations.

Practice Exercise 9–I: FTEs to Annualize Staffing

The office manager for a physicians’ group affiliated with Metropolis Health System is working on her budget for next year. She wants to annualize her staffing plan. To do so she needs to convert her staff’s net paid days worked to a factor. Their office is open and staffed seven days a week, per their agreement with two managed care plans.

The office manager has the MHS worksheet, which shows 9 holidays, 7 sick days, 15 va- cation days, and 3 education days, equaling 34 paid days per year not worked. The physi- cians’ group allows 8 holidays, 5 sick days, and 1 education day. An employee must work one full year to earn 5 vacation days. An employee must have worked full time for three full years before earning 10 annual vacation days. Because the turnover is so high, nobody on staff has earned more than 5 vacation days.

Require d

1. Compute net paid days worked for a full-time employee in the physicians’ group. 2. Convert net paid days worked to a factor so the office manager can annualize her

staffing plan.

Assignment Exercise 9–1: FTEs to Annualize Staffing

The Metropolis Health System managers are also working on their budgets for next year. Each manager must annualize his or her staffing plan, and thus must convert staff net paid days worked to a factor. Each manager has the MHS worksheet, which shows 9 holidays, 7 sick days, 15 vacation days, and 3 education days, equaling 34 paid days per year not worked.

The Laboratory is fully staffed seven days per week and the 34 paid days per year not worked is applicable for the lab. The Medical Records department is also fully staffed seven days per week. However, Medical Records is an outsourced department so the employee benefits are somewhat different. The Medical Records employees receive 9 holidays plus 21 personal leave days which can be used for any purpose.

Required

1. Compute net paid days worked for a full-time employee in the Laboratory and in Medical Records.

2. Convert net paid days worked to a factor for the Laboratory and for Medical Records so these MHS managers can annualize their staffing plans.

Example 9B

Review the chapter text about staffing requirements to fill a position. In particular review Exhibit 9-4, which contains (at the bottom of the exhibit) the staffing calculations. Re- member this method uses a basic work week as the standard.

Practice Exercise 9–II: FTEs to Fill a Position

Metropolis Health System (MHS) uses a basic work week of 40 hours throughout the sys- tem. Thus, one full-time employee works 40 hours per week. MHS also uses a standard 24- hour scheduling system of three 8-hour shifts. The Admissions manager needs to compute the staffing requirements to fill his departmental positions. He has more than one Admis- sions office staffed within the system. The West Admissions office typically has two Admis- sions officers on duty during the day shift, one Admissions officer on duty during the evening shift, and one Admissions officer on duty during the night shift. The day shift also has one clerical person on duty. Staffing is identical for all seven days of the week.

Require d

1. Set up a staffing requirements worksheet, using the format in Exhibit 9-4. 2. Compute the number of FTEs required to fill the Admissions officer position and the

clerical position at the West Admissions office.

Assignment Exercise 9–2: FTEs to Fill a Position

Metropolis Health System (MHS) uses a basic work week of 40 hours throughout the sys- tem. Thus, one full-time employee works 40 hours per week. MHS also uses a standard 24- hour scheduling system of three 8-hour shifts. The Director of Nursing needs to compute the staffing requirements to fill the Operating Room positions. Since MHS is a trauma cen- ter the OR is staffed 24 hours a day, 7 days a week. At present, staffing is identical for all seven days of the week, although the Director of Nursing is questioning the efficiency of this method.

The Operating Room department is staffed with two nursing supervisors on the day shift and one nursing supervisor apiece on the evening and night shifts. There are two techni- cians on the day shift, two technicians on the evening shift, and one technician on the night shift. There are three RNs on the day shift, two RNs on the evening shift, and one RN plus one LPN on the night shift. In addition, there is one aide plus one clerical worker on the day shift only.

Require d

1. Set up a staffing requirements worksheet, using the format in Exhibit 9-4. 2. Compute the number of FTEs required to fill the Operating Room staffing positions.

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CHAPTER 10

Practice Exercise 10–I: Components of Balance Sheet and Statement of Net Income

Financial statements for Doctors Smith and Brown are provided below. Use the doctors’ bal- ance sheet, statement of revenue and expenses, and statement of capital for this assignment.

Require d

Identify the following doctors’ balance sheet and statement of net income components. List the name of each component and its amount(s) from the appropriate financial statement.

Current Liabilities Total Assets Income from Operations Accumulated Depreciation Total Operating Revenue Current Portion of Long-Term Debt Interest Income Inventories

Assignment Exercise 10–1: Components of Balance Sheet and Statement of Net Income

Refer to the Metropolis Health System (MHS) supplemental information at the back of the Examples and Exercises section. Use the MHS comparative balance sheet, statement of rev- enue and expenses, and statement of fund balance for this assignment.

Require d

Identify the following MHS balance sheet components. List the name of each component and its amount(s) from the appropriate MHS financial statement.

Current Liabilities Total Assets Income from Operations Accumulated Depreciation Total Operating Revenue Current Portion of Long-Term Debt Interest Income Inventories

Doctors Smith and Brown: Statement of Net Income

for the Three Months Ended March 31, 2___

Revenue Net patient service revenue 180,000 Other revenue -0-

Total Operating Revenue 180,000

Expenses Nursing/PA salaries 16,650 Clerical salaries 10,150 Payroll taxes/employee benefits 4,800 Medical supplies and drugs 15,000 Professional fees 3,000 Dues and publications 2,400 Janitorial service 1,200 Office supplies 1,500 Repairs and maintenance 1,200 Utilities and telephone 6,000 Depreciation 30,000 Interest 3,100 Other 5,000

Total Expenses 100,000

Income from Operations 80,000

Nonoperating Gains (Losses) Interest Income -0-

Nonoperating Gains, Net -0-

Net Income 80,000

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374 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

Doctors Smith and Brown Balance Sheet March 31, 2___

Assets Current Assets

Cash and cash equivalents 25,000 Patient accounts receivable 40,000 Inventories—supplies and drugs 5,000

Total Current Assets 70,000

Property, Plant, and Equipment Buildings and Improvements 500,000 Equipment 800,000

Total 1,300,000 Less Accumulated Depreciation (480,000) Net Depreciable Assets 820,000 Land 100,000

Property, Plant, and Equipment, Net 920,000

Other Assets 10,000

Total Assets 1,000,000

Liabilities and Capital

Current Liabilities Current maturities of long-term 10,000

debt Accounts payable and accrued 20,000

expenses

Total Current Liabilities 30,000

Long-Term Debt 180,000 Less Current Portion of Long-Term Debt (10,000) Net Long-Term Debt 170,000

Total Liabilities 200,000

Capital 800,000

Total Liabilities and Capital 1,000,000

Doctors Smith and Brown Statement of Changes in Capital

for the Three Months Ended March 31, 2___

Beginning Balance $720,000 Net Income 80,000 Ending Balance $800,000

The MHS Balance Sheet

Example 10A: Components of Balance Sheet and Income Statement

The “Accounts Receivable (net)” in Exhibit 10-1 means the accounts receivable figure of $250,000 on the balance sheet is net of the allowance for bad debts. If the allowance for bad debts is raised on the balance sheet, then bad debt expense (a.k.a provision for doubtful ac- counts) on the income statement (a.k.a. statement of revenue and expense) also rises. Think of these two accounts as a pair.

Practice Exercise 10–II: Components of Balance Sheet and Income Statement

Refer to Doctors Smith and Brown’s balance sheet, where patient accounts receivable is stated at $40,000. Do you think this figure is net of an allowance for bad debts?

Assignment Exercise 10–2: Components of Balance Sheet and Income Statement

Refer to the Metropolis Health System (MHS) balance sheet and statement of revenue and expense in Chapter 25’s MHS Case Study. Patient accounts receivable of $7,400,000 is shown as net of $1,300,000 allowance for bad debts (8,700,000 � 1,300,000 � 7,400,000). (a) What percent of gross accounts receivable is the allowance for bad debts? (b) If the al- lowance for bad debts is raised to $1,500,000, where does the extra $200,000 go?

Example 10B: Components of Balance Sheet and Income Statement

Refer to Exhibit 10-1 and Exhibit 10-2’s Westside Clinic statements. The “Property, Plant, and Equipment (net)” total in Exhibit 10-1 means the property, plant, and equipment fig- ure of $360,000 on the balance sheet is net of the reserve for depreciation. If the reserve for depreciation is raised on the balance sheet, then the depreciation expense on the in- come statement (a.k.a. statement of revenue and expense) also rises. Think of these two ac- counts as another pair.

Practice Exercise 10–III: Components of Balance Sheet and Income Statement

Refer to Doctors Smith and Brown’s balance sheet, where buildings and equipment are both stated as net (the $820,000 figure), but land is not. Do you recall why this is so?

Assignment Exercise 10–3: Components of Balance Sheet and Income Statement

Refer to the Metropolis Health System (MHS) balance sheet and statement of revenue and expense in Chapter 25’s MHS Case Study. Property, plant, and equipment of $19,300,000 is shown as “net,” meaning net of the reserve for depreciation. If the $19,300,000 is reduced by $200,000 (meaning the reserve for depreciation has risen), what happens on the income statement?

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CHAPTER 11

Example 11A

To better understand how the information for the numerator and the denominator of each calculation is obtained, Figure 11-1, “Examples of Liquidity Ratio Calculations,” il- lustrates the process. This figure takes the balance sheet and the statement of revenue and expense that were discussed in the preceding chapter and illustrates the source of each fig- ure in the four liquidity ratios. The multiple computations in days cash on hand and in days receivables are further broken out into a three-step process to better illustrate sources of information.

Practice Exercise 11–I: Liquidity Ratios

Two of the liquidity ratios are illustrated in this practice exercise. Refer to Doctors Smith and Brown’s financial statements presented in the preceding Chapter 10.

Required

1. Set up a worksheet for the current ratio and the quick ratio. 2. Compute the ratios for Doctors Smith and Brown.

Assignment Exercise 11–1: Liquidity Ratios

Refer to the Metropolis Health System (MHS) case study in Chapter 25.

Require d

1. Set up a worksheet for the liquidity ratios. 2. Compute the four liquidity ratios using the Chapter 25 MHS financial statements.

Example 11B

To better understand how the information for the numerator and the denominator of each calculation is obtained, Figure 11-2, “Examples of Solvency and Profitability Ratio Calcula- tions,” illustrates the process. This figure takes the balance sheet and the statement of rev- enue and expense that were discussed in the preceding chapter and illustrates the source of each figure in the two solvency ratios. Any multiple computations are further broken out to better explain sources of information.

Practice Exercise 11–II: Solvency Ratios

Refer to Doctors Smith and Brown’s financial statements presented in the preceding Chap- ter 10.

Require d

1. Set up a worksheet for the solvency ratios.

2. Compute these ratios for Doctors Smith and Brown. To do so, you will need one ad- ditional piece of information that is not present on the doctors’ statements: their maximum annual debt service is $22,200.

Assignment Exercise 11–2: Solvency Ratios

Refer to the Metropolis Health System (MHS) case study in Chapter 25.

Require d

1. Set up a worksheet for the liquidity ratios. 2. Compute the solvency ratios using the Chapter 25 MHS financial statements.

Example 11C

To better understand how the information for the numerator and the denominator of each calculation is obtained, study Figure 11-2, “Examples of Solvency and Profitability Ratio Cal- culations.” This figure takes the balance sheet and the statement of revenue and expense that were discussed in the preceding chapter and illustrates the source of each figure in the two profitability ratios. Any multiple computations are further broken out to better explain sources of information.

Practice Exercise 11–III: Profitability Ratios

Refer to Doctors Smith and Brown’s financial statements presented in the preceding Chap- ter 10.

Require d

1. Set up a worksheet for the profitability ratios. 2. Compute these ratios for Doctors Smith and Brown. All the necessary information is

present on the doctors’ statements. [Hint: “Operating Income (Loss)” is also known as “Income from Operations.”]

Assignment Exercise 11–3: Profitability Ratios

Refer to the Metropolis Health System (MHS) case study in Chapter 25.

Require d

1. Set up a worksheet for the liquidity ratios. 2. Compute the profitability ratios using the Chapter 25 MHS financial statements.

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378 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

CHAPTER 12

Example 12A: Unadjusted Rate of Return

A s s u m p t i o n s :

• Average annual net income � $100,000 • Original investment amount � $1,000,000 • Unrecovered asset cost at the end of useful life (salvage value) � $100,000

Calculation using original investment amount:

$100,000

$1,000,000 � 10% Unadjusted Rate of Return

Calculation using average investment amount:

First Step: Compute average investment amount for total unrecovered asset cost.

At beginning of estimated useful life � $1,000,000 At end of estimated useful life � $ 100,000

Sum $1,100,000

Divided by 2 � $550,000 average investment amount

Second Step: Calculate unadjusted rate of return.

$100,000

$550,000 � 18.2% Unadjusted Rate of Return

Practice Exercise 12–I: Unadjusted Rate of Return

A s s u m p t i o n s :

• Average annual net income � $100,000 • Original investment amount � $500,000 • Unrecovered asset cost at the end of useful life (salvage value) � $50,000

Required

1. Compute the unadjusted rate of return using the original investment amount. 2. Compute the unadjusted rate of return using the average investment method.

Assignment Exercise 12–1: Unadjusted Rate of Return

Metropolis Health Systems’ Laboratory Director expects to purchase a new piece of equip- ment. The assumptions for the transaction are as follows:

• Average annual net income � $70,000 • Original investment amount � $410,000 • Unrecovered asset cost at the end of useful life (salvage value) � $41,000

Require d

1. Compute the unadjusted rate of return using the original investment amount. 2. Compute the unadjusted rate of return using the average investment method.

Example 12B: Finding the Future Value (with a Compound Interest Table)

Betty Dylan is Director of Nurses at Metropolis Health System. Her oldest son will be enter- ing college in five years. Today Betty is trying to figure what his college fund will amount to in five more years. (Hint: Compound interest means interest is not only earned on the prin- cipal, but also is earned on the previous interest earnings that have been left in the account. Interest is thus compounded.)

The college fund savings account presently has a balance of $9,000 and any interest earned over the next five years will be left in the account. Betty assumes the annual interest rate will be 6 percent. How much money will be in the account at the end of five more years?

Solution to Example

Step 1. Refer to the Compound Interest Table found in Appendix 12-B at the back of this chapter. Reading across, or horizontally, find the 6% column. Reading down, or vertically, find Year 5. Trace across the Year 5 line item to the 6% col- umn. The factor is 1.338.

Step 2. Multiply the current savings account balance of $9,000 times the factor of 1.338 to find the future value of $12,042. In five years at compound interest of 6% the college fund will have a balance of $12,042.

Practice Exercise 12–II: Finding the Future Value (with a Compound Interest Table)

Assume the college savings fund in the preceding example presently has a balance of $11,000 and any interest earned will be left in the account. Assume the annual interest rate will be 7%.

Required

Compute how much money will be in the account at the end of six more years. (Use the Fu- ture Value or Compound Interest Table found at the back of this chapter.)

Assignment Exercise 12–2: Finding the Future Value (with a Compound Interest Table)

John Whitten is one of the physicians on staff at Metropolis Health System. His practice is six years old. He has set up an office savings account to accumulate the funds to replace equipment in his practice. Today John is trying to figure what his equipment fund will amount to in four more years.

The equipment fund savings account presently has a balance of $63,500 and any interest earned over the next four years will be left in the account. John assumes the annual interest rate will be 5 percent. How much money will be in the account at the end of four more years?

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Required

Compute how much money will be in the account at the end of four more years. (Use the Future Value or Compound Interest Table found at the back of this chapter.)

Example 12C: Finding the Present Value (with a Present Value Table)

Betty Dylan is taking an adult education night course in personal finance at the community college. The class is presently studying retirement planning. Each student is to estimate the amount of funds (in addition to pension plans and social security) they believe will be needed at retirement. Then they are to make a retirement plan.

Betty has estimated she would need $100,000 fifteen years from now. In order to com- plete her assignment she needs to know the present value of the $100,000. Betty further as- sumes an interest rate of 6 percent.

Solution to Example

Step 1. Refer to the Present Value Table found in Appendix 12-A at the back of this chapter. Reading across, or horizontally, find the 6% column. Reading down, or vertically, find Year 15. Trace across the Year 15 line item to the 6% column. The factor is 0.4173.

Step 2. Multiply $100,000 times the factor of 0.4173 to find the present value of $41,730.

Practice Exercise 12–III: Finding the Present Value (with a Present Value Table)

Betty isn’t finished with her assignment. Now she wants to find the present value of $150,000 accumulated fifteen years from now. She further assumes a better interest rate of 7 percent.

Require d

Compute the present value of $150,000 accumulated fifteen years from now. Assume an in- terest rate of 7 percent. (Use the Present Value Table found at the back of this chapter.)

Assignment Exercise 12–3: Finding the Present Value (with a Present Value Table)

Part 1—Dr. John Whitten is still figuring out his equipment fund. According to his calcula- tions he needs $250,000 to be accumulated six years from now. John is now trying to find the present value of the $250,000. He continues to assume an interest rate of 5 percent.

Require d

Compute the present value of $250,000 accumulated fifteen years from now. Assume an in- terest rate of 5 percent. (Use the Present Value Table found at the back of this chapter.)

Part 2—John doesn’t like the answer he gets. What if he can raise the interest rate to 7 percent? How much difference would that make?

Require d

Compute the present value of $250,000 accumulated fifteen years from now assuming an interest rate of 7 percent. Compare the difference between this amount and the present value at 5 percent.

