Basic Mathematics for College Students

If an expression contains grouping symbols, we perform the operations within the
grouping symbols first.

3.7 Order of Operations and Complex Fractions 285

Self Check 2

Evaluate:

Now TryProblem 19

a

19

21



2

3

ba 2

1

7

b

Add the numerators: 18 (5) 13. Write

 (^) the sum over the common denominator 24.

13

24

Multiply the numerators: 3  6 18.

 (^) Multiply the denominators: 4  6 24.

18

24

a

5

24

b

EXAMPLE 2

Evaluate:

Strategy We will perform any operations within parentheses first.

WHY This is the first step of the order of operations rule.

Solution

We will begin by performing the subtraction within the first set of parentheses. The
second set of parentheses does not contain an operation to perform.

Within the first set of parentheses, prepare to

subtract the fractions: Their LCD is 8. Build so that

its denominator is 8.

Write the mixed number as an improper fraction.

Use the rule for division of fractions:

Multiply the first fraction by the reciprocal of.

Multiply the remaining factors in the numerator.

  (^) Multipy the remaining factors in the denominator.

2

7

To simplify, factor 16 as 2 8 and factor 35 as 5 7.

Remove the common factors of 5 and 8 from the

numerator and denominator.

 

5

1

 2  8

1

8

1

 5

1

 7

Multiply the numerators and multiply the denominators.

The product of two fractions with unlike signs is negative.

 5 ^16

8  35

 (^3516)



5

8

a

16

35

b



5

8

a

35

16

b

Subtract the numerators: 7  2 5.

Write the difference over the common denominator 8.



5

8

a 2

3

16

b

Multiply the numerators: 1  2 2.

Multiply the denominators: 4  2 8.

 a

7

8



2

8

ba 2

3

16

b

1

a 4

7

8



1

4



2

2

ba 2

3

16

b

a

7

8



1

4

ba 2

3

16

b

a

7

8



1

4

ba 2

3

16

b

EXAMPLE 3

Add to the difference of and.

Strategy We will translate the words of the problem to numbers and symbols.
Then we will use the order of operations rule to evaluate the resulting expression.

WHY Since the expression involves two operations, addition and subtraction, we
need to perform them in the proper order.

1

4

5

6

7

1

3

Self Check 3

Add to the difference of

and.

Now TryProblem 23

2

3

7

(^2 8)
1
4