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Arithmetic Review: Operations with Fractions

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. This chapter contains many examples of arithmetic techniques that are used directly or indirectly in algebra. Since the chapter is intended as a review, the problem-solving techniques are presented without being developed. Therefore, no work space is provided, nor does the chapter contain all of the pedagogical features of the text. As a review, this chapter can be assigned at the discretion of the instructor and can also be a valuable reference tool for the student.

Overview

  • Multiplication of Fractions
  • Division of Fractions
  • Addition and Subtraction of Fractions

Multiplication of Fractions

Multiplication of Fractions

To multiply two fractions, multiply the numerators together and multiply the denominators together. Reduce to lowest terms if possible.

Example 1

For example, multiply 3 4  ·  1 6 . 3 4  ·  1 6 .

3 4  ·  1 6 = 3 · 1 4 · 6 = 3 24 Now reduce. = 3 · 1 2 · 2 · 2 · 3 = 3  · 1 2 · 2 · 2 ·  3 3 is the only common factor. = 1 8 3 4  ·  1 6 = 3 · 1 4 · 6 = 3 24 Now reduce. = 3 · 1 2 · 2 · 2 · 3 = 3  · 1 2 · 2 · 2 ·  3 3 is the only common factor. = 1 8
Notice that we since had to reduce, we nearly started over again with the original two fractions. If we factor first, then cancel, then multiply, we will save time and energy and still obtain the correct product.

Sample Set A

Perform the following multiplications.

Example 2

1 4  ·  8 9 = 1 2 · 2  ·  2 · 2 · 2 3 · 3 = 1 2  ·  2  ·  2  ·  2  · 2 3 · 3 2 is a common factor. = 1 1  ·  2 3 · 3 = 1 · 2 1 · 3 · 3 = 2 9 1 4  ·  8 9 = 1 2 · 2  ·  2 · 2 · 2 3 · 3 = 1 2  ·  2  ·  2  ·  2  · 2 3 · 3 2 is a common factor. = 1 1  ·  2 3 · 3 = 1 · 2 1 · 3 · 3 = 2 9

Example 3

3 4  ·  8 9  ·  5 12 = 3 2 · 2  ·  2 · 2 · 2 3 · 3  ·  5 2 · 2 · 3 = 3 2  ·  2  ·  2  ·  2  ·  2 3  · 3  ·  5 2  · 2 · 3 2 and 3 are common factors. = 1 · 1 · 5 3 · 2 · 3 = 5 18 3 4  ·  8 9  ·  5 12 = 3 2 · 2  ·  2 · 2 · 2 3 · 3  ·  5 2 · 2 · 3 = 3 2  ·  2  ·  2  ·  2  ·  2 3  · 3  ·  5 2  · 2 · 3 2 and 3 are common factors. = 1 · 1 · 5 3 · 2 · 3 = 5 18

Division of Fractions

Reciprocals

Two numbers whose product is 1 are reciprocals of each other. For example, since 4 5  ·  5 4 =1, 4 5 4 5  ·  5 4 =1, 4 5 and 5 4 5 4 are reciprocals of each other. Some other pairs of reciprocals are listed below.

 2 7 , 7 2 3 4 , 4 3 6 1 , 1 6 2 7 , 7 2 3 4 , 4 3 6 1 , 1 6

Reciprocals are used in division of fractions.

Division of Fractions

To divide a first fraction by a second fraction, multiply the first fraction by the reciprocal of the second fraction. Reduce if possible.

This method is sometimes called the “invert and multiply” method.

Sample Set B

Perform the following divisions.

Example 4

1 3 ÷ 3 4 . The divisor is  3 4 . Its reciprocal is  4 3 . 1 3 ÷ 3 4 = 1 3  ·  4 3 = 1 · 4 3 · 3 = 4 9 1 3 ÷ 3 4 . The divisor is  3 4 . Its reciprocal is  4 3 . 1 3 ÷ 3 4 = 1 3  ·  4 3 = 1 · 4 3 · 3 = 4 9

