93.ART CLASSES Students in a painting class must pay an extra art supplies fee. On the first day of class, the instructor collected $28 in fees from several students. On the second day she collected $21 more from some different students, and on the third day she collected an additional $63 from other students.

a.What is the most the art supplies fee could cost a student?

a.Determine how many students paid the art supplies fee each day.

94.SHIPPING A toy manufacturer needs to ship 135 brown teddy bears, 105 black teddy bears, and

30 white teddy bears. They can pack only one type of teddy bear in each box, and they must pack the same number of teddy bears in each box. What is the greatest number of teddy bears they can pack in each box?

WRITING

95.Explain how to find the LCM of 8 and 28 using prime factorization.

96.Explain how to find the GCF of 8 and 28 using prime factorization.

97.The prime factorization of 12 is 2 2 3 and the prime factorization of 15 is 3 5. Explain why the LCM of 12 and 15 is not 2 2 3 3 5.

98.How can you tell by looking at the prime factorizations of two whole numbers that their GCF is 1?

REVIEW

Perform each operation.

S E C T I O N 1.9

Order of Operations

Recall that numbers are combined with the operations of addition, subtraction, multiplication, and division to create expressions. We often have to evaluate (find the value of) expressions that involve more than one operation. In this section, we introduce an order-of-operations rule to follow in such cases.

1 Use the order of operations rule.

Suppose you are asked to contact a friend if you see a Rolex watch for sale while you are traveling in Europe. While in Switzerland, you find the watch and send the following text message, shown on the left. The next day, you get the response shown on the right from your friend.

Objectives

1Use the order of operations rule.

2Evaluate expressions containing grouping symbols.

3Find the mean (average) of a set of values.

Self Check 1

Evaluate: 4 33 6

Now Try Problem 19

Something is wrong. The first part of the response (No price too high!) says to buy the watch at any price. The second part (No! Price too high.) says not to buy it, because it’s too expensive. The placement of the exclamation point makes us read the two parts of the response differently, resulting in different meanings. When reading a mathematical statement, the same kind of confusion is possible. For example, consider the expression

2 3 6

We can evaluate this expression in two ways. We can add first, and then multiply. Or we can multiply first, and then add. However, the results are different.

If we don’t establish a uniform order of operations, the expression has two different values. To avoid this possibility, we will always use the following order of operations rule.

Order of Operations

1.Perform all calculations within parentheses and other grouping symbols following the order listed in Steps 2–4 below, working from the innermost pair of grouping symbols to the outermost pair.

2.Evaluate all exponential expressions.

3.Perform all multiplications and divisions as they occur from left to right.

4.Perform all additions and subtractions as they occur from left to right. When grouping symbols have been removed, repeat Steps 2–4 to complete the calculation.

If a fraction bar is present, evaluate the expression above the bar (called the numerator) and the expression below the bar (called the denominator) separately. Then perform the division indicated by the fraction bar, if possible.

It isn’t necessary to apply all of these steps in every problem. For example, the expression 2 3 6 does not contain any parentheses, and there are no exponential expressions. So we look for multiplications and divisions to perform and proceed as follows:

2 3 6 2 18 Do the multiplication first.

Strategy We will scan the expression to determine what operations need to be performed. Then we will perform those operations, one at a time, following the order of operations rule.

WHY If we don’t follow the correct order of operations, the expression can have more than one value.

Solution

WHY The expression does not contain any parentheses, nor are there any exponents.

Solution

We will write the steps of the solution in horizontal form.

Caution! In Example 2, a common mistake is to forget to work from left to right and incorrectly perform the addition before the subtraction. This error produces the wrong answer, 58.

80 3 2 16 80 6 16

80 22

58

Remember to perform additions and subtractions in the order in which they occur. The same is true for multiplications and divisions.

EXAMPLE 3 Evaluate: 192 6 5(3)2

Strategy We will perform the division first.

WHY Although the expression contains parentheses, there are no calculations to perform within them. Since there are no exponents, we perform multiplications and divisions as they are occur from left to right.