Example 12D: Internal Rate of Return

Review the chapter text to follow the steps set out to compute the internal rate of return.

Practice Exercise 12–IV: Internal Rate of Return

Metropolis Health System (MHS) is considering purchasing a tractor to mow the grounds. It would cost $16,950 and have a 10-year useful life. It will have zero salvage value at the end of 10 years. The head of the MHS grounds crew estimates it would save $3,000 per year. He figures this savings because just one of the present maintenance crew would be driving the tractor, replacing the labor of several men now using small household-type lawn mowers. Compute the internal rate of return for this proposed acquisition.

Assignment Exercise 12–4: Computing an Internal Rate of Return

Dr. Whitten has decided to purchase equipment that has a cost of $60,000 and will produce a pretax net cash inflow of $30,000 per year over its estimated useful life of six years. The equipment will have no salvage value and will be depreciated by the straight-line method. The tax rate is 50%. Determine Dr. Whitten’s approximate after-tax internal rate of return.

Example 12E: Payback Period

Review the chapter text and follow the Doctor Green detailed example of payback period computation.

Practice Exercise 12–V: Payback

The MHS Chief Financial Officer is considering a request by the Emergency Room depart- ment for purchase of new equipment. It will cost $500,000. There is no trade-in. Its useful life would be 10 years. This type of machine is new to the department but it is estimated that it will result in $84,000 annual revenue and operating costs would be one quarter of that amount. The CFO wants to find the payback period for this piece of equipment.

Assignment Exercise 12–5: Payback Period

The MHS Chief Financial Officer is considering alternate proposals for the hospital Radi- ology department. The Director of Radiology has suggested purchasing one of two pieces of equipment. Machine A costs $15,000 and Machine B costs $12,000. Both machines are estimated to reduce radiology operating costs by $5,000 per year.

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382 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

Require d

Which machine should be purchased? Make your payback calculations to provide the answer.

CHAPTER 13

Example 13A: Common Sizing

Common sizing converts numbers to percentages so that comparative analysis can be per- formed. Reread the chapter text about common sizing and examine the percentages shown in Table 13-1.

Practice Exercise 13–I: Common Sizing

The worksheet below shows the assets of two hospitals.

Require d

Perform common sizing for the assets of the two hospitals.

Same Year for Both Hospitals Hospital A Hospital B

Current Assets $ 2,000,000 $ 8,000,000 Property, Plant, & Equipment 7,500,000 30,000,000 Other Assets 500,000 2,000,000 Total Assets $10,000,000 $40,000,000

Assignment Exercise 13–1: Common Sizing

Refer to the Metropolis Health System (MHS) comparative financial statements at the back of the Examples and Exercises section.

Require d

Common size the MHS statement of revenue and expenses.

Example 13B: Trend Analysis

Trend analysis allows comparison of figures over time. Reread the chapter text about trend analysis and examine the difference columns shown in Table 13-3.

Practice Exercise 13–II: Trend Analysis

The worksheet below shows the assets of Hospital A over two years.

Require d

Perform trend analysis for the assets of Hospital A.

Hospital A Year 1 Year 2

Current Assets $1,600,000 $ 2,000,000 Property, Plant, & Equipment 6,000,000 7,500,000 Other Assets 400,000 500,000 Total Assets $8,000,000 $10,000,000

Assignment Exercise 13–2: Trend Analysis

Refer to the Metropolis Health System (MHS) comparative financial statements at the back of the Examples and Exercises section.

Require d

Perform trend analysis on the MHS statement of revenue and expenses.

Practice Exercise 13–III: Contractual Allowance

A s s u m p t i o n s :

1. Your unit’s gross charges for the period to date amount to $200,000. 2. The uniform gross charge for each procedure in your unit is $100.00. 3. The unit receives revenue from four major payers. For purposes of this exercise, as-

sume the revenue volume from each represents 25% of the total. (The equal propor- tion is unrealistic, but serves the purpose for this exercise.)

4. The following contractual payment arrangements are in effect for the current period. The percentage of the gross charge that is currently paid by each payer is as follows: Payer 1 � 90% Payer 2 � 80% Payer 3 � 70% Payer 4 � 50%

Q: How many procedures has your unit recorded for the period to date?

Q: Of these, how many procedures are attributed to each payer?

Q: How much is the net revenue per procedure for each payer, and how much is the contractual allowance per procedure for each payer?

Assignment Exercise 13–3

As a follow-up to the Practice Exercise above, new assumptions are as follows: 1. Your unit’s gross charges for the period to date amount to $200,000. 2. The uniform gross charge for each procedure in your unit is $100.00.

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384 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

3. The unit receives revenue from four major payers. The number of procedures per- formed for the period totals 2,000. Of that total, the number of procedures per payer (stated as a percentage) is as follows: Payer 1 � 30% Payer 2 � 40% Payer 3 � 20% Payer 4 � 10%

4. The following contractual payment arrangements are in effect for the current period. The percentage of the gross charge that is currently paid by each payer is as follows: Payer 1 � 80% [Medicare] Payer 2 � 70% [Commercial managed care plans] Payer 3 � 50% [Medicaid] Payer 4 � 90% [Self-pay]

Q: How many procedures are attributed to each payer?

Q: How much is the net revenue per procedure for each payer, and how much is the con- tractual allowance per procedure for each payer?

Q: How much is the total net revenue for each payer, and how much is the total con- tractual allowance for each payer?

Assignment Exercise 13–4.1: Forecast Capacity Levels

Review the information in Exhibit 13-1 “Capacity Level Checkpoints for an Outpatient In- fusion Center.” The exhibit assumes three chairs and one 40-hour RN, for a realistic capac- ity level of seven patients infused per day.

Require d

Prepare another Infusion Center Capacity Level Forecast as follows: Assume the same three infusion chairs, but add another nurse for either four or six

hours per day. How would this change the daily capacity level for number of patients in- fused per day?

Assignment Exercise 13–4.2

Require d

Prepare another Infusion Center Capacity Level Forecast as follows: Increase the number of infusion chairs to four, and add another nurse for either four or

six hours per day. How would this change the daily capacity level for number of patients in- fused per day?

CHAPTER 14

Assignment Exercise 14–1: Comparable Data in a Graph

Review Figures 14-1 through 14-5. Each of the five figures presents a graph depicting some type of comparative data.

Required

Locate healthcare information that can reasonably be compared. (1) Prepare your com- parative data. (2) Using your data, create one or more graphs similar to those found in Fig- ures 14-1 through 14-5.

Assignment Exercise 14–2: Cumulative Inflation Factor for Comparable Data

Review Table 14-3 and the accompanying text.

A s s u m p t i o n s :

Two hospitals report their annual projected revenue for five years to the local newspaper for a story on the area’s future economic outlook. However, Hospital 1 has applied a cu- mulative inflation factor of five percent per year while Hospital 2 has not applied any infla- tion factor. Thus the information is not properly comparable.

Projected Revenue

Year 1 Year 2 Year 3 Year 4 Year 5

Hospital 1 $20,000,000 $22,500,000 $27,500,000 $27,500,000 $30,000,000

Hospital 2 $20,000,000 $21,000,000 $25,000,000 $24,000,000 $26,000,000

Require d

Revise Hospital 2’s projections by applying a cumulative inflation factor of five percent per year.

Assignment Exercise 14–3

The head of your department is a prominent researcher. A health research foundation has asked him travel to London to give an important speech at a conference. He will then travel to Paris to tour a research facility before returning home. Although his travel expenses are being funded by the foundation, he will still need to take along some personal money. Con- sequently, he asks you to figure the exchange rates for $500 and for $1,000 in both pounds and euros. He explains that he is trying to judge the spending power of U.S. dollars when converted to the other currencies so he can decide how much personal money to take on the trip.

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386 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

Require d

Locate the current exchange rates for pounds and euros and compute the currency con- version for $500 and for $1,000.

Assignment Exercise 14–4: The Discovery

The Chief Financial Officer at Sample Hospital has just discovered that the hospital’s Chief of the Medical Staff’s son Jason, a student at the local community college, is paid $100 per week year round for grounds maintenance at the hospital’s Outpatient Center.

The CFO, no fan of the Chief of Medical Staff, now wants you to prepare a report that compares the relative costs of lawn care at each of three locations; the hospital itself, the outpatient center, and the hospital-affiliated nursing home down the block.

Require d

Review the available information for grounds maintenance at the three facilities. Decide how to convert this information into comparable data. Then prepare a report, based on your assumptions, that presents comparable costs of grounds care. Also provide your as- sessment of what the best future course of action should be.

Relevant Information

So far you have assembled the following information. Now you need to decide how it can be converted into comparable data.

Introduction to the Three Facilities

Sample Hospital is an older 100-bed hospital. The new Outpatient Center, built last year, is across the street and the Golden Age Nursing Facility is down one block, on the corner. All three facilities are part of the Metropolis Health System. (Appendix 25-A contains some fi- nancial details about Sample Hospital.) The hospital is located in the midwestern sunbelt; there is occasional frost in the winter but no snow.

Grounds Maintenance Tasks That Should Be Performed at All Three Sites

• Mowing and edging • Walk sweeping • Raking leaves • Blowing off parking lot • Flower bed maintenance (where necessary) • Hedge trimming and minor tree pruning (major tree trimming is performed by a

contractor on an as-needed basis and thus should be disregarded)

Figure Ex-1 provides a map that illustrates the layout of the grounds for each facility and their proximity to each other.

Grounds Maintenance Arrangements for the Three Facilities

The current grounds maintenance arrangements vary among the three facilities as follows: 1. Sample Hospital uses its Maintenance department employees for grounds care. The

hospital pays these employees $15 per hour plus 15% employee benefits; it is estimated they spend 1,000 hours per year on grounds maintenance work. Another estimated 120 hours per year are spent on maintaining the lawn care equipment. The employees use a riding lawn mower, edger, and blower, all owned by the hospital. The hospital just bought a new mower for $2,995 less a ten percent discount. It is expected that the mower should last for five years.

2. The hospital’s Chief of the Medical Staff’s son Jason, a student at the local community college, is paid $100 per week year round for grounds maintenance at the hospital’s Out- patient Center. A friend sometimes helps, but when that happens Jason pays him out of his weekly $100. It takes about one-and-a-half hours to mow, edge, and blow. Jason uses his dad’s riding mower and blower, but Jason recently bought his own edger. Jason also buys fertilizer for the grass twice a year.

3. The Nursing Facility contracts with a landscape service on a seasonally adjusted sliding scale. The landscape service is paid $600 per month from April to October (mowing season); $400 per month for February, March, and November; and $200 per month for November,

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Parking

E n tra

n ce

Golden Age Center

Service

Jefferson Street

Washington Street

Jackson Street

7 th

A ve

n u e

8 th

A ve

n u e

9 th

A ve

n u e

Outpatient

Parking

Sample Hospital

Parking

T re

e s

Figure Ex–1 Sample Hospital Map

388 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

December, and January. The landscape service provides all their own equipment. They also provide fertilizer and provide annuals to plant in the flower beds every quarter.

Sample Hospital Property Description

The grounds to be maintained are as follows:

• The front lawn is grass in two sections on either side of the front entrance. Each sec- tion is about 50� by 60�.

• There is a hedge along the front of the building that is about 50� on either side of the front entrance.

• There are two small matching flower beds on either side of the front entrance. • Another strip of grass alongside of the building is 30� by 100�. • A third small strip of grass about 5� by 25� is by the Emergency entrance. • The walkway dimensions are as follows: about 50� of front walk; about 30� of staff en-

trance walk, both of which are 5� wide. • The Emergency Department’s paved patient drop-off area is about 25� by 30�. • The parking lot surface is about 200� by 250�. Along one side are overhanging trees

that drop leaves and debris and are a constant sweeping problem. These are the only trees on the hospital site.

Outpatient Center Property Description

The grounds to be maintained are as follows:

• There is a strip of grass at the front of the building that is 12� wide and 65� long, split in the middle by a walkway 5� wide.

• There is a strip of grass at the back of the building between the building and the park- ing lot that is 5� wide and 50� long

• All the rest of the property is paved.

Nursing Center Property Description

Golden Age Nursing Center occupies one whole block. The grounds have many large trees. Flowerbeds have been planted around the trees as well as along the front walk and en- trance. There are also two secured patio areas at the side of the building, screened by hedges, and each has a small bed of annuals. Because of the unique design of the building, grounds maintenance requires considerable handwork such as edging with a weed eater.

CHAPTER 15

Example 15A: Budgeting

A static budget is based on a single level of operations, which is never adjusted. Therefore, the static budgeted expense amounts will not change, even though actual volume does change during the year.

The computation of a static budget variance only requires one calculation, as follows:

Actual Static Budget Static Budget Results

minus Amount

equals Variance

We can set up the example in the chapter text in this format as follows: Use patient days as an example of level of volume, or output. Assume that the budget an-

ticipated 40,000 patient days this year at an average of $600 revenue per day, or $2,400,000. Further assume that expenses were budgeted at $560 per patient day, or $22,400,000. The budget would look like this:

As Budgeted

Revenue $24,000,000 Expenses 22,400,000 Excess of Revenue over Expenses $1,600,000

Now assume that only 36,000, or 90%, of the patient days are going to actually be achieved for the year. The average revenue of $600 per day will be achieved for these 36,000 days (thus 36,000 times 600 equals 21,600,000). Further assume that, despite the best ef- forts of the Chief Financial Officer, the expenses will amount to $22,000,000. The actual re- sults would look like this:

Actual

Revenue $21,600,000 Expenses 22,000,000 Excess of Expenses over Revenue $ (400,000)

The budgeted revenue and expenses still reflect the original expectation of 40,000 pa- tient days; the budget report would look like this:

Static Budget Actual Budget Variance

Revenue $21,600,000 $24,000,000 $(2,400,000) Expenses 22,000,000 22,400,000 (400,000) Excess of Expenses over Revenue $ (400,000) $ 1,600,000 $(2,000,000)

Note: The negative actual result of (400,000) combined with the positive budget expecta- tion of 1,600,000 amounts to the negative net variance of (2,000,000).

This example has shown a static budget, geared toward only one level of activity and re- maining constant or static.

Practice Exercise 15–I: Budgeting

Budget assumptions for this exercise include both inpatient and outpatient revenue and ex- pense. Assumptions are as follows:

As to the initial budget:

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390 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

• The budget anticipated 30,000 inpatient days this year at an average of $650 revenue per day.

• Inpatient expenses were budgeted at $600 per patient day. • The budget anticipated 10,000 outpatient visits this year at an average of $400 revenue

per visit. • Outpatient expenses were budgeted at $380 per visit.

As to the actual results:

• Assume that only 27,000, or 90%, of the inpatient days are going to actually be achieved for the year.

• The average revenue of $650 per day will be achieved for these 270,000 inpatient days. • The outpatient visits will actually amount to 110%, or 11,000 for the year. • The average revenue of $400 per visit will be achieved for these 11,000 visits. • Further assume that, due to the heroic efforts of the Chief Financial Officer, the ac-

tual inpatient expenses will amount to $11,600,000 and the actual outpatient expenses will amount to $4,000,000.

Required

1. Set up three worksheets that follow the format of those in Example 15A. However, in each of your worksheets make two lines for revenue; label one as Revenue-Inpatient and the other Revenue-Outpatient. Add a Revenue Subtotal line. Likewise, make two lines for expense; label one as Expense-Inpatient and the other Expense-Outpatient. Add an Expense Subtotal line.

2. Using the new assumptions, complete the first worksheet for “As Budgeted.” 3. Using the new assumptions, complete the second worksheet for “Actual.” 4. Using the new assumptions, complete the third worksheet for “Static Budget Vari-

ance.”

Assignment Exercise 15–1: Budgeting

Select an organization; either Metropolis Health System from the Chapter 25 Case Study or one of the organizations presented in the Mini-Case Studies in Chapters 26–28.

Require d

1. Using the organization selected, create a budget for the next fiscal year. Set out the details of all assumptions you needed in order to build this budget.

2. Use the “Checklist for Building a Budget” (Exhibit 15-2) and critique your own budget.

Assignment Exercise 15–2: Budgeting

Find an existing budget from a published source. Detail should be extensive enough to present a challenge.

Require d

1. Using the existing budget, create a new budget for the next fiscal year. Set out the de- tails of all the assumptions you needed in order to build this budget.

2. Use the “Checklist for Building a Budget” (Exhibit 15-2) and critique your own effort. 3. Use the “Checklist for Reviewing a Budget” (Exhibit 15-1) and critique the existing

budget.

Assignment Exercise 15–3: Transactions outside the Operating Budget

Review Figure 15-2 and the accompanying text. Metropolis Health System has received a wellness grant from the charitable arm of an

area electronics company. The grant will run for twenty-four months, beginning at the first of the next fiscal year. Two therapists and two registered nurses will each be spending half of their time working on the wellness grant. All four individuals are full-time employees of MHS. The electronics company has only recently begun to operate the charitable organi- zation that awarded the grant. While they have gained all the legal approvals necessary, they have not yet provided the manuals and instructions for grant transactions that MHS usually receives when grants are awarded. Consequently, guidance about separate accounting is not yet forthcoming from the grantor.

Require d

How would you handle this issue on the MHS operating budget for next year?

