3.8: Dividing Common Fractions

Think over what you know about dividing:

When we divide, we take the total amount and separate (divide it) into equal parts or groups.

Remember:

[latex]8 \div 4=2[/latex]

  • 8 is the dividend
  • 4 is the divisor
  • 2 is the quotient

Example A

[latex]8 \div 4 =[/latex]

  • The total amount is 8.
  • The divisor is 4. How many groups of 4 are in 8? Yes, 2.

[latex]8 \div 4 = 2[/latex]

Example B

[latex]3 \div \dfrac{1}{2}=[/latex]

  • The total amount is 3.
  • The total amount is [latex]\dfrac{1}{2}[/latex].
  • How many [latex]\dfrac{1}{2}[/latex]‘s are in 3?
Three circles that are each cut in half.
Figure 3.8.1
  • There are 6 halves.

[latex]3 \div \dfrac{1}{2}=6[/latex]

Example C

[latex]2\div \dfrac{2}{3}[/latex]

  • The total amount is 2.
  • The total amount is [latex]\dfrac{2}{3}[/latex].
  • How many [latex]\dfrac{2}{3}[/latex]‘s are in 2?
  • Use different colours to shade in each group of two that you can find.
Two rectangles where each are split into three parts.
Figure 3.8.2

[latex]2 \div \dfrac{2}{3} = 3[/latex]

Example D

[latex]1\div \dfrac{1}{4}[/latex]

  • The total amount is 1. Divisor is [latex]\dfrac{1}{4}[/latex]
  • How many [latex]\dfrac{1}{4}[/latex]‘s in 1?
  • Draw a shape. Divide it into quarters.
  • How many [latex]\dfrac{1}{4}[/latex]‘s are there?
  • There are 4 quarters.

[latex]1 \div \dfrac{1}{4} = 4[/latex]

Example E

[latex]3\div\dfrac{3}{8} =[/latex]

  • How many [latex]\dfrac{3}{8}[/latex] in 3.
  • Use different colors to shade in each group of 3 that you can find.
Three squares that are each divided into eight parts.
Figure 3.8.2
  • Did you find 8 groups of [latex]\dfrac{3}{8}[/latex]?

[latex]3\div \dfrac{3}{8} = 8[/latex]

The division of fractions by a fraction is difficult to picture, probably because it is not often used in everyday life. Here are some everyday examples for you to think about.

Example 1: You have half a dollar. Someone asks you to change it for quarters. How many quarters are there in half a dollar?

[latex]\dfrac{3}{8}\div\dfrac{1}{4}=2[/latex]

2 quarters in one half a dollar

Examples 2: It takes [latex]\tfrac{1}{4}[/latex] hour to solve a math problem. How many problems can you solve in [latex]\tfrac{3}{4}[/latex] of an hour?

[latex]\dfrac{3}{4}\div\dfrac{1}{4}=3[/latex]

3 problems in one [latex]\tfrac{3}{4}[/latex] of an hour

Reciprocals

Dividing by a number is the same as multiplying by its reciprocal. We use reciprocals when we divide fractions. Two numbers are reciprocals if they have a product of 1.

To find the reciprocal of a fraction, turn the fraction upside down (flip it over). This is called “inverting the fraction.”

Some people remember this by thinking of reciprocals as “refliprocals”!

Table 3.8.1 shows some fractions and their reciprocals.

Table 3.8.1: Fractions and Their Reciprocals
Fraction Reciprocal
[latex]\dfrac{1}{2}[/latex] [latex]\dfrac{2}{1}[/latex]
[latex]\dfrac{3}{4}[/latex] [latex]\dfrac{4}{3}[/latex]
[latex]\dfrac{7}{8}[/latex] [latex]\dfrac{8}{7}[/latex]
[latex]\dfrac{2}{3}[/latex] [latex]\dfrac{3}{2}[/latex]
[latex]\dfrac{1}{4}[/latex] [latex]\dfrac{4}{1}[/latex]

To find the reciprocal of a whole number:

  1. Rename the whole number as a fraction with a denominator of 1.
  2. Invert the fraction.
  3. Check the reciprocal by multiplying the fraction by the reciprocal. The product will be one.

Table 3.8.2 shows some whole numbers with their reciprocals.