Solution

We will write the steps of the solution in horizontal form.

We will use the five-step problem solving strategy introduced in Section 1.6 and the order of opertions rule to solve the following application problem.

Success Tip Calculations that you cannot perform in your head should be shown outside the steps of your solution.

Self Check 2

Evaluate: 60 2 3 22

Now Try Problem 23

Self Check 3

Evaluate: 144 9 4(2)3

Now Try Problem 27

104 Chapter 1 Whole Numbers

Self Check 4

LONG-DISTANCE CALLS A newspaper reporter in Chicago made a 90-minute call to Afghanistan, a 25-minute call to Haiti, and a 55-minute call to Russia. What was the total cost of the calls?

Now Try Problem 105

Form We translate the words of the problem to numbers and symbols. Since the word per indicates multiplication, we can find the cost of each call by multiplying the length of the call (in minutes) by the rate charged per minute (in cents). Since the word total indicates addition, we will add to find the total cost of the calls.

Solve To evaluate this expression (which involves multiplication and addition), we apply the order of operations rule.

State The total cost of the overseas calls is 1,800¢, or $18.00.

Check We can check the result by finding an estimate using front-end rounding. The total cost of the calls is approximately 60(2¢) 50(10¢) 30(40¢) 120¢ 500¢ 1,200¢ or 1,820¢. The result of 1,800¢ seems reasonable.

2 Evaluate expressions containing grouping symbols.

Grouping symbols determine the order in which an expression is to be evaluated. Examples of grouping symbols are parentheses ( ), brackets [ ], braces { }, and the fraction bar .

Self Check 5

Evaluate each expression:

a.20 7 6

b.20 (7 6)

Now Try Problem 33

EXAMPLE 5 Evaluate each expression: a. 12 3 5 b. 12 (3 5)

Strategy To evaluate the expression in part a, we will perform the subtraction first. To evaluate the expression in part b, we will perform the addition first.

WHY The similar-looking expression in part b is evaluated in a different order because it contains parentheses. Any operations within parentheses must be performed first.

Solution

Strategy We will perform the multiplication within the parentheses first.

WHY When there is more than one operation to perform within parentheses, we follow the order of operations rule. Multiplication is to be performed before subtraction.

Self Check 7

Evaluate: 50 4(12 5 2)

Now Try Problem 39

Self Check 8

Evaluate:

130 7[22 3(6 2)]

Now Try Problem 43

Self Check 9

3(14) 6

Evaluate:

2(32)

Now Try Problem 47

If an expression contains more than one pair of grouping symbols, we always begin by working within the innermost pair and then work to the outermost pair.

Innermost parentheses

16 6[42 3(5 2)]

Outermost brackets

The Language of Mathematics Multiplication is indicated when a number is next to a parenthesis or a bracket. For example,

16 6[42 3(5 2)]

Multiplication Multiplication

EXAMPLE 8 Evaluate: 16 6[42 3(5 2)]

Strategy We will work within the parentheses first and then within the brackets. Within each set of grouping symbols, we will follow the order of operations rule.

WHY By the order of operations, we must work from the innermost pair of grouping symbols to the outermost.

Solution

Caution! In Example 8, a common mistake is to incorrectly add 16 and 6 instead of correctly multiplying 6 and 7 first. This error produces a wrong answer, 154.

16 6[42 3(5 2)] 16 6[42 3(3)]

16 6[16 3(3)]

16 6[16 9]

16 6[7]

22[7]

154

Evaluate:

3(23)

Strategy We will evaluate the expression above and the expression below the fraction bar separately. Then we will do the indicated division, if possible.

WHY Fraction bars are grouping symbols. They group the numerator and denominator. The expression could be written [2(13) 2)] [3(23)].

Self Check 10

NFL DEFENSIVE LINEMEN The weights of the 2008–2009 New York Giants starting defensive linemen were 273 lb, 305 lb,

317 lb, and 265 lb. What was their mean (average) weight? (Source: nfl.com/New York Giants depth chart)

Now Try Problems 51 and 113