Assignment Exercise 15–4: Identified versus Allocated Costs in Budgeting

Review Figure 15-3 and the accompanying text. Metropolis Health System is preparing for a significant upgrade in both hardware and

software for its information systems. As part of the project, the Chief of Information Oper- ations (CIO) has indicated that the Information Systems (IS) department can change the format of the MHS operating budgets and related reports before the operating budget is constructed for the coming fiscal year. The Chief Financial Officer (CFO) has long wanted to modify what costs are identified and what costs are allocated (along with the method of allocation). This is a golden opportunity to do so. To gain ammunition for the change, the CFO is preparing to conduct a survey. The survey will obtain a variety of suggestions for po- tential changes in allocation methods for the new operating budget report formats. You have been selected as one of the employees who will be surveyed.

Require d

You may choose your role for this assignment, as follows: Refer to the “MHS Executive-Level Organization Chart” (Figure 25-2 in the MHS Case

Study). (1) Either (a) choose any type of patient service that would be under the direction

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392 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

of the Senior Vice President of Service Delivery Operations or (b) choose any other func- tion shown on the organization chart. (Your function could be a whole department or a di- vision or unit of that department. For example, you might choose Community Outreach or Human Resources Operations or the Emergency department, etc.) (2) Make up your own organization chart for other employee levels within the function you have chosen. (3) Now make up another chart that indicates the operating budget costs you think would be mostly identifiable for the department or unit or division you have chosen and what other operat- ing budget costs you think would be mostly allocated to it. (You may use Figure 15-3 as a rough guide, but do not let it limit your imagination. Model the detail on your “identifiable versus allocated costs” chart after a real department if you so choose.) Use MHS hospital statistics shown in Exhibit 25-8 of the MHS Case Study as a basis for allocation if these sta- tistics are helpful. If they are not, make a note of what other statistics you would like to have.

Note: As an alternative approach, you may choose a function from the “Nursing Practice and Administration Organization Chart” as shown in Figure 25-1 of the MHS Case Study in- stead of choosing from the Executive-Level Organization Chart.

CHAPTER 16

Example 16A: Description of Capital Expenditure Proposals Scoring System

Worthwhile Hospital has a total capital expenditure budget for next year of five million dol- lars. Of this amount, three million is already committed as spending for capital assets that have already been acquired and are in place. The remaining two million dollars is available for new assets and for new projects or programs.

Worthwhile Hospital typically divides the available capital expenditure funds into monies available for inpatient purposes and monies available for outpatient purposes. This year the split is proposed to be 50-50.

The hospital’s CFO is also proposing that a scoring system be used to evaluate this year’s proposals. She has set up a scoring system that allows a maximum of five points. Thus the low is a score of one point and the high is a score of five points.

In addition to the points earned by a funding proposal, the CFO will allow one “bonus point” for upgrading existing equipment and one “bonus point” for funding expansion of existing programs.

Practice Exercise 16–I: Capital Expenditure Proposals

Jody Smith, the director who supervises the Intensive Care Units, wants to secure as much of the one million dollars available for inpatient purposes as is possible for the ICU. At the same time Ted Jones, the director who supervises the Surgery Unit, also wants to secure as much of the one million dollars available for inpatient purposes as is possible for his Surgery Unit.

Given the CFO’s new scoring system, how should Jody go about choosing exactly what to request?

Assignment Exercise 16–1: Capital Expenditure Proposals

Ted Jones, the Surgery Unit Director, is about to choose his strategy for creating a capital expenditure funding proposal for the coming year. Ted’s unit needs more room. The Surgery Unit is running at over 90% capacity. In addition, a prominent cardiology surgeon on staff at the hospital wants to create a new cardiac surgery program that would require extensive funding for more space and for new state-of-the-art equipment. The surgeon has been campaigning with the hospital board members.

Require d

What should Ted decide to ask for? How should he go about crafting a strategy to justify his request, given the hospital’s new scoring system?

CHAPTER 17

Example 17A: Variance Analysis

Our variance analysis example and practice exercise use the flexible budget approach. A flexible budget is one that is created using budgeted revenue and/or budgeted cost amounts. A flexible budget is adjusted, or flexed, to the actual level of output achieved (or perhaps expected to be achieved) during the budget period. A flexible budget thus looks toward a range of activity or volume (versus only one level in the static budget).

Examples of how the variance analysis works are contained in Figure 17-1 (the ele- ments), in Figure 17-2 (the composition), and in Figures 17-3 and 17-4 (the calculation). Study these examples before undertaking the Practice Exercise.

Practice Exercise 17–I: Variance Analysis

Exhibit 17-4 presents a summary variance report for the nursing activity center of St. Joseph Hospital for the month of September. For our practice exercise we will duplicate this report for the month of March.

Assumptions are as follows:

• Actual Activity Level is 687,000. • Budgeted Activity Level is 650,000. • Actual Cost per RVU is $4.70. • Budgeted Cost per RVU is $5.00. • Actual Overhead Costs are $3,228,900. • Budgeted Overhead Costs are $3,250,000.

Require d

1. Set up a worksheet for the month of March like that shown in Exhibit 17-4 for the month of September.

2. Insert the March input data (per assumptions given above) on the worksheet. 3. Complete the “Actual Costs,” “Flexible Budget,” and “Budgeted Costs” sections at the

top of the worksheet.

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394 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

4. Compute the price variance and the quantity variance in the middle of the worksheet. 5. Indicate whether the price and the quantity variances are favorable or unfavorable

for March.

O p t i o n a l

Can you compute how the $3,228,900 actual overhead costs and the $3,250,000 budgeted overhead costs were calculated?

Assignment Exercise 17–1: Variance Analysis

Greenview Hospital operated at 120% of normal capacity in two of its departments during the year. It operated 120% times 20,000 normal capacity direct labor nursing hours in rou- tine services and it operated 120% times 20,000 normal capacity equipment hours in the laboratory. The lab allocates overhead by measuring minutes and hours the equipment is used; thus equipment hours.

Assumptions: For Routine Services Nursing:

• 20,000 hours � 120% � 24,000 direct labor nursing hours. • Budgeted Overhead at 24,000 hours � $42,000 fixed plus $6,000 variable � $48,000

total. • Actual Overhead at 24,000 hours � $42,000 fixed plus $7,000 variable � $49,000 total. • Applied Overhead for 24,000 hours at $2.35 � $56,400.

For Laboratory:

• 20,000 hours � 120% � 24,000 equipment hours. • Budgeted Overhead at 24,000 hours � $59,600 fixed plus $11,400 variable � $71,000

total. • Actual Overhead at 24,000 hours � $59,600 fixed plus $11,600 variable � $71,200

total. • Applied Overhead for 24,000 hours at $3.455 � $82,920.

Require d

1. Set up a worksheet for applied overhead costs and volume variance with a column for Routine Services Nursing and a second column for Laboratory.

2. Set up a worksheet for actual overhead costs and budget variance with a column for Routine Services Nursing and a second column for Laboratory.

3. Set up a worksheet for volume variance and budget variance totaling net variance with a column for Routine Services Nursing and a second column for Laboratory.

4. Insert input data from Assumptions. 5. Complete computations for all three worksheets.

Example 17B

Review the “Sensitivity Analysis Overview” section and Figure 17-5 in Chapter 17.

Assignment Exercise 17–2: Three-Level Revenue Forecast

Three eye-ear-nose-and-throat physicians decide to hire an experienced audiologist in order to add a new service line to their practice.* They ask the practice manager to prepare a three-level volume forecast as a first step in their decision-making.

Assumptions: for the base level (most likely) revenue forecast, assume $200 per proce- dure times four procedures per day times five days equals 20 procedures per week times 50 weeks per year equals 1,000 potential procedures per year.

For the best case revenue forecast, assume an increase in volume of one procedure per day average, for an annual increase of 250 procedures (5 days per week times 50 weeks equals 250). (The best case is if the practice gains a particular managed care contract.)

For the worst case revenue forecast, assume a decrease in volume of two procedures per day average, for an annual decrease of 500 procedures. (The worst case is if the practice loses a major payer.)

*Audiologists were designated as “eligible for physician and other prescriber incentives” as discussed in Chapter 20. Thus the new service line was a logical move.

Require d

Using the above assumptions, prepare a three-level forecast similar to the example in Fig- ure 17-5 and document your calculations.

Practice Exercise 17–II

Closely study the chapter text concerning target operating income. The necessary inputs for target operating income include the following:

• Desired (target) operating income amount � $20,000 • Unit price for sales � $500 • Variable cost per unit � $300 • Total fixed cost � $10,000

Compute the required revenue to achieve the target operating income and compute a contribution income statement to prove the totals.

Assignment Exercise 17–3: Target Operating Income

Acme Medical Supply Company desires a target operating income amount of $100,000, with assumption inputs as follows:

• Desired (target) operating income amount � $100,000 • Unit price for sales � $80 • Variable cost per unit � $60 • Total fixed cost � $60,000

Compute the required revenue to achieve the target operating income and compute a contribution income statement to prove the totals.

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396 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

CHAPTER 18

Assignment Exercise 18–1: Estimate of Loss

You are the practice manager for a four-physician office. You arrive on Monday morning to find the entire office suite flooded from overhead sprinklers that malfunctioned over the weekend. Water stands ankle-deep everywhere. The computers are fried and the contents of all the filing cabinets are soaked. Your own office, where most of the records were stored, has the worst damage.

The practice carries valuable papers insurance coverage for an amount up to $250,000. It is your responsibility to prepare an estimate of the financial loss so that a claim can be filed with the insurance company. How would you go about it? What would your summary of the losses look like?

Assignment Exercise 18–2: Estimate of Replacement Cost

The landlord carries contents insurance that should cover the damage to the furnishings, equipment, and to the computers, and the insurance company adjuster will come tomor- row to assess the furnishings and equipment damage. However, your boss is sure that the in- surance settlement will not cover replacement costs. Consequently, you have been instructed to prepare an estimate of what has been lost and/or damaged plus an estimate of what the replacement cost might be. How would you go about it? What would your sum- mary of these losses look like?

Assignment Exercise 18–3: Benchmarking

Review the chapter text about benchmarking.

Require d

1. Select either the MHS case study in Chapter 25 or one of the organizations repre- sented by a mini-case study in Chapters 27, 28, or 29.

2. Prepare a list of measures that could be benchmarked for this organization. Com- ment on why these items are important for benchmarking purposes.

3. Find another example of benchmarking for a healthcare organization. The example can be an organization report or it can be taken from a published source such as a journal article.

Assignment Exercise 18–4: Pareto Rule

Review the chapter text about the Pareto rule and examine Figure 18-4. Note that the text says Pareto diagrams are often drawn to reflect before and after results.

Assume that Figure 18-4 is the before diagram for the Billing department. Further as- sume that the after results are as follows:

Activity Activity Code Number

Process Denied Bills PDB 12 Review with Supervisor RWS 10 Locate Documentation LD 6 Copy Documentation CD 5

Require d

1. Redo the Pareto diagram with the after results. (Use Figure 18-4 as a guide.) 2. Comment on the before and after results for the Billing department.

CHAPTER 19

Assignment Exercise 19–1: Physician Incentive Payments and Costs under ARRA

Refer to the description of physician incentives under the American Recovery and Reinvest- ment Act of 2009 (ARRA) within the chapter text. See, for example, that the maximum amount a physician can receive in year 1 is $15,000, except if the first year is 2011 or 2012 the year 1 payment is $18,000, and so on. See also the definition of physicians who are “mean- ingful electronic health records users” (and thus eligible for payment under this program).

Require d

Locate additional information about electronic health records systems that are being sold to physicians based on their qualification under the ARRA program. Attempt to determine what the net cost of hardware, software, and installation would be for an average physician practice. Compare this cost with the payments that an eligible physician who is a meaning- ful electronic health records user would receive over a five-year period. Determine the ap- proximate net technology cost to the physician after such incentive payments.

As an extension of this assignment you might also determine what start-up costs other than the technology costs may be incurred by the physician.

Assignment Exercise 19–2: Hospital Conversion to ICD-10

Try to locate sufficient detail about a healthcare organization; enough that you can per- form a make-believe SWOT analysis about a conversion of electronic systems to ICD-10 as required. Write a description of the organization’s background, including its information system. (Add imaginary details if you need to.) This description will then lead into the ICD- 10 conversion’s situation analysis. Perform the make-believe SWOT analysis, using the four- part format (internal strengths and weaknesses and external opportunities and threats).

As an alternative approach, you can use the Sample Hospital information in Appendix 25-A as a starting point and use your personal experience and observations to fill out the rest of the details you would need in order to commence a make-believe SWOT analysis for this hospital’s ICD-10 conversion.

Chapter 19 397

398 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

CHAPTER 20

Practice Exercise 20–I

The productivity loss section of Chapter 20 describes how the dollar amount of such loss was computed after determining that it took coders an extra 1.7 minutes per claim in the first month of ICD-10 transition.

Assume a hospital’s coders are dealing with 1,500 claims within a certain period. What would the dollar amount of productivity loss be if the coders took an extra two minutes (in- stead of 1.7 minutes) per claim?

Assignment Exercise 20–1: Information about the ICD-10-CM and ICD-10-PCS Transition

Require d

Locate some articles and/or government Web sites that describe the ICD-10-CM and ICD- 10-PCS diagnosis codes and tools for their implementation over a period of years. Write a summary of whether the materials you have found fully explain the breadth and depth of the transition challenge for managers who must live through the transition.

Assignment Exercise 20–2: Hospital Costs to Implement

Refer to the scenario entitled “ICD-10 Conversion Costs for a Midwestern Community Hos- pital,” located in the Supplemental Materials section of this book.

Require d

Within this scenario the productivity loss for the six-month learning period is calculated to be $1,233. Beginning with month one at $353 ($1.41 times 250 equals $353), compute the cost of productivity loss for the remaining five months as explained in the scenario, to total an overall amount of $1,233. Be prepared to show and explain your computations.

Later in the scenario CMS states that the hospital’s total cost amounts to $303,990. Study the explanation and summarize the totals of each type of cost discussed. When you are fin- ished your total should amount to $303,990. Be prepared to show and explain how you ar- rived at this total.

CHAPTER 21

Assignment Exercise 21–1

Review the information about public companies and stock exchanges in the Chapter 21 text.

Require d

Obtain a copy of the Wall Street Journal. Locate the “Stock Tables” section of the Journal. Re- view the column headings in the tables and locate the names of various stock exchanges

that are included in the findings. See if you can find the abbreviated names and the stock exchange symbols for healthcare companies that are publicly held.

Alternatively, explore the Web sites of three or four publicly held healthcare organiza- tions. Somewhere on the Web site they should identify their stock exchange symbol. Then go onto a Web-based stock exchange listing of the market for the day, locate the symbols, and determine their current stock prices according to the listing.

CHAPTER 22

Example 22A: Loan Amortization

This example illustrates the initial monthly payments of a loan with a principal balance of 50,000, an interest rate of ten percent, and a payment period of three years or thirty-six months. Loan Amortization Schedule Principal borrowed: 50,000 Total payments: 36 Annual interest rate 10.00% (monthly rate � 0.8333%)

Principal Interest Expense Remaining Payment Total Portion of Portion of Principal

# Payment Payment Payment Balance

Beginning balance � 50,000.00

1 1,613.36 1,196.69 416.67 48,403.31 2 1,613.36 1,206.67 406.69 47,596.64 3 1,613.36 1,216.72 396.64 46,379.92 4 1,613.36 1,226.86 386.50 45,153.06 5 1,613.36 1,237.08 376.28 43,915.98 6 1,613.36 1,247.39 365.97 42,668.58

Practice Exercise 22–I: Loan Amortization

This exercise illustrates a different principal amount than Example 22A, but computed at the same monthly interest rate and the same number of payments.

Require d

Compute the first six months of a loan amortization schedule with a principal balance of 60,000, an interest rate of ten percent, and a payment period of three years or thirty-six months. Loan Amortization Schedule Principal borrowed: 60,000 Total payments: 36 Annual interest rate 10.00% (monthly rate � 0.8333%)

Chapter 22 399

400 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

Principal Interest Expense Remaining Payment Total Portion of Portion of Principal

# Payment Payment Payment Balance

Beginning balance � 60,000.00

1 2 3 4 5 6

Assignment Exercise 22–1: Financial Statement Capital Structures

Require d

Find three different financial statements that have varying capital structures. Write a para- graph about each that explains the debt-equity relationship and that computes the percent of debt and the percent of equity represented.

Also note whether the percent of annual interest on debt is revealed in the notes to the financial statements. If so, do you believe the interest rate is fair and equitable? Why?

CHAPTER 23

Practice Exercise 23–I: Cost of Leasing

A cost of leasing table is reproduced below.

Require d

Using the appropriate table from the Chapter 12 Appendices, record the present value fac- tor at 6% for each year and compute the present value cost of leasing.

Cost of Leasing: Suburban Clinic—Comparative Present Value

Not-for-Profit Cost of Leasing: Year 0 Year 1 Year 2 Year 3 Year 4 Year 5

Net Cash Flow (11,000) (11,000) (11,000) (11,000) (11,000) —- Present value factor (at 6%) Present value answer � Present value cost of leasing �

Assignment Exercise 23–1: Cost of Owning and Cost of Leasing

Cost of owning and cost of leasing tables are reproduced below.

Require d

Using the appropriate table from the Chapter 12 Appendices, record the present value fac- tor at 10% for each year and compute the present value cost of owning and the present value of leasing. Which alternative is more desirable at this interest rate? Do you think your answer would change if the interest rate was six percent instead of ten percent?