Example 5

3 8 ÷ 5 4 . The divisor is  5 4 . Its reciprocal is  4 5 . 3 8 ÷ 5 4 = 3 8  ·  4 5 = 3 2 · 2 · 2  ·  2 · 2 5 = 3 2  ·  2  · 2  ·  2  ·  2 5 2 is a common factor. = 3 · 1 2 · 5 = 3 10 3 8 ÷ 5 4 . The divisor is  5 4 . Its reciprocal is  4 5 . 3 8 ÷ 5 4 = 3 8  ·  4 5 = 3 2 · 2 · 2  ·  2 · 2 5 = 3 2  ·  2  · 2  ·  2  ·  2 5 2 is a common factor. = 3 · 1 2 · 5 = 3 10

Example 6

5 6 ÷ 5 12 . The divisor is  5 12 . Its reciprocal is  12 5 . 5 6 ÷ 5 12 = 5 6  ·  12 5 = 5 2 · 3  ·  2 · 2 · 3 5 = 5 2  ·  3  ·  2  · 2 ·  3 5 = 1 · 2 1 = 2 5 6 ÷ 5 12 . The divisor is  5 12 . Its reciprocal is  12 5 . 5 6 ÷ 5 12 = 5 6  ·  12 5 = 5 2 · 3  ·  2 · 2 · 3 5 = 5 2  ·  3  ·  2  · 2 ·  3 5 = 1 · 2 1 = 2

Addition and Subtraction of Fractions

Fractions with Like Denominators

To add (or subtract) two or more fractions that have the same denominators, add (or subtract) the numerators and place the resulting sum over the common denominator. Reduce if possible.

CAUTION

Add or subtract only the numerators. Do not add or subtract the denominators!

Sample Set C

Find the following sums.

Example 7

3 7 + 2 7 . The denominators are the same. Add the numerators and place the sum over 7. 3 7 + 2 7 = 3+2 7 = 5 7 3 7 + 2 7 . The denominators are the same. Add the numerators and place the sum over 7. 3 7 + 2 7 = 3+2 7 = 5 7

Example 8

7 9 − 4 9 . The denominators are the same. Subtract 4 from 7 and place the difference over 9. 7 9 − 4 9 = 7−4 9 = 3 9 = 1 3 7 9 − 4 9 . The denominators are the same. Subtract 4 from 7 and place the difference over 9. 7 9 − 4 9 = 7−4 9 = 3 9 = 1 3

Fractions can only be added or subtracted conveniently if they have like denominators.

Fractions with Unlike Denominators

To add or subtract fractions having unlike denominators, convert each fraction to an equivalent fraction having as the denominator the least common multiple of the original denominators.

The least common multiple of the original denominators is commonly referred to as the least common denominator (LCD). See Section ((Reference)) for the technique of finding the least common multiple of several numbers.

Sample Set D

Find each sum or difference.

Example 9

1 6 + 3 4 . The denominators are not alike. Find the LCD of 6 and 4. { 6=2 · 3 4= 2 2 The LCD is  2 2  · 3=4 · 3=12. Convert each of the original fractions to equivalent fractions having the common denominator 12. 1 6 = 1 · 2 6 · 2 = 2 12 3 4 = 3 · 3 4 · 3 = 9 12 Now we can proceed with the addition. 1 6 + 3 4 = 2 12 + 9 12 = 2+9 12 = 11 12 1 6 + 3 4 . The denominators are not alike. Find the LCD of 6 and 4. { 6=2 · 3 4= 2 2 The LCD is  2 2  · 3=4 · 3=12. Convert each of the original fractions to equivalent fractions having the common denominator 12. 1 6 = 1 · 2 6 · 2 = 2 12 3 4 = 3 · 3 4 · 3 = 9 12 Now we can proceed with the addition. 1 6 + 3 4 = 2 12 + 9 12 = 2+9 12 = 11 12

Example 10

5 9 − 5 12 . The denominators are not alike. Find the LCD of 9 and 12. { 9= 3 2 12= 2 2  · 3 The LCD is  2 2  ·  3 2 =4 · 9=36. Convert each of the original fractions to equivalent fractions having the common denominator 36. 5 9 = 5 · 4 9 · 4 = 20 36 5 12 = 5 · 3 12 · 3 = 15 36 Now we can proceed with the subtraction. 5 9 − 5 12 = 20 36 − 15 36 = 20−15 36 = 5 36 5 9 − 5 12 . The denominators are not alike. Find the LCD of 9 and 12. { 9= 3 2 12= 2 2  · 3 The LCD is  2 2  ·  3 2 =4 · 9=36. Convert each of the original fractions to equivalent fractions having the common denominator 36. 5 9 = 5 · 4 9 · 4 = 20 36 5 12 = 5 · 3 12 · 3 = 15 36 Now we can proceed with the subtraction. 5 9 − 5 12 = 20 36 − 15 36 = 20−15 36 = 5 36

Exercises

For the following problems, perform each indicated operation.

