Basic Mathematics for College Students

284 Chapter 3 Fractions and Mixed Numbers

We have seen that the order of operations rule is used to evaluate expressions that

contain more than one operation. In Chapter 1, we used it to evaluate expressions

involving whole numbers, and in Chapter 2, we used it to evaluate expressions

involving integers. We will now use it to evaluate expressions involving fractions and

mixed numbers.

Use the order of operations rule.

Recall from Section 1.9 that if we don’t establish a uniform order of operations, an

expression can have more than one value. To avoid this possibility, we must always use

the following 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.

1

SECTION 3.7

Order of Operations and Complex Fractions

Objectives

1 Use the order of operations

rule.

2 Solve application problems by

using the order of operations rule.

3 Evaluate formulas.

4 Simplify complex fractions.

EXAMPLE 1

Evaluate:

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

Although the expression contains parentheses, there are no calculations to perform

withinthem. We will begin with step 2 of the rule: Evaluate all exponential

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

Evaluate:.

Multiply:.

Prepare to add the fractions: Their LCD

is 24. To build the first fraction so that its

denominator is 24, multiply it by a form of 1.



3

4



6

6

a

5

24

b

5

31 ^

1

82 ^

5  1

3  8 ^

5

 (^24)

3

4

a

5

24

b

1  212

3

(^)  1  1221  2121  212  81

3

4



5

3

a

1

2

b

3



3

4



5

3

a

1

8

b

3

4



5

3

a

1

2

b

Self Check 1 3

Evaluate:

Now TryProblem 15

7

8



3

2

a

1

4

b

2