Cost of Owning: Anywhere Clinic—Comparative Present Value

For-Profit Cost of Owning: Year 0 Year 1 Year 2 Year 3 Year 4 Year 5

Net Cash Flow (48,750) 2,500 2,500 2,500 2,500 5,000 Present value factor Present value answers � Present value cost of owning �

Cost of Leasing: Anywhere Clinic—Comparative Present Value

Line For-Profit Cost # of Leasing: Year 0 Year 1 Year 2 Year 3 Year 4 Year 5

19 Net Cash Flow (8,250) (8,250) (8,250) (8,250) (8,250) —- 20 Present value factor 21 Present value answers � 22 Present value cost of leasing �

Assignment Exercise 23–2

Great Docs, a three-physician practice with two office sites, is considering whether to buy or lease a new computer system. Currently they own a low-tech (and low-cost) information sys- tem. The new system will have to meet all government specifications for an electronic health record system and will also have to connect the two office sites. It will be consider- ably more sophisticated than the current hardware and software and thus will require train- ing for office staff, clinical staff, and the physicians. Everyone agrees there will be a learning curve in order to reach the system’s full potential.

Doctor Smith, the majority owner of the practice, wants to buy a medical records system from Sam’s Club. He argues that the package is supposed to electronically prescribe, track billings, set appointments, and keep records, so it should meet their needs. The cost of the first installed system is supposed to be $25,000, plus $10,000 for each additional system.* The doctors are not sure if this means $25,000 for one office site plus $10,000 for the (con- nected) second office site for a total of $35,000, or if this means $25,000 for the first in- stalled system plus $10,000 each for three more doctors, for a total of $55,000. There is also supposed to be $4,000 to $5,000 in maintenance costs each year as part of the purchased

Chapter 23 401

*Details about this system were announced in a Wall Street Journal story as quoted in Chapter 20. The prices in this exercise are fictitous.

402 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

package. Doctor Smith proposes to pay twenty percent down and obtain a five-year install- ment loan from the local bank for the remaining eighty percent at an interest rate of eight percent.

Doctor Jones, the youngest of the three physicians, has been recently added to the prac- tice. A computer nerd, he wants to lease a complete system from the small company his col- lege roommate began last year. While he has received a quote of $20,000 for the entire system including first year maintenance, it does not meet the government requirements for an electronic health record system. Consequently, the other two doctors have outvoted Doc- tor Jones and this system will not be seriously considered.

Doctor Brown, the usual peace-maker between Doctor Smith and Doctor Jones, wants to lease a system. He argues that leasing will place the responsibility for upgrades and mainte- nance upon the lessor company, and that removing the responsibilities of ownership is ad- vantageous. He has received a quote of $20,000 per year for a five-year lease that includes hardware and software for both offices, that meets the government requirements for an electronic health record system, and that includes training, maintenance, and upgrades.

Require d

Summarize the costs to the practice of owning a system (per Doctor Smith) versus leasing (per Doctor Brown). Include a computation of comparative present value. (Refer to As- signment 23-1 for setting up a comparative present value table.)

Assignment Exercise 23–3

Metropolis Health System has to do something about their ambulance situation. They have to (a) buy a new ambulance; (b) lease a new one; or (c) renovate an existing ambulance that MHS already owns. Rob Lackey, the Assistant Controller, has been asked to gather per- tinent information in order to make a decision. So far Rob has found these facts:

1. It will cost at least $250,000 to purchase a new ambulance, although the cost varies widely depending upon the quantity and sophistication of the emergency equipment con- tained on the vehicle.

2. In order to renovate the existing vehicle, it will cost at least $100,000 to purchase and install a new “box.” (In other words, a new emergency-equipped body is installed on the ex- isting chassis.) Rob has found this existing ambulance has an odometer reading of 80,000 miles. The vehicle will also need a new fuel pump and new tires, but he believes these items would be recorded as repair and maintenance operating expenses and thus would not be included in his calculations.

3. Lease terms for ambulances also vary widely, but so far Rob believes a cost of $60,000 per year is a ball-park figure.

Require d

How much more information should Rob have before he begins to make any calculations? Make a list. Which alternative do you believe would be best? Give your reasons.

CHAPTER 24

Example 24A: Assumptions

Types of assumptions required for the financial portion of a business plan typically include answers to the following questions: Example of Typical Income Statement Assumption Information Requirements:

• What types of revenue? • How many services will be offered to produce the revenue? (by month) • How much labor will be required? (FTEs) • What will the labor cost? • How many and what type of supplies, drugs, and/or devices will be required to offer

the service? • What will the supplies, drugs, and/or devices cost? • How much space will be required? • What will the required space occupancy cost? • Is special equipment required? • If so, how much will it cost? • Is staff training required to use the special equipment? • If so, how much time is required, and what will it cost?

Practice Exercise 24–I: Assumptions

Refer to the proposal to add a retail pharmacy Mini-Case Study in Chapter 26.

Require d

Identify how many of the assumption items listed in the example above can be found in the retail pharmacy proposal worksheets.

Assignment Exercise 24–1: Business Plan

Refer to the proposal to add a retail pharmacy Mini-Case Study in Chapter 26.

Require d

Build a business plan for this proposal. Prepare the service description using your con- sumer knowledge of a retail pharmacy (if necessary). Of course this retail pharmacy will be located within the hospital, but its purpose is to dispense prescriptions to carry off-site and use at home. Thus, it operates pretty much like the neighborhood retail pharmacy that you use yourself.

Use the information provided in Chapter 26 to prepare the financial section of the busi- ness plan. Use your imagination to create the marketing segment and the organization segment.

Chapter 24 403

404 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

SUPPLEMENTAL MATERIALS

Present Value of an Annuity of $1

Periods 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% Periods

1 .980 .962 .943 .926 .909 .893 .877 .862 .848 .833 1 2 1.942 1.886 1.833 1.783 1.736 1.690 1.647 1.605 1.566 1.528 2 3 2.884 2.775 2.673 2.577 2.487 2.402 2.322 2.246 2.174 2.107 3 4 3.808 3.630 3.465 3.312 3.170 3.037 2.914 2.798 2.690 2.589 4 5 4.713 4.452 4.212 3.993 3.791 3.605 3.433 3.274 3.127 2.991 5 6 5.601 5.242 4.917 4.623 4.355 4.111 3.889 3.685 3.498 3.326 6 7 6.472 6.002 5.582 5.206 4.868 4.564 4.288 4.039 3.812 3.605 7 8 7.325 6.733 6.210 5.747 5.335 4.968 4.639 4.344 4.078 3.837 8 9 8.162 7.435 6.802 6.247 5.759 5.328 4.946 4.607 4.303 4.031 9

10 8.983 8.111 7.360 6.710 6.145 5.650 5.216 4.833 4.494 4.193 10 15 12.849 11.118 9.712 8.560 7.606 6.811 6.142 5.576 5.092 4.676 15 20 16.351 13.590 11.470 9.818 8.514 7.469 6.623 5.929 5.353 4.870 20 25 19.523 15.622 12.783 10.675 9.077 7.843 6.873 6.097 5.467 4.948 25

Metropolis Health System Balance Sheet

March 31, 20X3 and 20X2

Assets

Current Assets Cash and cash equivalents 1,150,000 400,000 Assets whose use is limited 825,000 825,000 Patient accounts receivable 8,700,000 8,950,000 Less allowance for bad debts (1,300,000) (1,300,000) Other receivables 150,000 100,000 Inventories of supplies 900,000 850,000 Prepaid expenses 200,000 150,000

Total Current Assets 10,625,000 9,975,000

Assets Whose Use Is Limited Corporate funded depreciation 1,950,000 1,800,000 Under bond indenture agreements— held by trustee 1,425,000 1,475,000

Total Assets Whose Use Is Limited 3,375,000 3,275,000 Less Current Portion (825,000) (825,000) Net Assets Whose Use Is Limited 2,550,000 2,450,000

Property, Plant, and Equipment, Net 19,300,000 19,200,000

Other Assets 325,000 375,000

Total Assets 32,800,000 32,000,000

Metropolis Health System Balance Sheet

March 31, 20X3 and 20X2

Liabilities and Fund Balance

Current Liabilities Current maturities of long-term debt 525,000 500,000 Accounts payable and accrued expenses 4,900,000 5,300,000 Bond interest payable 300,000 325,000 Reimbursement settlement payable 100,000 175,000

Total Current Liabilities 5,825,000 6,300,000

Long-Term Debt 6,000,000 6,500,000 Less Current Portion of Long-Term Debt (525,000) (500,000) Net Long-Term Debt 5,475,000 6,000,000

Total Liabilities 11,300,000 12,300,000

Fund Balances General Fund 21,500,000 19,700,000

Total Fund Balances 21,500,000 19,700,000

Total Liabilities and Fund Balances 32,800,000 32,000,000

Supplemental Materials 405

406 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

Metropolis Health System Statement of Revenue and Expenses

for the Years Ended March 31, 20X3 and 20X2

Revenue Net patient service revenue 34,000,000 33,600,000 Other revenue 1,100,000 1,000,000

Total Operating Revenue 35,100,000 34,600,000

Expenses Nursing services 5,025,000 5,450,000 Other professional services 13,100,000 12,950,000 General services 3,200,000 3,220,000 Support services 8,300,000 8,340,000 Depreciation 1,900,000 1,800,000 Amortization 50,000 50,000 Interest 325,000 350,000 Provision for doubtful accounts 1,500,000 1,600,000

Total Expenses 33,400,000 33,760,000

Income from Operations 1,700,000 840,000

Nonoperating Gains (Losses) Unrestricted gifts and memorials 20,000 70,000 Interest income 80,000 40,000

Nonoperating Gains, Net 100,000 110,000

Revenue and Gains in Excess of Expenses and Losses 1,800,000 950,000

Metropolis Health System Statement of Changes in Fund Balance

for the Years Ended March 31, 20X3 and 20X2

General Fund Balance April 1st $19,700,000 $18,750,000

Revenue and Gains in Excess of Expenses and Losses 1,800,000 950,000

General Fund Balance March 31st $21,500,000 $19,700,000

Metropolis Health System Schedule of Property, Plant, and Equipment

for the Years Ended March 31, 20X3 and 20X2

Buildings and Improvements 14,700,000 14,000,000

Land Improvements 1,100,000 1,100,000

Equipment 28,900,000 27,600,000

Total 44,700,000 42,700,000

Less Accumulated Depreciation (26,100,000) (24,200,000)

Net Depreciable Assets 18,600,000 18,500,000

Land 480,000 480,000

Construction in Progress 220,000 220,000

Net Property, Plant, and Equipment 19,300,000 19,200,000

Supplemental Materials 407

408 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

Metropolis Health System Schedule of Patient Revenue

for the Years Ended March 31, 20X3 and 20X2

Patient Services Revenue Routine revenue 9,850,000 9,750,000 Laboratory 7,375,000 7,300,000 Radiology and CT scanner 5,825,000 5,760,000 OB–nursery 450,000 445,000 Pharmacy 3,175,000 3,140,000 Emergency service 2,200,000 2,180,000 Medical and surgical supply and IV 5,050,000 5,000,000 Operating rooms 5,250,000 5,200,000 Anesthesiology 1,600,000 1,580,000 Respiratory therapy 900,000 890,000 Physical therapy 1,475,000 1,460,000 EKG and EEG 1,050,000 1,040,000 Ambulance services 900,000 890,000 Oxygen 575,000 570,000 Home health and hospice 1,675,000 1,660,000 Substance abuse 375,000 370,000 Other 775,000 765,000

Subtotal 48,500,000 48,000,000

Less Allowances and Charity Care 14,500,000 14,400,000

Net Patient Service Revenue 34,000,000 33,600,000

Metropolis Health System Schedule of Operating Expenses

for the Years Ended March 31, 20X3 and 20X2

Nursing Services Routine Medical-Surgical 3,880,000 4,200,000 Operating Room 300,000 325,000 Intensive Care Units 395,000 430,000 OB–Nursery 150,000 165,000 Other 300,000 330,000 Total 5,025,000 5,450,000

Other Professional Services Laboratory 2,375,000 2,350,000 Radiology and CT Scanner 1,700,000 1,680,000 Pharmacy 1,375,000 1,360,000 Emergency Service 950,000 930,000 Medical and Surgical Supply 1,800,000 1,780,000 Operating Rooms and Anesthesia 1,525,000 1,515,000 Respiratory Therapy 525,000 530,000 Physical Therapy 700,000 695,000 EKG and EEG 185,000 180,000 Ambulance Services 80,000 80,000 Substance Abuse 460,000 450,000 Home Health and Hospice 1,295,000 1,280,000 Other 130,000 120,000 Total 13,100,000 12,950,000

General Services Dietary 1,055,000 1,060,000 Maintenance 1,000,000 1,010,000 Laundry 295,000 300,000 Housekeeping 470,000 475,000 Security 50,000 50,000 Medical Records 330,000 325,000 Total 3,200,000 3,220,000

Support Services General 4,600,000 4,540,000 Insurance 240,000 235,000 Payroll Taxes 1,130,000 1,180,000 Employee Welfare 1,900,000 1,950,000 Other 430,000 435,000 Total 8,300,000 8,340,000

Supplemental Materials 409

410 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

Depreciation 1,900,000 1,800,000

Amortization 50,000 50,000

Interest Expense 325,000 350,000

Provision for Doubtful Accounts 1,500,000 1,600,000

Total Operating Expenses 33,400,000 33,760,000

EXCERPTS FROM METROPOLITAN HEALTH SYSTEM NOTES TO FINANCIAL STATEMENTS

Note 1—Nature of Operations and Summary of Significant Accounting Policies

General

Metropolitan Hospital System (Hospital) currently operates as a general acute care hospi- tal. The hospital is a municipal corporation and body politic created under the hospital dis- trict laws of the state.

Cash and Cash Equivalents

For purposes of reporting cash flows, the hospital considers all liquid investments with an original maturity of three months or less to be cash equivalents.

I n v e n t o r y

Inventory consists of supplies used for patients and is stated as the lower of cost or market. Cost is determined on the basis of most recent purchase price.

I n v e s t m e n t s

Investments, consisting primarily of debt securities, are carried at market value. Realized and unrealized gains and losses are reflected in the statement of revenue and expenses. In- vestment income from general fund investments is reported as nonoperating gains.

Income Ta x e s

As a municipal corporation of the state, the hospital is exempt from federal and state in- come taxes under Section 115 of the Internal Revenue Code.

P ro p e r t y, Plant, and Equipment

Expenditures for property, plant, and equipment, and items that substantially increase the useful lives of existing assets are capitalized at cost. The hospital provides for depreciation on the straight-line method at rates designed to depreciate the costs of assets over estimated useful lives as follows:

Years

Equipment 5 to 20 Land Improvements 20 to 25 Buildings and Improvements 40

Funded Depreciation

The hospital’s Board of Directors has adopted the policy of designating certain funds that are to be used to fund depreciation for the purpose of improvement, replacement, or ex- pansion of plant assets.

Unamortized Debt Issue Costs

Revenue bond issue costs have been deferred and are being amortized.

Revenue and Gains in Excess of Expenses and Losses

The statement of revenue and expenses includes revenue and gains in excess of expenses and losses. Changes in unrestricted net assets that are excluded from excess of revenue over expenses, consistent with industry practice, would include such items as contributions of long-lived assets (including assets acquired using contributions that by donor restriction were to be used for the purposes of acquiring such assets) and extraordinary gains and losses. Such items are not present on the current financial statements.

Net Patient Service Revenue

Net patient service revenue is reported as the estimated net realizable amounts from pa- tients, third-party payers, and others for services rendered, including estimated retroactive adjustments under reimbursement agreements with third-party payers. Retroactive adjust- ments are accrued on an estimated basis in the period the related services are rendered and adjusted in future periods as final settlements are determined.

Contractual Agreements with Third-Party Payers

The hospital has contractual agreements with third-party payers, primarily the Medicare and Medicaid programs. The Medicare program reimburses the hospital for inpatient ser- vices under the Prospective Payment System, which provides for payment at predetermined amounts based on the discharge diagnosis. The contractual agreement with the Medicaid program provides for reimbursement based upon rates established by the state, subject to state appropriations. The difference between established customary charge rates and re- imbursement is accounted for as a contractual allowance.

Gifts and Bequests

Unrestricted gifts and bequests are recorded on the accrual basis as nonoperating gains.

Donated Services

No amounts have been reflected in the financial statements for donated services. The hos- pital pays for most services requiring specific expertise. However, many individuals volun- teer their time and perform a variety of tasks that assists the hospital with specific assistance programs and various committee assignments.

Excerpts from Metropolitan Health System Notes to Financial Statements 411

412 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

Note 2—Cash and Investments

Statutes require that all deposits of the hospital be secured by federal depository insurance or be fully collateralized by the banking institution in authorized investments. Authorized investments include those guaranteed by the full faith and credit of the United States of America as to principal and interest; or in bonds, notes, debentures, or other similar obli- gations of the United States of America or its agencies; in interest-bearing savings accounts, interest-bearing certificates of deposit; or in certain money market mutual funds.

At March 31, 20X3, the carrying amount and bank balance of the hospital’s deposits with financial institutions were $190,000 and $227,000, respectively. The difference between the carrying amount and the bank balance primarily represents checks outstanding at March 31, 20X3. All deposits are fully insured by the Federal Deposit Insurance Corporation or collateralized with securities held in the hospital’s name by the hospital agent.