Exercise 1

1 3  ·  4 3 1 3  ·  4 3

Solution

4 9 4 9

Exercise 2

1 3  ·  2 3 1 3  ·  2 3

Exercise 3

2 5  ·  5 6 2 5  ·  5 6

Solution

1 3 1 3

Exercise 4

5 6  ·  14 15 5 6  ·  14 15

Exercise 5

9 16  ·  20 27 9 16  ·  20 27

Solution

5 12 5 12

Exercise 6

35 36  ·  48 55 35 36  ·  48 55

Exercise 7

21 25  ·  15 14 21 25  ·  15 14

Solution

9 10 9 10

Exercise 8

76 99  ·  66 38 76 99  ·  66 38

Exercise 9

3 7  ·  14 18  ·  6 2 3 7  ·  14 18  ·  6 2

Solution

1

Exercise 10

14 15  ·  21 28  ·  45 7 14 15  ·  21 28  ·  45 7

Exercise 11

5 9 ÷ 5 6 5 9 ÷ 5 6

Solution

2 3 2 3

Exercise 12

9 16 ÷ 15 8 9 16 ÷ 15 8

Exercise 13

4 9 ÷ 6 15 4 9 ÷ 6 15

Solution

10 9 10 9

Exercise 14

25 49 ÷ 4 9 25 49 ÷ 4 9

Exercise 15

15 4 ÷ 27 8 15 4 ÷ 27 8

Solution

10 9 10 9

Exercise 16

24 75 ÷ 8 15 24 75 ÷ 8 15

Exercise 17

57 8 ÷ 7 8 57 8 ÷ 7 8

Solution

57 7 57 7

Exercise 18

7 10 ÷ 10 7 7 10 ÷ 10 7

Exercise 19

3 8 + 2 8 3 8 + 2 8

Solution

5 8 5 8

Exercise 20

3 11 + 4 11 3 11 + 4 11

Exercise 21

5 12 + 7 12 5 12 + 7 12

Solution

1

Exercise 22

11 16 − 2 16 11 16 − 2 16

Exercise 23

15 23 − 2 23 15 23 − 2 23

Solution

13 23 13 23

Exercise 24

3 11 + 1 11 + 5 11 3 11 + 1 11 + 5 11

Exercise 25

16 20 + 1 20 + 2 20 16 20 + 1 20 + 2 20

Solution

19 20 19 20

Exercise 26

3 8 + 2 8 − 1 8 3 8 + 2 8 − 1 8

Exercise 27

11 16 + 9 16 − 5 16 11 16 + 9 16 − 5 16

Solution

15 16 15 16

Exercise 28

1 2 + 1 6 1 2 + 1 6

Exercise 29

1 8 + 1 2 1 8 + 1 2

Solution

5 8 5 8

Exercise 30

3 4 + 1 3 3 4 + 1 3

Exercise 31

5 8 + 2 3 5 8 + 2 3

Solution

31 24 31 24

Exercise 32

6 7 − 1 4 6 7 − 1 4

Exercise 33

8 15 − 3 10 8 15 − 3 10

Solution

5 6 5 6

Exercise 34

1 15 + 5 12 1 15 + 5 12

Exercise 35

25 36 − 7 10 25 36 − 7 10

Solution

−1 180 −1 180

Exercise 36

9 28 − 4 45 9 28 − 4 45

Exercise 37

8 15 − 3 10 8 15 − 3 10

Solution

7 30 7 30

Exercise 38

1 16 + 3 4 − 3 8 1 16 + 3 4 − 3 8

Exercise 39

8 3 − 1 4 + 7 36 8 3 − 1 4 + 7 36

Solution

47 18 47 18

Exercise 40

3 4 − 3 22 + 5 24 3 4 − 3 22 + 5 24

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