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4.5: Multiply and Divide Mixed Numbers and Complex Fractions (Part 1)

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Learning Objectives
  • Multiply and divide mixed numbers
  • Translate phrases to expressions with fractions
  • Simplify complex fractions
  • Simplify expressions written with a fraction bar
be prepared!

Before you get started, take this readiness quiz.

  1. Divide and reduce, if possible: (4+5)÷(107). If you missed this problem, review Example 3.2.8.
  2. Multiply and write the answer in simplified form: 1823. If you missed this problem, review Example 4.2.7.
  3. Convert 235 into an improper fraction. If you missed this problem, review Example 4.1.11.

Multiply and Divide Mixed Numbers

In the previous section, you learned how to multiply and divide fractions. All of the examples there used either proper or improper fractions. What happens when you are asked to multiply or divide mixed numbers? Remember that we can convert a mixed number to an improper fraction. And you learned how to do that in Visualize Fractions.

Example 4.5.1: multiply

Multiply: 31358

Solution

Convert 313 to an improper fraction. 10358
Multiply. 10538
Look for common factors. 255324
Remove common factors. 5534
Simplify. 2512

Notice that we left the answer as an improper fraction, 2512, and did not convert it to a mixed number. In algebra, it is preferable to write answers as improper fractions instead of mixed numbers. This avoids any possible confusion between 2112 and 2112.

Exercise 4.5.1

Multiply, and write your answer in simplified form: 523617.

Answer

2

Exercise 4.5.2

Multiply, and write your answer in simplified form: 37514.

Answer

94

HOW TO: MULTIPLY OR DIVIDE MIXED NUMBERS

Step 1. Convert the mixed numbers to improper fractions.

Step 2. Follow the rules for fraction multiplication or division.

Step 3. Simplify if possible.

Example 4.5.2:

Multiply, and write your answer in simplified form: 245(178).

Solution

Convert mixed numbers to improper fractions. 145(178)
Multiply. 141558
Look for common factors. 2753524
Remove common factors. 734
Simplify. 214
Exercise 4.5.3

Multiply, and write your answer in simplified form. 557(258).

Answer

15

Exercise 4.5.4

Multiply, and write your answer in simplified form. 325416.

Answer

856

Example 4.5.3: divide

Divide, and write your answer in simplified form: 347÷5.

Solution

Convert mixed numbers to improper fractions. 257÷51
Multiply the first fraction by the reciprocal of the second. 25715
Multiply. 25175
Look for common factors. 55175
Remove common factors. 517
Simplify. 57
Exercise 4.5.5

Divide, and write your answer in simplified form: 438÷7.

Answer

58

Exercise 4.5.6

Divide, and write your answer in simplified form: 258÷3.

Answer

78

Example 4.5.4: divide

Divide: 212÷114.

Solution

Convert mixed numbers to improper fractions. 52÷54
Multiply the first fraction by the reciprocal of the second. 5245
Multiply. 5425
Look for common factors. 522215
Remove common factors. 21
Simplify. 2
Exercise 4.5.7

Divide, and write your answer in simplified form: 223÷113.

Answer

2

Exercise 4.5.8

Divide, and write your answer in simplified form: 334÷112.

Answer

52

Translate Phrases to Expressions with Fractions

The words quotient and ratio are often used to describe fractions. In Subtract Whole Numbers, we defined quotient as the result of division. The quotient of a and b is the result you get from dividing a by b, or ab. Let’s practice translating some phrases into algebraic expressions using these terms.

Example 4.5.5: translate

Translate the phrase into an algebraic expression: “the quotient of 3x and 8.”

Solution

The keyword is quotient; it tells us that the operation is division. Look for the words of and and to find the numbers to divide.

The quotient of 3x and 8.

This tells us that we need to divide 3x by 8. 3x8

Exercise 4.5.9

Translate the phrase into an algebraic expression: the quotient of 9s and 14.

Answer

9s14

Exercise 4.5.10

Translate the phrase into an algebraic expression: the quotient of 5y and 6.

Answer

5y6

Example 4.5.6:

Translate the phrase into an algebraic expression: the quotient of the difference of m and n, and p.

Solution

We are looking for the quotient of the difference of m and n, and p. This means we want to divide the difference of m and n by p.

mnp

Exercise 4.5.11

Translate the phrase into an algebraic expression: the quotient of the difference of a and b, and cd.

Answer

abcd

Exercise 4.5.12

Translate the phrase into an algebraic expression: the quotient of the sum of p and q, and r.

Answer

p+qr

Simplify Complex Fractions

Our work with fractions so far has included proper fractions, improper fractions, and mixed numbers. Another kind of fraction is called complex fraction, which is a fraction in which the numerator or the denominator contains a fraction. Some examples of complex fractions are:

6733458x256

To simplify a complex fraction, remember that the fraction bar means division. So the complex fraction 3458 can be written as 34÷58.

Example 4.5.7: simplify

Simplify: 3458.

Solution

Rewrite as division. 34÷58
Multiply the first fraction by the reciprocal of the second. 3485
Multiply. 3845
Look for common factors. 34245
Remove common factors and simplify. 65
Exercise 4.5.13

Simplify: 2356.

Answer

45

Exercise 4.5.14

Simplify: 37611.

Answer

1114

HOW TO: SIMPLIFY A COMPLEX FRACTION

Step 1. Rewrite the complex fraction as a division problem.

Step 2. Follow the rules for dividing fractions.

Step 3. Simplify if possible.

Example 4.5.8: simplify

Simplify: 673.

Solution

Rewrite as division. 67÷3
Multiply the first fraction by the reciprocal of the second. 6713
Multiply; the product will be negative. 6173
Look for common factors. 32173
Remove common factors and simplify. 27
Exercise 4.5.15

Simplify: 874.

Answer

27

Exercise 4.5.16

Simplify: 3910.

Answer

103

Example 4.5.9: simplify

Simplify: x2xy6.

Solution

Rewrite as division. x2÷xy6
Multiply the first fraction by the reciprocal of the second. x26xy
Multiply. x62xy
Look for common factors. x322xy
Remove common factors and simplify. 3y
Exercise 4.5.17

Simplify: a8ab6.

Answer

34b

Exercise 4.5.18

Simplify: p2pq8.

Answer

4q

Example 4.5.10: simplify

Simplify: 23418.

Solution

Rewrite as division. 234÷18
Change the mixed number to an improper fraction. 114÷18
Multiply the first fraction by the reciprocal of the second. 11481
Multiply. 11841
Look for common factors. 114241
Remove common factors and simplify. 22
Exercise 4.5.19

Simplify: 57125.

Answer

2549

Exercise 4.5.20

Simplify: 85315.

Answer

12

Contributors and Attributions


This page titled 4.5: Multiply and Divide Mixed Numbers and Complex Fractions (Part 1) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax.

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