Carrying Amount

20X3 20X2

U.S. Government Securities or U.S. Government Agency Securities 4,325,000 3,575,000 Total Investments 4,325,000 3,575,000 Petty Cash 3,000 3,000 Deposits 190,000 93,000 Accrued Interest 7,000 4,000 Total 4,525,000 3,675,000 Consisting of Cash and Cash Equivalents—General Fund 1,150,000 400,000 Assets Whose Use Is Limited Corporate Funded Depreciation 1,950,000 1,800,000 Held by Trustee under Bond Indenture Agreements 1,425,000 1,475,000 Total 4,525,000 3,675,000

Note 3—Charity Care

The hospital voluntarily provides free care to patients who lack financial resources and are deemed to be medically indigent. Such care is in compliance with the hospital’s mission. Because the hospital does not pursue collection of amounts determined to qualify as char- ity care, they are not reported as revenue.

The hospital maintains records to identify and monitor the level of charity care it pro- vides. These records include the amount of charges forgone for services and supplies fur- nished under its charity care policy. During the years ended March 31, 20X3, and 20X2 such charges forgone totaled $395,000 and $375,000, respectively.

Note 4—Net Patient Service Revenue

The hospital provides healthcare services through its inpatient and outpatient care facili- ties. The mix of receivables from patients and third-party payers at March 31, 20X3, and 20X2 is as follows:

20X3 20X2

Medicare 30.0% 28.5% Medicaid 15.0 16.0 Patients 13.0 12.5 Other third-party payers 42.0 43.0

Total 100.0% 100.0%

The hospital has agreements with third-party payers that provide for payments to the hospital at amounts different from its established rates. Contractual adjustments under third-party reimbursement programs represent the difference between the hospital’s es- tablished rates for services and amounts paid by third-party payers. A summary of the pay- ment arrangements with major third-party payers follows:

Medicare

Inpatient acute care rendered to Medicare program beneficiaries is paid at prospectively de- termined rates-per-discharge. These rates vary according to a patient classification system that is based on clinical, diagnostic, and other factors. Inpatient nonacute care services and certain outpatient services are paid based upon either a cost reimbursement method, established fee screens, or a combination thereof. The hospital is reimbursed for cost reimbursable items at a tentative rate with final settlement determination after submission of annual cost reports by the hospital and audits by the Medicare fiscal intermediary. At the current year end, all Medicare settlements for the previous two years are subject to audit and retroactive adjustments.

Medicaid

Inpatient services rendered to Medicaid program beneficiaries are reimbursed at prospec- tively determined rates-per-day. Outpatient services rendered to Medicaid program benefi- ciaries are reimbursed at prospectively determined rates-per-visit.

Blue Cro s s

Inpatient services rendered to Blue Cross subscribers are reimbursed under a cost reimburse- ment methodology. The hospital is reimbursed at a tentative rate with final settlement deter- mined after submission of annual cost reports by the hospital and audits by Blue Cross. The Blue Cross cost report for the prior year end is subject to audit and retroactive adjustment.

The hospital has also entered into payment agreements with certain commercial insur- ance carriers, health maintenance organizations, and preferred provider organizations. The bases for payment under these agreements include discounts from established charges and prospectively determined daily rates.

Gross patient service revenue for services rendered by the hospital under the Medicare, Medicaid, and Blue Cross payment agreements for the years ended March 31, 20X3, and 20X2 is approximately as follows:

Excerpts From Metropolitan Health System Notes to Financial Statements 413

414 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

20X3 20X2

Amount % Amount %

Medicare $20,850,000 43.0 $19,900,000 42.0 Medicaid 10,190,000 21.0 10,200,000 21.5 All other payers 17,460,000 36.0 17,300,000 36.5

$48,500,000 100.0 $47,400,000 100.0

Note 5—Property, Plant, and Equipment

The hospital’s property, plant, and equipment at March 31, 20X3, and 20X2 are as follows:

20X3 20X2

Buildings and improvements $14,700,000 $14,000,000 Land improvements 1,100,000 1,100,000 Equipment 28,900,000 27,600,000 Total $44,700,000 $42,700,000 Accumulated depreciation (26,100,000) (24,200,000) Net Depreciable Assets $18,600,000 $18,500,000 Land 480,000 480,000 Construction in progress 220,000 220,000 Net Property, Plant, Equipment $19,300,000 $19,200,000

Construction in progress, which involves a renovation project, has not progressed in the last twelve-month period because of a zoning dispute. The project will not require signifi- cant outlay to reach completion, as anticipated additional expenditures are currently esti- mated at $100,000.

Note 6—Long-Term Debt

Long-term debt consists of the following: Hospital Facility Revenue Bonds (Series 1995) at varying interest rates from 4.5% to 5.5%, depending on date of maturity through 2020. 20X3 20X2

$6,000,000 $6,500,000

The future maturities of long-term debt are as follows:

Years Ending March 31

20X2 $ 475,000 20X3 500,000 20X4 525,000 20X5 550,000 20X6 575,000 20X7 600,000 Thereafter 3,750,000

Under the terms of the trust indenture the following funds (held by the trustee) were established:

Interest Fund

The hospital deposits (monthly) into the interest fund an amount equal to one sixth of the next semi-annual interest payment due on the bonds.

Bond Sinking Fund

The hospital deposits (monthly) into the bond sinking fund an amount equal to one twelfth of the principal due on the next July 1.

Debt Service Reserve Fund

The debt service reserve fund must be maintained at an amount equal to 10 percent of the aggregate principal amount of all bonds then outstanding. It is to be used to make up any deficiencies in the interest fund and bond sinking fund.

Assets held by the trustee under the trust indenture at March 31, 20X3, and 20X2 are as follows:

20X3 20X2 Interest Fund $ 300,000 $ 325,000 Bond Sinking Fund 525,000 500,000 Debt Service Reserve 600,000 650,000 Total $1,425,000 $1,475,000

Note 7—Commitments

At March 31, 20X3, the hospital had commitments outstanding for a renovation project at the hospital of approximately $100,000. Construction in progress on the renovation has not progressed in the last twelve-month period because of a zoning dispute. Upon resolution of the dispute, remaining construction costs will be funded from corporate funded deprecia- tion cash reserves.

ICD-10 CONVERSION COSTS FOR A MIDWESTERN COMMUNITY HOSPITAL

Authors’ Note This CMS example illustrates the computation of hospital training costs and productivity

loss costs and estimates a cost for system changes and upgrades in order to arrive at a total hospital ICD-10 conversion cost. We have numbered the paragraphs for easy reference. (And FYI, when the scenario below says “we” it means CMS, not the authors.)

Introduction

To further illustrate the computation of hospital ICD-10 conversion costs, CMS staff devel- oped a scenario for a typical community hospital in the Midwest. The material presented below was published in the proposed rule as an example of costs that might be incurred by a hospital. The data were drawn from the American Hospital Directory, available at

Solution to Practice Exercise 5–I 415

416 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

www.AHD.com. While based on an actual hospital in a midwestern state, the data have been altered to make calculations simpler.

The Scenario

1. The hospital has 100 beds, 4,000 discharges annually, and gross revenues of $200 mil- lion. Using the factors presented in the impact analysis, we estimated training costs (including the cost of the actual training as well as lost time away from the job), pro- ductivity loss for the first six months resulting from becoming familiar with the diag- nostic and procedure codes, and the cost of system changes.

2. For our scenario, we assumed that the hospital employs three full-time coders who will require eight hours of training at $500 per coder for $1,500 ($500 times 3). While they are in training, the hospital will have to substitute other staff, either by hiring temporary coders if possible, or shifting staff. The estimated cost at $50 per hour is $1,200 (8 hours times 3 staff times $50 per hour).

3. In estimating the productivity loss, we are only looking at the initial six months after implementation. Therefore, we divided the annual number of discharges of 4,000 by 2 to equal 2,000. We assume that three quarters of the discharges are surgical, giving us 1,500 discharges requiring use of PCS codes. Dividing this by six months yields an average monthly discharge rate of 250.

4. We performed a similar calculation for outpatient claims. Of the 13,000 outpatient claims, the monthly average is 1,083 (we do not distinguish between medical and sur- gical outpatient claims).

5. Applying the 1.7 extra minutes per discharge, we estimated it would take an extra 425 minutes (1.7 times 250) to code the discharges in the first month. At $50 per hour, the cost per minute is $0.83 ($50 divided by 60 minutes) and the cost per claim is $1.41 ($0.83 times 1.7). For the first month, the productivity loss for inpatient coding is $353 ($1.41 times 250). Assuming for simplicity’s sake that the resumption of pro- ductivity over the six month period would increase in a straight line, we divide the $353 by six to come up with $59. We reduce the productivity loss by this amount each month through the sixth month. The total loss for the six-month period is $1,233.

6. We apply the same method to determine the outpatient productivity loss. Based on our assumption that outpatient claims will require one hundredth of the time for hos- pital inpatient claims, when applying the 0.17 extra minutes per claim, we estimate it would take an extra 18.41 minutes (0.017 times 1,083) to code the discharges in the first month. At $50 per hour, the cost per minute is $0.83 ($50 divided by 60 minutes) and the cost per claim is $0.14 ($0.83 times 0.017). For the first month, the produc- tivity loss for outpatient coding is $15.28 ($0.014 times 1,083). Assuming for simplic- ity sake that the resumption of productivity over the six-month period would increase in a straight line, we divide the $15.28 by six; to come up with $2.55. We reduce the productivity loss by this amount each month through the sixth month. Thus the total loss for the first six months will equal $53.

7. In estimating the cost of system changes and software upgrades, we deliberately chose a value that we think overstates the cost. We assumed that the hospital will have to spend $300,000 on its data infrastructure to accommodate the new codes. Summing

the training costs, productivity losses, and system upgrades, we estimate the total cost to the hospital will equal approximately $303,990. Finally, in order to determine the percent of the hospital’s revenue that would be diverted to funding the conversion to the ICD-10 we compared the estimated cost associated with the conversion to ICD-10 to the total hospital revenue of $200 million. The costs amount to 0.15% of the hos- pital’s annual revenues.

8. We note that although the impact in our scenario of 0.15% is significantly larger than the estimated impact of 0.03% for inpatient facilities (set out in the rule), it is still sig- nificantly below the threshold the Department considers a significant economic im- pact. We are of the opinion that, for most providers and suppliers, payers and computer firms involved in facilitating the transition, the costs will be relatively small.

Source: 73 Federal Register 49830 (August 22, 2008).

SOLUTIONS TO PRACTICE EXERCISES

SOLUTION TO PRACTICE EXERCISE 3-I

Short-term assets: cash on hand; accounts receivable; inventory Long-term assets: land; buildings Short-term liabilities: payroll taxes due; accounts payable Long-term liabilities: mortgage payable (non-current); note payable (due in 24 months)

SOLUTION TO PRACTICE EXERCISE 5–I

Intensive Care Unit Laboratory Laundry

Drugs requisitioned X Pathology supplies X Detergents and bleach X Nursing salaries X Clerical salaries X X X Uniforms (for laundry aides) X Repairs (parts for microscopes) X

Note: If no clerical salaries are assigned to Laundry, this is an acceptable alternative solution.

SOLUTION TO PRACTICE EXERCISE 6–I

Direct Cost Indirect Cost

Managed care marketing expense X Real estate taxes X Liability insurance X Clinic telephone expense X Utilities (for the entire facility) X Emergency room medical supplies X

Solution to Practice Exercise 6–I 417

418 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

SOLUTION TO PRACTICE EXERCISE 6–II

In real life the solution to this exercise will depend upon factors unique to the particular organization. The following solution is a generic one.

Responsibility Center Support Center

Security X Communications X Ambulance services X Medical records X Educational resources X Human resources X

Reporting: Each responsibility center has a manager. All report to the director.

SOLUTION TO PRACTICE EXERCISE 7–I

Step 1. Find the highest volume of 1,100 packs at a cost of $7,150 in September and the lowest volume of 100 packs at a cost of $1,010 in August.

Step 2. Compute the variable rate per pack as:

# of Packs Training Pack Cost

Highest volume 1,100 $7,150 Lowest volume 100 1,010 Difference 1,000 $6,140

Step 3. Divide the difference in cost ($6,140) by the difference in # of packs (1,000) to arrive at the variable cost rate:

$6,140 divided by 1,000 packs � $6.14 per pack

Step 4. Compute the fixed overhead rate as follows: At the highest level: Total cost $7,150 Less: Variable portion [1,100 packs � $6.14] (6,754) Fixed Portion of Cost $ 396

At the lowest level: Total cost $1,010 Less: Variable portion [100 packs � $6.14] (614) Fixed Portion of Cost $ 396

Proof totals: $396 fixed portion at both levels.

SOLUTION TO PRACTICE EXERCISE 7–II

Step 1. Divide costs into variable and fixed portions. In this case $3,450,000 times 40% equals $1,380,000 variable cost and $3,450,000 times 60% equals $2,070,000 fixed cost.

Step 2. Compute the contribution margin:

Amount

Revenue $3,500,000 Less variable cost (1,380,000) Contribution margin $2,120,000 Less fixed cost 2,070,000 Operating income $ 50,000

SOLUTION TO PRACTICE EXERCISE 7–III

Amount %

Revenue $1,210,000 100.00 Less variable cost (205,000) 16.94 Contribution margin $1,005,000 83.06 � PV or CM Ratio Less fixed cost (1,100,000) 90.91 Operating loss $(95,000) 7.85

SOLUTION TO PRACTICE EXERCISE 8–I

1. Straight-line depreciation would amount to $54,000 per year for 10 years. This amount is computed as follows: Step 1. Compute the cost net of salvage or trade-in value: 600,000 less 10% salvage

value or 60,000 equals 540,000. Step 2. Divide the resulting figure by the expected life (also known as estimated use-

ful life): 540,000 divided by 10 equals 54,000 depreciation per year for 10 years.

2. Double declining depreciation is computed as follows: Step 1. Compute the straight-line rate: 1 divided by 10 equals 10%. Step 2. Now double the rate (as in “double declining method”): 10% times 2 equals

20%. Step 3. Compute the first year’s depreciation expense: 600,000 times 20% � 120,000. Step 4. Compute the carry-forward book value at the beginning of the second year:

600,000 book value beginning year 1 less year 1 depreciation of 120,000 equals book value at beginning of the second year of 480,000.

Step 5. Compute the second year’s depreciation expense: 480,000 times 20% � 96,000.

Solution to Practice Exercise 8–I 419

420 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

Step 6. Compute the carry-forward book value at the beginning of the third year: 480,000 book value beginning year 2 less year 2 depreciation of 96,000 equals book value at beginning of the third year of 384,000. —Continue until the asset’s salvage or trade-in value has been reached.

Book Value at Beginning of Year Depreciation Expense Book Value at End of Year

600,000 600,000 � 20% � 120,000 600,000 � 120,000 � 480,000 480,000 480,000 � 20% � 96,000 480,000 � 96,000 � 384,000 384,000 384,000 � 20% � 76,800 384,000 � 76,800 � 307,200 307,200 307,200 � 20% � 61,440 307,200 � 61,440 � 245,760 245,760 245,760 � 20% � 49,152 245,760 � 49,152 � 196,608 196,608 196,608 � 20% � 39,322 196,608 � 39,322 � 157,286 157,286 157,286 � 20% � 31,457 157,286 � 31,457 � 125,829 125,829 125,829 � 20% � 25,166 125,829 � 25,166 � 100,663 100,663 100,663 � 20% � 20,132 100,663 � 20,132 � 80,531 80,531 80,561 at 10th year: 80,561 � 20,561 � 60,000

—Balance remaining at end of tenth year represents the salvage or trade-in value.

Note: Under the double declining balance method, book value never reaches zero. Therefore, a company typically adopts the straight-line method at the point where straight line would exceed the double declining balance.

SOLUTION TO PRACTICE EXERCISE 8–II

Straight-line depreciation would amount to $9,000 per year for five years. This amount is computed as follows:

Step 1. Compute the cost of salvage or trade-in value: 50,000 less 10% salvage value or 5,000 equals 45,000.

Step 2. Divide the resulting figure by the expected life (also known as the estimated use- ful life): 45,000 divided by 5 equals 9,000 depreciation per year for 5 years

SOLUTION TO PRACTICE EXERCISE 9–I

1. Compute Net Paid Days Worked

Total days in business year 364 Less two days off per week 104 # Paid days per year 260

Less paid days not worked Holidays 8 Sick days 5 Education day 1 Vacation days 5

19 Net paid days worked 241

2. Convert Net Paid Days Worked to a Factor

Total days in business year divided by net paid days worked equals factor

364/241 � 1.510373

SOLUTION TO PRACTICE EXERCISE 9–II

24-Hour Shift 1 Shift 2 Shift 3 � Scheduling

Day Evening Night Total

Position: Admissions officer 2 1 1 four 8-hour shifts FTEs—to cover position 7 days/week equals 2.8 1.4 1.4 5.6 FTEs

Position: Clerical 1 0 0 one 8-hour shift FTEs—to cover position 7 days/week equals 1.4 0 0 1.4 FTEs

SOLUTION TO PRACTICE EXERCISE 10–I

Current Liabilities 30,000 Total Assets 1,000,000 Income from Operations 80,000 Accumulated Depreciation 480,000 Total Operating Revenue 180,000 Current Portion of Long-Term Debt 10,000 Interest Income -0- Inventories 5,000

SOLUTION TO PRACTICE EXERCISE 10–II

No, Doctors Smith and Brown’s patient accounts receivable does not appear to be net of an allowance for bad debts, because we cannot find an equivalent bad debt expense on their statement of net income. Do you think the doctors should have an allowance for bad debts on their statement? Why do you think they do not?

Solution to Practice Exercise 10–II 421

422 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

SOLUTION TO PRACTICE EXERCISE 10–III

As mentioned in the chapter text, land is not stated at “net” because land is never depreciated.

SOLUTION TO PRACTICE EXERCISE 11–I

Current Ratio

The current ratio is represented as Current Ratio = Current Assets divided by Current Lia- bilities. This ratio is considered to be a measure of short-term debt-paying ability. However, it must be carefully interpreted.

Current Ratio Computation

Current Assets �

$70,000 � 2.33 to 1

Current Liabilities $30,000

Quick Ratio

The quick ratio is represented as Quick Ratio � Cash � Short-Term Investments � Net Re- ceivables divided by Current Liabilities. This ratio is considered to be an even more severe test of short-term debt-paying ability (even more severe than the current ratio). The quick ratio is also known as the acid-test ratio, for obvious reasons.

Cash & Cash Equivalents � Net Receivables �

$65,000 � 2.167 to 1

Current Liabilities $30,000

SOLUTION TO PRACTICE EXERCISE 11–II

Solvency Ratios

Debt Service Coverage Ratio (DSCR)

The Debt Service Coverage Ratio (DSCR) is represented as change in unrestricted net as- sets (net income) plus interest, depreciation, and amortization divided by maximum an- nual debt service. This ratio is universally used in credit analysis.

Change in Unrestricted Net Assets (net income) plus Interest, Depreciation, Amortization

� $113,100

� 5.1 Maximum Annual Debt Service $22,200

Note: $80,000 � $3,100 � $30,000 � $113,100.

Liabilities to Fund Balance (or Debt to Net Worth)

The liabilities to fund balance or net worth computation is represented as total liabilities di- vided by unrestricted net assets (fund balances or net worth) � total debt divided by tangi- ble net worth. This figure is a quick indicator of debt load.

Total Liabilities �

$200,000 � 2.5

Unrestricted (Fund Balance) $800,000

SOLUTION TO PRACTICE EXERCISE 11–III

Profitability Ratios

Operating Margin

The operating margin, which is generally expressed as a percentage, is represented as op- erating income (loss) divided by total operating revenues. This ratio is used for a number of managerial purposes and also sometimes enters into credit analysis. It is therefore a multi- purpose measure.

Operating Income (Loss) �

$80,000 � 44.4%

Total Operating Revenues $180,000

Return on To t a l A s s e t s The return on total assets is represented as earnings before interest and taxes (EBIT) di- vided by total assets. This is a broad measure in common use.

EBIT (Earnings before Interest & Taxes) �

$83,100 � 8.3%

Total Assets $1,000,000

Note: $80,000 � $3,100 � $83,100.

SOLUTION TO PRACTICE EXERCISE 12–I: UNADJUSTED RATE OF RETURN

1. Calculation using original investment amount:

$100,000

$500,000 � 20% Unadjusted Rate of Return

2. Calculation using average investment amount: Step 1. Compute average investment amount for total unrecovered asset cost:

At beginning of estimated useful life � $500,000 At end of estimated useful life � $ 50,000 Sum $550,000

Divided by 2 � $275,000 average investment amount

Step 2. Calculate unadjusted rate of return:

$100,000

$275,000 � 36.4% Unadjusted Rate of Return

Solution to Practice Exercise 12–I: Unadjusted Rate of Return 423

424 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

SOLUTION TO PRACTICE EXERCISE 12–II: FINDING THE FUTURE VALUE (WITH A COMPOUND INTEREST TABLE)

Step 1. Refer to the Compound Interest Table found in Appendix 12-B at the back of this chapter. Reading across, or horizontally, find the 7% column. Reading down, or vertically, find Year 6. Trace across the Year 6 line item to the 7% col- umn. The factor is 1.501.

Step 2. Multiply the current savings account balance of $11,000 times the factor of 1.501 to find the future value of $16,511. In six years at compound interest of 7%, the college fund will have a balance of $16,511.

SOLUTION TO PRACTICE EXERCISE 12–III: FINDING THE PRESENT VALUE

Step 1. Refer to the Present Value Table found in Appendix 12-A at the back of this chapter. Reading across, or horizontally, find the 7% column. Reading down, or vertically, find Year 15. Trace across the Year 15 line item to the 7% column. The factor is 0.3624.

Step 2. Multiply $150,000 times the factor of 0.3624 to find the present value of $54,360.

SOLUTION TO PRACTICE EXERCISE 12–IV

Assemble the assumptions in an orderly manner:

Assumption 1: Initial cost of the investment � $16,950. Assumption 2: Estimated annual net cash inflow the investment will generate � $3,000. Assumption 3: Useful life of the asset � 10 years.

Perform calculation:

Step 1. Divide the initial cost of the investment ($16,950) by the estimated annual net cash inflow it will generate ($3,000). The answer is a ratio amounting to 5.650.

Step 2. Now use the abbreviated look-up table for the Present Value of an Annuity of $1, which is found at the back of the Examples and Exercises section. Find the line item for the number of periods that matches the useful life of the asset (10 years in this case).

Step 3. Look across the 10 year line on the table and find the column that approximates the ratio of 5.650 (as computed in Step 1). That column contains the interest rate representing the rate of return. In this case the rate of return is 12%.

SOLUTION TO PRACTICE EXERCISE 12–V

Assemble assumptions in an orderly manner:

Assumption 1: Purchase price of the equipment � $500,000. Assumption 2: Useful life of the equipment � 10 years.

Assumption 3: Revenue the machine will generate per year � $84,000. Assumption 4: Direct operating costs associated with earning the revenue � $21,000. Assumption 5: Depreciation expense per year (computed as purchase price per assump-

tion 1 divided by useful life per assumption 2) � $50,000. Perform computation: Step 1. Find the machine’s expected net income after taxes:

Revenue (Assumption 3) $84,000 Less Direct operating costs (Assumption 4) $21,000 Depreciation (Assumption 5) 50,000

71,000 Net income $13,000

Note: No income taxes for this hospital.

Step 2. Find the net annual cash inflow the machine is expected to generate (in other words, convert the net income to a cash basis).

Net income $13,000 Add back depreciation (a noncash expenditure) 50,000 Annual net cash inflow after taxes $63,000

Step 3. Compute the payback period:

Investment �

$500,000 Machine Cost* � 7.9 Year Payback Period

Net Annual Cash Inflow $63,000

*Assumption 1 above. **Per Step 2 above.

The machine will pay back its investment under these assumptions in 7 9⁄10 years.

SOLUTION TO PRACTICE EXERCISE 13–I

Common sizing for the assets of the two hospitals appears on the worksheet below. Note that their gross numbers are very different, yet the proportionate relationships of the per- centages (20%, 75%, and 5%) are the same for both hospitals.

Same Year for Both Hospitals Hospital A Hospital B

Current assets $ 2,000,000 20% $ 8,000,000 20% Property, plant, and equipment 7,500,000 75% 30,000,000 75% Other assets 500,000 5% 2,000,000 5% Total assets $10,000,000 100% $40,000,000 100%

Solution to Practice Exercise 13–I 425

426 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

SOLUTION TO PRACTICE EXERCISE 13–II

Hospital A Year 1 Year 2 Difference

Current assets $1,600,000 $ 2,000,000 $ 400,000 25% Property, plant, and equipment 6,000,000 7,500,000 1,500,000 25% Other assets 400,000 500,000 100,000 25% Total assets $8,000,000 $10,000,000 $2,000,000 —

Note: The worksheet below shows Hospital A with both common sizing and trend analysis:

Hospital A Year 1 Year 2 Difference

Current assets $1,600,000 20% $ 2,000,000 20% $ 400,000 25% Property, plant, and equipment 6,000,000 75% 7,500,000 75% 1,500,000 25% Other assets 400,000 5% 500,000 5% 100,000 25% Total assets $8,000,000 100% $10,000,000 100% $2,000,000 —

SOLUTION TO PRACTICE EXERCISE 13–III

Q: How many procedures has your unit recorded for the period to date?

Solution: The unit has recorded 2,000 procedures ($200,000 divided by $100 apiece equals 2,000 procedures).

Q: Of these, how many procedures are attributed to each payer?

Solution: At 25% of the volume per payer, each payer accounts for 500 procedures (2,000 times 25% equals 500 procedures). Proof total: 500 procedures apiece times four payers equals 2,000 procedures.

Q: How much is the net revenue per procedure for each payer, and how much is the con- tractual allowance per procedure for each payer?

Solution: The computation is as follows:

Payer # Gross % Paid by Net Revenue Contractual Allowance Charges Each Payer per Procedure per Procedure

1 $100.00 90% $90.00 $10.00

2 $100.00 80% $80.00 $20.00

3 $100.00 70% $70.00 $30.00

4 $100.00 50% $50.00 $50.00

SOLUTION TO PRACTICE EXERCISE 15–I

Your initial budget assumptions were as follows: Assume the budget anticipated 30,000 inpatient days this year at an average of $650 rev-

enue per day, or $19,500,000. Further assume that inpatient expenses were budgeted at $600 per patient day, or $18,000,000. Also assume the budget anticipated 10,000 outpatient visits this year at an average of $400 revenue per visit, or $4,000,000. Further assume that outpatient expenses were budgeted at $380 per visit, or $3,800,000. The budget worksheet would look like this:

As Budgeted

Revenue—Inpatient $19,500,000 Revenue—Outpatient 4,000,000 Subtotal $23,500,000 Expenses—Inpatient $18,000,000 Expenses—Outpatient 3,800,000 Subtotal $21,800,000 Excess of revenue over expenses $1,700,000

Now assume that only 27,000, or 90%, of the patient days are going to actually be achieved for the year. The average revenue of $650 per day will be achieved for these 27,000 days (thus 27,000 times 650 equals 17,550,000). Also assume that outpatient visits will actu- ally amount to 110%, or 11,000 for the year. The average revenue of $400 per visit will be achieved for these 11,000 visits (thus 11,000 times 400 equals 4,400,000). Further assume that, due to the heroic efforts of the Chief Financial Officer, the actual inpatient expenses will amount to $11,600,000 and the actual outpatient expenses will amount to $4,000,000. The actual results would look like this:

Actual

Revenue—Inpatient $17,550,000 Revenue—Outpatient 4,400,000 Subtotal $21,950,000 Expenses—Inpatient 16,100,000 Expenses—Outpatient 4,000,000 Subtotal $20,100,000 Excess of revenue over expenses $1,850,000

Since the budgeted revenues and expenses still reflect the original expectations of 30,000 inpatient days and 10,000 outpatient visits, the budget report would look like this:

Solution to Practice Exercise 15–I 427

428 EXAMPLES AND EXERCISES, SUPPLEMENTAL MATERIALS, AND SOLUTIONS

Static Budget Actual Budget Variance

Revenue—Inpatient $17,550,000 $19,500,000 $(1,950,000) Revenue—Outpatient 4,400,000 4,000,000 400,000 Subtotal $21,950,000 $23,500,000 $(1,550,000)

Expenses—Inpatient $16,100,000 $18,000,000 $(1,900,000) Expenses—Outpatient 4,000,000 3,800,000 200,000 Subtotal $20,100,000 $21,800,000 $(1,700,000)

Excess of revenue over expenses $ 1,850,000 $ 1,700,000 $ 150,000

Note: The negative effect of the $1,550,000 net drop in revenue is offset by the greater effect of the $1,700,000 net drop in expenses, resulting in a positive net effect of $150,000.

PRACTICE EXERCISE 16-I

Because there is no one right answer, students will approach this exercise in different ways.

REQUIRED SOLUTION TO PRACTICE EXERCISE 17–II

The Price Variance is $206,100 (3,435,000 less 3,228,900 equals 206,100). The Quantity Variance is $185,000 (3,435,000 less 3,250,000 equals 185,000).

OPTIONAL SOLUTION TO PRACTICE EXERCISE 17–I

The $3,228,900 actual overhead costs represent 687,000 RVUs times $4.70 per RVU. The $3,250,000 budgeted overhead costs represent 650,000 RVUs times $5.00 per RVU.

SOLUTION TO PRACTICE EXERCISE 17-II

The required revenue to achieve a target operating income of $20,000 amounts to revenue of $75,000.

The contribution income statement to prove the formula results is as follows: Revenue $500/unit x 150 units � $75,000 Variable costs $300/unit x 150 units � 45,000_______ Contribution margin $30,000 Fixed costs 10,000_______ Desire (Target) Operating Income � $20,000

SOLUTION TO PRACTICE EXERCISE 20–I

$50 per hour divided by 60 minutes equals $0.8333; thus 2 minutes equals $1.6667. If a hos- pital’s coders are dealing with 1,500 claims, then the dollar amount of productivity loss is $2,499 (1.6667 per claim times 1,500 claims equals $2,499).

SOLUTION TO PRACTICE EXERCISE 22-I

With beginning principal of $60,000, the monthly payment is $1,936.03 and the remaining principal balance at the end of six payments is $51,202.30.

SOLUTION TO PRACTICE EXERCISE 23-I

The present value cost of leasing for Suburban Clinic amounts to $49,116.

SOLUTION TO PRACTICE EXERCISE 24-I

All of the assumption items listed in Example 23A are present in the retail pharmacy mini- case study in Chapter 26.

Solution to Practice Exercise 24-I 429

Index

Accelerated book depreciation methods, 84–85, 87–91

Accounting recording inventory in, 76 types of, 7, 9

Accounting income inputs, 186 Accounting rate of return, 179, 186 Account numbers, 15 Accounts, groupings of, 14 Accounts payable, 27, 42 Accounts receivable, 26, 33 Acid-test ratio, 117 Admissions

analyzing referral activity, 331–332 areas to automate, 330 assessing, 329–330 case studies, 290, 329–332 fax and document management, 330 referral tracking and approval, 331 saving time with automated, 332

Allocation basis, 52 Alternative Depreciation System, 86 Ambulatory care systems, 12 American Hospital Association, 165 American Recovery and Reinvestment Act of

2009 (ARRA) hospital incentives under, 219–220 physician incentives under, 220

Amortization schedules, 258, 259, 261–262 Annual management cycles, 19–20 Annual year-end reports, 19–20 Annuities, 133

431

ARRA. See American Recovery and Reinvestment Act of 2009 (ARRA)

Assets on balance sheets, 107–108 capital, 178, 182, 183 description of, 25 examples, 26 fixed, and depreciation expense, 81 lease-purchase agreements, 263–264 long-term, 26, 108 return on, 120–121 short-term, 26

Assisted living services, 37 Assumptions, forecasts and, 141–144 Authority, lines of, 7, 42, 52–55, 145 Averages, 186, 213–214

Bad debts, 33 Balance sheets, 107–108

in business plans, 275–276 case study, 284 checklist for review of, 113 liquidity ratio calculations, 119 profitability ratios, 121 solvency ratios, 121

Behavioral health, 39 Benchmarking, 210–211

case study, 293–298 financial, 211

Billing departments, 212 Board of Trustees, 306 Bonds, 251–252, 258

432 INDEX

Bonds payable, 27 Books and records, 18–19 Book values, 27, 81–83 Break even points, 68–70 Budgets, 165–176

building, 168–171 capital expenditure, 177–184 checklists, 165, 176 comparative budgets, 152–154 construction tools, 178–180 forecasts and, 141 managerial accounting and, 9 reviewing, 166–168, 175–176 types of, 165–168

Buildings, 26 Burden approach, 96 Bureau of Economic Analysis, 254 Business loans, 257–262 Business plans, 271–278

assembly of, 276–277 cash flow assumptions, 305 elements of, 271 executive summaries, 276 financial analysis segment, 273–276 formats, 277 “knowledgeable reader” approach, 276 marketing segment, 273 monthly income statements, 304 organization segment, 272–273 planning stage, 271–272 presentation, 277 for retail pharmacies, 301–306

Buy-or-lease decisions, 264–269

Capacity issues, staffing and, 145, 147 Capital

asset acquisition, 183 costs of, 185, 188 definition, 257 sources of, 257–258

Capital budgets, 185–188 Capital expenditures

budgets, 177–184 evaluating proposals, 182–183 spending plans, 177–178 types of proposals, 181–182

Capital Stock, 28 Capital structures, 257

Care settings, grouping by, 36–37, 45–46 Case mix adjustments, 210 Case studies

admissions, 290, 329–332 balance sheets, 284 benchmarking, 293–298 expansions, 301–306 financial ratios, 293–298 improving quality of care, 308–314 Metropolis Health System, 281–298 pharmacy addition, 301–306 physicians offices, 308–322 resource misallocation, 323–328 Schedule of Operating Expenses, 289 Schedule of Patient Revenue, 288 Schedule of Property, Plant, and Equip-

ment, 287 solvency ratios, 295, 297 statement of revenue and expense, 284, 285 technology in healthcare, 229, 245–246

Cash, as asset, 26 Cash equivalents, 249–250 Cash flow

budgeting and, 178 cumulative, 178 discounted, 125 equipment purchase and, 185 monthly detail and assumptions, 301 net annual inflow, 126 owning v. leasing equipment, 265, 267 payback methods, 179 projections, 274 reporting methods, 179 for retail pharmacies, 302 statement of, 111–113

Cash flow statement, 286 Cash/noncash concept, 111 CDs (certificates of deposit), 250 Centers for Disease Control and Prevention

(CDC), 221 Centers for Medicare and Medicaid Services

(CMS), 221, 222, 317. See also Medicaid Program (Title XIX); Medicare Advantage; Medicare Program (Title XVIII)

Certificates of deposit (CDs), 250 Changes in Fund Balance, 287 Charity services, 33

Charts of accounts, 14–17 Chief executive officers, 42 Chief financial officers, 257 Clinical viewpoint, 4 Closing costs, 259 Coders, and ICD-10 training, 234–235, 236 Code users, and ICD-10 training, 235, 236 Coinsurance, 35 Commercial insurers, 36 Common costs, 49–50. See also Indirect costs Common sizing, 137–138 Common stock, 252 Communication

in admissions, 330–331 financial information, 20 lines of, 7

Community health programs, 281–282 Company ownership, 253 Comparative analysis, 137–138, 175, 293–298 Comparative budget review, 152–153 Comparative data, 151–162

annualizing expenses, 158–159 comparability requirements, 151–152 comparing current expenses to current

budget, 153–154 comparing organization’s current actual

expenses to prior periods, 154–155 comparing to industry standards, 157–158 comparing to other organizations, 155–157 currency measures, 161 inflation factors, 159–160 manager’s view of, 152–153 standardized measures, 161 uses of, 153–154

Competition, 309 Competitors, 210, 293 Compound interest, 124, 131–132 Continuing care retirement communities

(CCRCs), 37–38 Contracts, types of, 35 Contractual allowances, 32–33 Contribution income statement, 199–200 Contribution margins, 68–73, 199–202

break-even point using, 201–202 contribution income statement and,

199–200 target operating income using, 200–201 worksheet example, 201

Index 433

Controller’s office, 49 Controlling, purpose of, 5 Cost centers, 42–44, 47 Cost objects, 49, 50, 55 Cost of goods sold, 76–77 Costs

assignment to cost objects, 50 of capital, 15, 185 case study, 294–296 classification, 49–56, 59–62 definition, 41 direct, 49, 50, 52 e-prescribing, 239 expired, 41 of financing, 258–259 fixed, 59–62 grouping, 168 ICD-10 training and lost productivity costs,

234–237 indirect, 49, 50, 52 of leasing/owning equipment, 266 reports, 46–48 of sales, 55 semifixed, 61–62 semivariable, 59–62 staffing and, 99–100 unexpired, 42 variable, 59–62

Cost-volume profit (CVP) ratios, 68–70 Credit analysis, 121 Credit losses, 33 Currency measures, 161 Current ratio, 116, 117

Daily operating reports, 19 Data repositories, 12 Days cash on hand (DCOH), 116, 118 Days receivables, 116, 118 Debentures, 252 Debt-equity relationships, 257 Debts

as liabilities, 27 long-term, 108, 258 short-term, 258 uncollectible, 33

Debt service coverage ratio (DSCR), 116, 118–119

Debt to fund balance, 120

434 INDEX

Decentralization, 7 Decision making

about business loans, 260 budgeting and, 178 in business plans, 272 controllable costs and, 144–145 lines of authority and, 6 purpose of, 6 rationing capital, 15

Deductibles, 35 Deflation, 254 Dell, Inc., 239 Demographics, 310 Department of Veterans’ Affairs, 35 Depreciation

of assets, 27 assumptions, 265 book value of fixed asset and reserve for,

81–83 calculations, 265 computing tax depreciation, 85–86 expenses, 112 interrelationship of expense of and reserve

for, 82 methods of computing, 83–85 overview, 81

Diagnoses expense grouping by, 44–45 hospital department codes by, 46 technology issues and problems, 232–234

Diagnosis-related groups (DRGs) expense grouping by, 44 flexible budgets and, 173 hospital department codes by, 46 profitability matrix, 72

Dialysis centers, 52 Direct costs, 49, 50, 52 Directing, purpose of, 5 Disbursements for services, 42 Discharges, case study, 290 Discounted cash flow, 125 Discounted fees for services, 31–32 Disease management, 39 Dispenser, defined, 237 Document management, 330 Donations, 36, 258 Double-Declining Balance (DDB), 84–85, 87,

89, 90

Doubtful accounts, 33 Drug and alcohol services, 35 Drug costs, 73

Earnings, time pattern of, 125–126 Earnings before interest and taxes (EBIT),

120–121 Economic measures, 211 Efficiency, specialization and, 3 Efficiency variance, 191 Electronic data information (EDI) links, 12 Electronic documents, 330 Electronic intercompatibility, 221 Electronic prescribing. See E-prescribing Electronic records, 217–230

compliance requirements, 218–219 definitions, 217–218 slow adoption of, 218 standardized input and, 161

Emergency services, 39 Employee welfare cost centers, 43 E-prescribing

adoption rates, 238 benefits, 238 costs, 239 definitions, 237–238 “eligible professionals,” 241–242 implementation of, 240–241 incentives program, 241–242, 243 manner of reporting, 243–245 penalties, 242–243 qualified system of, 242 technical input example, 245 transactions, 237–238 Wal-Mart and, 239

Equipment acquisition, 181, 276 as asset, 26 budgeting for, 178 decision making, 185 descriptions, 272 funding requests, 181–182 leasing, 263 owning v. leasing, 265, 267 purchasing, 263 replacing, 181–182 upgrading, 181

Equity, definition, 27–28

E-records. See Electronic records E-referrals, 331 Expansions, 178, 301–306 Expenses (outflow), 41–48

administrative, 55 annualizing, 158–159 budgeted, 175 comparative analysis, 137 controllable, 144–145 definition, 41 depreciation as, 265 general, 55 grouping, 42–46 noncontrollable, 144–145 operating, 108, 110 physician practices, 316–322 for retail pharmacies, 302 statement of, 108, 110 trend analysis, 138, 139

Expired costs, 41

Fax management, 330 Federal Deposit Insurance Corporation

(FDIC), 250 Feedback, purpose of, 5 Fee schedules, 316–322 Fees for services, 31–32 FIFO inventory method, 77, 78, 151 Financial accounting, 7 Financially indigent patients, 33 Financial management

concept, 4 elements of, 5–6 history, 3–4 viewpoints, 4–5

Financial ratios, 293–298 Financial systems, 11 Financial viewpoint, 4 Financing, costs of, 258–259 First-In, First-Out (FIFO) inventory method,

77, 78, 151 Fiscal years, 20 Fixed assets, 81–84 Fixed costs

analysis, 72 budgeting, 168 description, 59–60 examples, 52–55

Index 435

high-low method, 66–68 operating room, 63–64

Flexible budgets, 168, 173–174, 196–198 Flexible costs, 168 Flowsheets, 12–14 Food costs, 60, 66, 73 Forecasts. See also Projections

building budgets, 169 in business plans, 273–274 equipment acquisition, 276 use of, 144–147

Foreign currency measures, 161 For-profit organizations

income taxes, 264–265 net worth terminology, 27–28 owning v. leasing decisions, 266, 267 revenue retention, 258

Freight in, 76 Full-time equivalent staff (FTES)

annualizing positions, 95–98 calculation of, 95–98 costs, 63–64

Fund balances, 28 on balance sheets, 107–108 revenues and, 41 statement of changes, 111

Fund designators, 17 Funding requests, 180–182

GDP (gross domestic product), 254 General Depreciation System (GDS), 85–86 General ledgers, 18–19 Generally accepted accounting principles

(GAAPs), 108 General obligation bonds, 251 General Purpose Simulation System, 325 General services, 43 Geographic practice cost indices (GPCIs),

320–322 Goods sold, inventory and costs of, 76–77 Government

perspectives of, 312 reporting to, 7 revenue sources, 33–35

Government securities, 250 Graphs

high-low method, 66–68 scatter, 66, 70, 72–73

436 INDEX

Gross domestic product (GDP), 254 Gross margins, 77, 303 Gross revenue, 32

Health Insurance for the Aged and Disabled. See Medicare Program

Health Insurance Portability and Accountability Act of 1996 (HIPAA), 221–223

Health maintenance organizations (HMOs). See also Managed care

case study, 308–322 organizational perspectives, 312–314 plan descriptions, 35

Health systems, case study, 281–298 High-low graphs, 63, 66–68 Historical costs, 27 Home care systems, 13–14, 38–39 Horizontal analyses, 138–139, 141 Hospital care codes, 319 Hospital department codes, 45, 46 Hourly base rates, staff, 99 Housing and Urban Development (HUD)

subsidized independent housing, 38

ICD-9-CM, 232–234 ICD-10 e-records, 221–225

benefits and costs of, 223–225 developmental steps, 227–228 ICD-10-CM and ICD-10-PCs, 221 impact in the U.S., 221–223 managers and, 231 overview and impact of, 221–223 providers and suppliers impacted by

transition to, 223 situational analysis, 226–228 strategic steps, 226–227 system implementation planning,

225–226 training and lost productivity costs,

234–237 Income. See also Revenues (inflow)

accounting inputs, 186 monthly statements, 304 operating, 108 for retail pharmacies, 302

Independent living facilities, 37–38 “Indexed to inflation,” 254

Indian healthcare services, 35 Indirect costs

definition, 49 dialysis center example, 52 examples, 50, 52

Individual practice associations (IPAs) models, 35

Inflation comparative data and, 159–160 vs. deflation, 254

Inflow (revenues), 31–39 Information systems

ambulatory care example, 12 basic elements, 14–19 manager’s challenge in changes of,

231–246 reports and, 19 structure of, 11–12

Inpatient care settings, 45–46 Input-output analysis, 191–192 Inputs, into information systems, 12 Inside-out perspectives, 312 Insurance companies

databases, 12 perspectives, 314 verification systems, 12

Insurance cost centers, 43 Insurance verification, 331 Interest, 112

compound, 124 expenses, 258–259 points, 259

Interim reports, 19 Internal rates of return (IRRs), 125, 179–180,

187–188 Inventory, 26, 55–56

calculating turnover, 80–81 costs of goods sold and, 76–77 methods, 77–78 necessary adjustments, 80 overview, 75 tracking, 78–80 types of, 75

Investment indicators, 253–254 Investments

payback periods, 125–126 rates of return on, 123, 188 terminology of, 249–255

Joint costs, 49–50. See also Indirect costs

Labor market forecasts, 145 Land, as asset, 26 Last-In, First-Out (LIFO) inventory method,

77, 79, 151 Lease-purchase agreements, 263–264 Leases, capitalizing, 263–264 Least squares method, 73 Lengths of stay, 290, 294, 295, 296 Leverage ratios, 295, 297 Liabilities

on balance sheets, 107–108 comparative analysis, 138 description of, 25 examples of, 27 long- and short-term, 27 trend analysis, 139 types, 108

Liabilities to fund balance, 116, 120 LIFO inventory method, 77, 79, 151 Lines of authority

cost center designations, 42 nursing staff classification by, 145 organization charts and, 6–7 reports and, 52–55

Liquidity ratios, 117–118, 295, 297 Loans

amortization schedules, 259 costs of, 259 decision-making about, 260 short-term, 258

Long-term assets, 81 Long-term borrowing, 258 Long-term care, 37–38 Long-term debt instruments, 251

Major diagnostic categories (MDCs), 37, 44, 46

Malpractice insurance, 316–317 Managed care. See also Health maintenance

organizations (HMOs) case study, 308–322 contractual allowances, 32–33 revenue sources, 35 standardized measures and, 161

Management, organization charts, 291 Management responsibilities, 52–55, 95–104

Index 437

Managerial accounting, 9 Managers

comparative data and, 152–153 ICD-10 and, 234 information systems changes and, 231–246

Market studies, 301–306 Master Staffing Plans, 98 Maternal and child health services, 35 Measurement tools, 211–213 Medicaid Program (Title XIX), 34–35, 46 Medical records division, 42 Medicare Advantage, 34 Medicare Program, 33–34 Medicare Program (Title XVIII), 46 Mental health services, 35, 282 Migrant healthcare services, 35 Mixed costs, 62

analysis of, 65–68 characteristics, 65 scatter graph analysis, 70, 72–73 step methods, 65

Modified Accelerated Cost Recovery System (MACRS), 85

Monetary unit measurement, 152, 161 Money, time value of, 124 Money-market funds, 250 Mortgage bonds, 252 Mortgages, 27, 258 Municipal bonds, 251

National Center for Health Statistics, 221 Net book value, 82. See also Book values Net cash flow, 178, 265 Net present value (NPV), 179, 187 Net purchases, 76 Net rates, staffing, 99 Net worth

on balance sheets, 107–108 definition of, 25 forms of, 27–28 statement of changes, 111 terminology, 28

No method inventory, 78 Noncompetitors, 210 Nonoperating gains (losses), 110 Nonproductive time, 96–97, 99 No-show rates, 310 Notes payable, 27

438 INDEX

Notes receivable, 26 Not-for-profit organizations

church-affiliated, 258 donations from, 36 governmental, 6 net worth terminology, 28 owning v. leasing decisions, 266, 267 revenue retention, 258 voluntary, 6

Nurse practitioners, 313 Nursing facilities, 37 Nursing practice and administration, 290 Nursing services cost centers, 43

Obsolete items, in inventory, 80 Occupancy rates, 294 Occupational health services, 282 Office visit costs, 65 150% Declining Balance accelerated

depreciation method, 85, 90–91 Operating data, analysis, 139–141 Operating expenses, 108, 110, 289 Operating income, 108, 110 Operating leases, 264 Operating margins, 116, 120 Operating reports, 19 Operating revenue, 108, 110 Operating rooms

comparative data analysis, 140 costs, 63–64

Opportunity costs, 183 Organization charts, 6–7

case study, 290, 291 charts of accounts and, 15 health system example, 8 physician’s office example, 7

Organizations, structure and types of, 6–7 Organizing, purpose of, 5 Original records, 18 Outflow (expenses), 41–48 Outpatient care settings, 45–46, 308–322 Outputs, information systems, 12 Outside-in perspectives, 312 Overhead, 275 Owner’s Equity, 28

Parametric analysis, 210 Pareto analyses, 211–213

Patient days, 290 Patient information, confirming, 331 Patient revenue, 288 Patients

information verification, 13, 14 satisfaction surveys, 311

Payback methods, 179–180, 185–186 Payback periods, 125–126 Payer mix, 33 Payers

changes, 144 definition, 32 private, 36

Payrolls register, 101 taxes, 27 transaction records, 101

Peer groups, 210 Pension cost centers, 43 Percentile rankings, 293 Performance gaps, 210 Performance measures, 115–122

adjusted over time, 209 benchmarking, 210–211 variety of, 209–210

Period costs, 55–56 Periodic inventory system, 79–80 Perpetual inventory system, 79 Petty cash funds, 42 Pharmacy additions, 301–306 Physician fee schedules, geographical

adjustments, 320–322 Physician groups, 39 Physicians

e-prescribing for, 237–240 ICD-10 training and, 236

Physicians Current Procedural Terminology (CPT), 44

Physicians offices, 33, 308–322 Planning, purpose of, 5 Point-of-service models, 35 Points, on loans, 259 Predetermined per-person payments, 32 Preferred provider organizations (PPOs), 35 Preferred stock, 252 Prescriber, defined, 237 Prescription Drug Benefit, 34 Prescriptions. See E-prescribing

Presentations, tips, 277 Present-value analysis, 124 Present value costs, 267–268 Present values, net, 179–180 Present value tables, 124, 129–130, 133, 187 Price variance, 192 Privately held companies, 253 Private payers, 36 Procedure codes, 46, 52

ICD-9-CM and ICD-10-CM comparison, 232–233

Procedures, expense grouping, 44–45 Process viewpoint, 4 Product costs, 55–56 Productive time, 96–97, 100 Professional services, 43 Profitability matrix, 72, 302–305 Profitability ratios

case study, 295, 297 description, 120–121 examples, 121 types of, 116

Profit centers, 53 Profit-volume (PV) ratios, 70, 71 Program cost center, 47 Programs

definition, 46 existing, 182 expenses grouped by, 46 new, 182

Projections, 9. See also Forecasts Proprietary organizations, 6 Prospective payment reimbursement

methodology, 44 Providers, perspective of, 314 Public companies, 253 Public health clinics, 35

Quantity variance, 191–192 Quarterly reports, 19 Quartile competition, 213–214 Quick ratio, 117

Radiology departments, 46, 50, 51 Rate setting, 32 Rates of growth, 304 Rates of return

accounting, 179

Index 439

actual, 188 internal, 125, 179–180 unadjusted, 123

Rates per day, 42 Rate variance, 192 Ratio analysis, 115, 116 Real estate taxes, 259 Record-keeping systems, 3 Recovery services, 282 Referrals

analyzing referral activity, 331–332 assessing admissions process, 329 e-referrals, 331 HMOs and, 309–311 tracking and approval, 331

Regression lines, 72 Rehabilitation centers, 281 Relative value units (RVUs), 194, 195

facility total weight, 317 fee schedules and, 316–322 nonfacility total weight, 318 physician fee schedule reporting, 319

Remaining value, 185 Renovations, budgeting for, 178 Rent, 59 Reports

Director’s summary, 53, 54 external, 7 financial statements, 107–114 information systems and, 19 lines of, 7 staffing costs, 99–100

Reserve for depreciation, 82 Resource-Based Relative Value Scale

(RBRVS), 316 Resource utilization, 294, 295

case study, 295, 296 changes in, 143 evaluation of, 127 peak-load problem, 324–325 WIC program, 323–328

Responsibility centers, 36, 52–55 Retained Earnings, 28 Return on total assets, 116, 120–121 Returns on investment, 183 Revenue bonds, 251 Revenue centers, 36 Revenues (inflow), 31–39

440 INDEX

contractual allowances, 32–33 deductions from, 32–33 equity and, 41 grouping, 36–39 operating, 108 projections of, 274 for services, 31–33 sources, 33–36, 274 statement by source, 37 statement of, 108 types, 274

Revenue stream, 31

Salaries, 43 Salvage value, 81, 84, 185 Same-day surgery, 281 Sam’s Club, 239 Scatter graphs, 70, 72–73 Scheduled-position method, 98–99 Schedule of Operating Expenses, 289 Schedule of Patient Revenue, 288 Schedule of Property, Plant, and Equipment,

287 Scheduling systems, 12 School health programs, 35 Seawell’s Chart of Accounts for Hospitals, 15 Securities and Exchange Commission (SEC),

253 Security, in business plans, 275 Semifixed costs, 61–62 Semivariable costs, 59–65 Sensitivity analysis, 199–203

overview, 199 tools, 199–202

Service lines, 37–39, 45–46 Services

cost of, 55 descriptions, 262–263 disbursements for, 42 revenue for, 31–33

Shortages, of inventory, 80 Short-term borrowing, 258 Skilled nursing facilities, 37, 281 Social Security Act, 33 Social security taxes, 43 Solvency ratios, 118–120

case study, 295, 297 examples, 121 types of, 116

Space occupancy, 275 Specialization, efficiency and, 3 Spending variance, 192 St. Joseph Hospital Nursing Center variance

analysis, 195–196 Staffing

annualizing positions, 95–98 calculating FTEs, 95–98 capacity issues and, 145 classification by lines of authority, 145 comparative hours report, 103 costs, 61, 63–64, 65, 99–100 definition, 97 forecasts, 144–147 labor market and, 145 manager responsibility, 95–104 operating room example, 100 requirements, 95, 99 resource utilization and, 294

Standardized measures, 161 Startup cost concepts, 180 Statement of cash flows, 286 Statement of revenue and expense, 108

case study, 284, 285 checklist for review of, 113 liquidity ratio calculations, 119 profitability ratios, 121 solvency ratios, 121

State-only general assistance programs, 35 Static budgets, 172, 196 Statistical support, 12 Stockholders, 7 Stock investments, 252 Stock warrants, 252 Straight-line depreciation method, 83–84 Strategic planning, 7, 12, 37 Stub periods, 20 Subsidiary journals, 18–19, 99–100 Subsidiary reports, 113 Sum-of-the-Year’s Digits (SYD) depreciation

method, 84, 87, 88 Supplemental medical insurance (SMI), 34 Supplies, 65, 275 Support center reports, 54 Support services, 43, 44 SWOT analysis, 228–229

Target operating income, 200–201 Tax depreciation

computing, 85–86 defined, 83

Taxpayers, reporting to, 7 Tax revenues, 36 Tests, costs, 65 Three-variance analysis, 192–194 Time cards, 100, 102 Time periods, 55 Time value of money, 124 Title XIX. See Medicaid Program

(Title XIX) Title XVIII. See Medicare program Traffic flow, example, 326 Transaction records, 100 Transactions, progress of, 18 Treasury bills, 250 Treasury notes, 250 Trend analysis, 138–139, 144, 293 Trial balances, 18 Two-variance analysis, 192–194

Unadjusted rates of return, 123, 186–187 Units of Service or Units of Production

depreciation method, 85, 91, 93, 94 U.S. Securities and Exchange Commission

(SEC), 253 Useful life, 81, 185 Use variance, 191–192

Variable costs behavior, 65 description, 59–62 examples, 52–55 operating room, 63 views of, 62–63

Index 441

Variance analysis, 191–199 calculation of, 192–193 composition, 192–193 elements of, 191–192 examples, 194–198 percentile rankings and, 293 price (or spending) variance, 192 quantity (or use) variance, 191–192 summary, 198–199 two-variance/three-variance analysis com-

pared, 192–194 volume variance, 191

Vendor software, and ICD-9 to ICD-10 transitions, 232

Vertical analysis, 137, 139, 141 Volume variance, 191 Voluntary organizations, 6 Vouchers, WIC, 323

Wages, 43 Wal-Mart, 239 Weekly operating reports, 19 Weighted average inventory method, 77 Wellness programs, 281–282 Wharton, Joseph, 3 Wharton School, 3 Women, Infants, and Children Public Health

Program (WIC) appointments, 324 environment, 323 peak-load problem, 324–325 resource utilization, 324–325, 328 traffic flow, 325, 326 vouchers, 323

Workers’ compensation programs, 35, 39

Judith J. Baker, PhD, CPA, is Executive Director of Resource Group, Ltd., a Dallas-based health care consulting firm. She earned her Bachelor of Science degree in Business Ad- ministration at the University of Missouri, Columbia and her Master of Liberal Studies with a concentration in Business Management at the University of Oklahoma, Norman. She earned her Master of Arts and Doctorate in Human and Organizational Systems, with a concentration in costing systems, at the Fielding Institute, Santa Barbara, California. She is an adjunct faculty member at the Case Western Reserve University Frances Payne Bolton School of Nursing.

Judith has over thirty years experience in health care and consults on numerous health care systems and costing problems. She has worked with health care systems, costing, and reimbursement throughout her career. As a CMS contractor she has assisted in validation of costs for new programs and for rate setting and consults on cost report design.

Judith has written over 40 articles, manuals, and books. She served as Consulting Editor for Aspen Publishers, Inc. Her books include Activity-Based Costing and Activity-Based Management for Health Care, Prospective Payment for Long-Term Care: An Annual Guide, and Prospective Payment for Home Health Agencies. She is editor emeritus of the quarterly Journal of Healthcare Finance.

R.W. Baker, JD, is Managing Partner of Resource Group, Ltd., a Dallas-based health care consulting firm. He has more than 30 years of experience in health care and has designed, directed, and administered numerous financial impact studies for health care providers. His recent studies have centered around facility-specific MDS data collection and analysis. He and his firm subcontracted to the HCFA/CMS Nursing Home Case Mix and Quality Demonstration for over nine years.

R.W. is the editor of continuing professional education seminar manuals and training manuals for facility personnel and for research staff members. He served as a Consulting Editor with Aspen Publishers, Inc. and is co-author of A Step-by-Step Guide to the Mini- mum Data Set (Aspen Publishers, Inc.).

443443

About the Authors

What Does the Healthcare

Manager Need to K n o w ?

2 C H A P T E R

HOW THE SYSTEM WORKS IN HEALTH CARE

The information that you, as a manager, work with is only one part of an overall system. To understand financial management, it is essential to recognize the overall sys- tem in which your organization operates. An order exists within the system, and it is generally up to you to find that order. Watch for how the information fits together. The four segments that make a healthcare financial sys- tem work are (1) the original records, (2) the informa- tion system, (3) the accounting system, (4) and the reporting system. Generally speaking, the original records provide evidence that some event has occurred; the information system gathers this evidence; the ac- counting system records the evidence, and the reporting system produces reports of the effect. The healthcare manager needs to know that these separate elements exist and that they work together for an end result.

THE INFORMATION FLOW

Structure of the Information System

Information systems can be simplistic or highly com- plex. They can be fully automated or semiautomated. Occasionally—even today—they can still be generated by hand and not by computer. (This last instance is be- coming rare and can happen today only in certain small and relatively isolated healthcare organizations that are not yet required to electronically submit their billings.)

We will examine a particular information system and point out the basics that a manager should be able to

After completing this chapter, you should be able to

1. Understand that four segments make a financial management system work.

2. Follow an information flow. 3. Recognize the basic system

elements. 4. Follow the annual management

cycle.

P r o g r e s s N o t e s

recognize. Figure 2-1 shows information system components for an ambulatory care setting. This complex system uses a clinical and financial data repository; in other words, both clin- ical and financial data are fed into the same system. An automated medical record is also linked to the system. These are basic facts that a manager should recognize about this am- bulatory information system.

In addition, the financial information, both outpatient and any relevant inpatient, is fed into the data repository. Scheduling-system data also enter the data repository, along with any relevant inpatient care plan and nursing information. Again, all of these are basic facts that a manager should recognize about this ambulatory care information system.

These items have all been inputs. One output from the clinical and financial data repos- itory (also shown in Figure 2-1) is insurance verification for patients through an electronic data information (EDI) link to insurance company databases. Insurance verification is daily operating information. Another output is decision-making information for managed care strategic planning, including support for demand, utilization, enrollment, and eligibility, plus some statistical support. The manager does not have to understand the specifics of all the inputs and outputs of this complex system, but he or she should recognize that these outputs occur when this ambulatory system is activated.

Function of Flowsheets

Flowsheets illustrate, as in this case, the flow of activities that capture information.1 Flow- sheets are useful because they portray who is responsible for what piece of information as it

12 CHAPTER 2 What Does the Healthcare Manager Need to Know?

Managed Care Systems –Enrollment/eligibility –Utilization management –Demand management –Algorithmic scheduling support

Insurance Verification

EDI Link

Enterprise-Wide Master Patient Index

Interface Engine

Clinical and Financial Data Repository

Practice Management System Scheduling

Ancillary Scheduling

Patient Accounting System Financial (OP and IP)

OR Scheduling

Other Provider Inpatient System

–Clinical order entry –IP care plans –IP nursing

Chart Tracking

(IP and OP)

A u t o m a t e d

M e d i c a l

R e c o r d

Figure 2–1 Information System Components for an Ambulatory Care Setting; OP, Outpatient; IP, Inpa- tient; OR, Operating Room.

The Information Flow 13

enters the system. The manager needs to realize the significance of such information. We give, as an example, obtaining confirmation of a patient’s correct address. The manager should know that a correct address for a patient is vital to the smooth operation of the sys- tem. An incorrect address will, for example, cause the billing to be rejected. Understand- ing this connection between deficient data (e.g., a bad address) and the consequences (the bill will be rejected by the payer and thus not be paid) illustrates the essence of good fi- nancial management knowledge.

We can examine two examples of patient information flows. The first, shown in Figure 2-2, is a physician’s office flowsheet for address confirmation. Four different personnel are involved, in addition to the patient. This physician has computed the cost of a bad address as $12.30 to track down each address correction. He pays close attention to the handling of this information because he knows there is a direct financial management consequence in his operation.

The second example, shown in Figure 2-3, is a health system flowsheet for verification of patient information. This flowsheet illustrates the process for a home care system. In this case, the flow begins not with a receptionist, as in the physician office example, but with a central database. This central database downloads the information and generates a sum- mary report to be reviewed the next day. Appropriate verification is then made in a series of steps, and any necessary corrections are made before the form goes to the billing depart- ment. The object of the flow is the same in both examples: that is, the billing must have a

Initiates Call

Mark Superbill if Change

Patient

Enter Corrected Address in Computer

Copy Insurance

Card

Instruct Patient

to Correct

Financial Data

Check Address

Insurance

Review Patient

Records

Record Message

Receive Call

Phone Intake Entry/Exit

Receptionist

Ask Address Change

Medical Assistant

Ask Address Change

Doctor

Type Chart Label

Coder

Ask Insurance Change

Figure 2–2 Physician’s Office Flowsheet for Address Confirmation.

14 CHAPTER 2 What Does the Healthcare Manager Need to Know?

correct address to receive payment. But the flow is different within two different systems. A manager must understand how the system works to understand the consequences—then good financial management can prevail.

BASIC SYSTEM ELEMENTS

To understand financial management, it is essential to decipher the reports provided to the manager. To comprehend these reports, it is helpful to understand certain basic system el- ements that are used to create the information contained in the reports.

Chart of Accounts—The Map

The chart of accounts is a map. It outlines the elements of your company in an organized manner. The chart of accounts maps out account titles with a method of numeric coding. It is designed to compile financial data in a uniform manner that the user can decode.

The groupings of accounts in the chart of accounts should match the groupings of the organization. In other words, the classification on the organization chart (as discussed in the previous chapter) should be compatible with the groupings on the chart of accounts. Thus, if there is a human resources department on your facility’s organization chart, and if

Central intake enters demographics at time

referral received

Patient Accounts Clerk generates

Patient Information Summary next day

Data downloads to CDB overnight

Information on Patient Information Summary

verified by care manager at next

visit

Correct information written in by staff

Information Correct?

Form is placed in appropriate

care manager’s mailbox

Form turned in to Patient Accounts Representative

Data in CDB’s central intake updated

by Patient Accounts Representative

Form placed in billing folder

New labels generated if necessary

YesNo

Figure 2–3 Health System Flowsheet for Verification of Patient Information.

expenses are grouped by department in your facility, then we would expect to find a human resources grouping in the chart of accounts.

The manager who is working with financial data needs to be able to read and compre- hend how the dollars are laid out and how they are gathered together, or assembled. This assembly happens through the guidance of the chart of accounts. That is why we compare it to a map.

Basic guidance for healthcare charts of accounts is set out in publications such as that of Seawell’s Chart of Accounts for Hospitals.2 However, generic guides are just that—generic. Every organization exhibits differences in its own chart of accounts that express the unique aspects of its structure. We examine three examples to illustrate these differences. Remem- ber, we are spending time on the chart of accounts because your comprehension of de- tailed financial data may well depend on whether you can decipher your facility’s own chart of accounts mapping in the information forwarded for your use.

The first format, shown in Exhibit 2-1, is a basic use, probably for a smaller organization. The exhibit is in two horizontal segments, “Structure” and “Example.” There are three parts to the account number. The first part is one digit and indicates the financial statement element. Thus, our example shows “1,” which is for “Asset.” The second part is two digits and is the primary subclassification. Our example shows “10,” which stands for “Current Asset” in this case. The third and final part is also two digits and is the secondary subclassi- fication. Our example shows “11,” which stands for “Petty Cash—Front Office” in this case. On a report, this account number would probably appear as 1-10-11.

The second format, shown in Exhibit 2-2, is full use and would be for a large organiza- tion. The exhibit is again in two horizontal segments, “Structure” and “Example,” and there

Basic System Elements 15

Exhibit 2–1 Chart of Accounts, Format 1

Structure X XX XX Financial Primary Secondary Statement Subclassification Subclassification Element

Example 1 10 11 Asset Current Petty Cash—

Asset Front Office (Financial (Primary (Secondary Statement Subclassification) Subclassification) Element)

16 CHAPTER 2 What Does the Healthcare Manager Need to Know?

are now two line items appearing in the Example section. This full-use example has five parts to the account number. The first part is two digits and indicates the entity designator number. Thus, we conclude that there is more than one entity within this system. Our ex- ample shows “10,” which stands for “Hospital A.” The second part is two digits and indicates the fund designator number. Thus, we conclude that there is more than one fund within this system. Our example shows “10,” which stands for “General Fund.”

The third part of Exhibit 2-2 is one digit and indicates the financial statement element. Thus, the first line of our example shows “4,” which is for “Revenue,” and the second line of our example shows “6,” which is for “Expense.” (The third part of this example is the first part of the simpler example shown in Exhibit 2-1.) The fourth part is four digits and is the primary subclassification. Our example shows 3125, which stands for “Lab—Microbiology.” The number 3125 appears on both lines of this example, indicating that both the revenue and the expense belong to Lab—Microbiology. (The fourth part of this example is the sec- ond part of the simpler example shown in Exhibit 2-1. The simpler example used only two digits for this part, but this full-use example uses four digits.) The fifth and final part is two digits and is the secondary subclassification. Our example shows “03” on the first line, the revenue line, which stands for “Payer: XYZ HMO” and indicates the source of the revenue. On the second line, the expense line, our example shows “10,” which stands for “Clerical Salaries.” Therefore, we understand that these are the clerical salaries belonging to Lab—

Exhibit 2–2 Chart of Accounts, Format 2

Structure XX XX X XXXX XX Entity Fund Financial Primary Secondary Designator Designator Statement Subclassification Subclassification

Element

Example 10 10 4 3125 03 Hospital General Revenue Lab—Microbiology Payer: XYZ HMO A Fund

10 10 6 3125 10 Hospital General Expense Lab—Microbiology Clerical Salaries A Fund

(Entity (Fund (Financial (Primary (Secondary Designator) Designator) Statement Subclassification) Subclassification)

Element)

Basic System Elements 17

Microbiology in Hospital A. (The fifth part of this example is the third and final part of the simpler example shown in Exhibit 2-1.) On a report, these account numbers might appear as 10-10-4-3125-03 and 10-10-6-3125-10. Another optional use that is easier to read at a glance is 10104-3125-03 and 10106-3125-10.

Because every organization is unique and because the chart of accounts reflects that uniqueness, the third format, shown in Exhibit 2-3, illustrates a customized use of the chart of accounts. This example is adapted from a large hospital system. There are four parts to its chart of accounts number. The first part is an entity designator and designates a com- pany within the hospital system. The fund designator two-digit part, as traditionally used (see Exhibit 2-2), is missing here. The financial statement element one-digit part, as tradi- tionally used (see Exhibit 2-2), is also missing here. Instead, the second part of Exhibit 2-3 represents the primary classification, which is shown as an expense category (“Payroll”) in the example line. The third part of Exhibit 2-3 is the secondary subclassification, repre- senting a labor subaccount expense designation (“Regular per-Visit RN”). The fourth and final part of Exhibit 2-3 is another subclassification that indicates the department within the company (“Home Health”). On a report for this organization, therefore, the account num- ber 21-7000-2200-7151 would indicate the home care services company’s payroll for regu- lar per-visit registered nurses (RNs) in the home health department. Finally, remember that time spent understanding your own facility’s chart of accounts will be time well spent.

Exhibit 2–3 Chart of Accounts, Format 3

Structure XX XXXX XXXX XXXX Company Expense Subaccount Department

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