Table 3.8.2: Products of Whole Numbers With Fractions and Reciprocals
Whole Number Fraction Reciprocal Check
3 [latex]\dfrac{3}{1}[/latex] [latex]\dfrac{1}{3}[/latex] [latex]\dfrac{\cancel{3}1}{1}\times\dfrac{1}{\cancel{3}1}=1[/latex]
6 [latex]\dfrac{6}{1}[/latex] [latex]\dfrac{1}{6}[/latex] [latex]\dfrac{\cancel{6}1}{1}\times\dfrac{1}{\cancel{6}1}=1[/latex]
10 [latex]\dfrac{10}{1}[/latex] [latex]\dfrac{1}{10}[/latex] [latex]\dfrac{\cancel{10}1}{1}\times\dfrac{1}{\cancel{10}1}=1[/latex]

To find the reciprocal of a mixed number:

  1. Rename the mixed number as an improper fraction.
  2. Invert the fraction.

Table 3.8.3 shows some mixed numbers with their reciprocals.

Table 3.8.3: Mixed Numbers With Their Fractions and Reciprocals
Mixed Number Fraction Reciprocal Check
[latex]1\dfrac{1}{2}=[/latex] [latex]\dfrac{3}{2}[/latex] [latex]\dfrac{2}{3}[/latex] [latex]\dfrac{\cancel{3}1}{\cancel{2}1}\times\dfrac{\cancel{2}1}{\cancel{3}1}[/latex] = [latex]\dfrac{6}{6}=1[/latex]
[latex]2\dfrac{1}{3}[/latex] [latex]\dfrac{7}{3}[/latex] [latex]\dfrac{3}{7}[/latex] [latex]\dfrac{\cancel{7}1}{\cancel{3}1}\times\dfrac{\cancel{3}1}{\cancel{7}1}[/latex] = 1
[latex]4\dfrac{3}{8}[/latex] [latex]\dfrac{35}{8}[/latex] [latex]\dfrac{8}{35}[/latex] [latex]\dfrac{\cancel{35}1}{\cancel{8}1}\times\dfrac{\cancel{8}1}{\cancel{35}1}[/latex] = 1

Exercise 1

Write the reciprocal of these numbers.

Examples:

  • [latex]\dfrac{2}{5}[/latex]
    The reciprocal  is [latex]\dfrac{5}{2}[/latex]
  • [latex]5=\dfrac{5}{1}[/latex]
    The reciprocal  is [latex]\dfrac{1}{5}[/latex]
  • [latex]2\dfrac{1}{2}=\dfrac{5}{2}[/latex]
    The reciprocal  is [latex]\dfrac{2}{5}[/latex]

 

  1. [latex]\dfrac{5}{8}[/latex]
  2. [latex]\dfrac{1}{2}[/latex]
  3. 9
  4. 2
  5. [latex]2\dfrac{1}{2}=\dfrac{5}{2}[/latex]
  6. [latex]1\dfrac{1}{4}[/latex]
  7. [latex]8\dfrac{1}{3}[/latex]

Exercise 1 Answers

  1. [latex]\dfrac{8}{5}[/latex]
  2. 2
  3. [latex]\dfrac{1}{9}[/latex]
  4. [latex]\dfrac{1}{2}[/latex]
  5. [latex]\dfrac{4}{5}[/latex]
  6. [latex]\dfrac{3}{25}[/latex]

Multiplying by the Reciprocal

To divide fractions, multiply by the reciprocal of the divisor:

  1. Rewrite the division question.
    • Rename all mixed numbers as improper fractions.
    • Give any whole numbers a denominator of 1
  2. Change the ÷ sign to a × sign.
    • Invert (turn upside down) the divisor to make the reciprocal.
    • Remember: the divisor is always the number after the sign.
  3. Simplify (cancel) and then multiply to find the answer.
  4. Write the answer in the lowest terms.

Example F

[latex]\dfrac{3}{4}\div\dfrac{1}{2}=[/latex]

Step 1: No whole numbers or mixed numbers.

Step 2:

[latex]\dfrac{3}{4}\div\dfrac{1}{2}=\dfrac{3}{4}\times\dfrac{2}{1}=[/latex]

Step 3 & 4:

[latex]\dfrac{3}{\cancel{4}2}\times\dfrac{\cancel{2}1}{1}=\dfrac{3}{2} = 1\dfrac{2}{1}[/latex]

Example G

[latex]\dfrac{7}{8}\div\dfrac{1}{4}=[/latex]

Step 1: No whole numbers or mixed numbers.

Step 2:

[latex]\dfrac{7}{8}\div\dfrac{1}{4}=\dfrac{7}{8}\times\dfrac{4}{1}=[/latex]

Step 3 & 4:

[latex]\dfrac{7}{\cancel{8}2}\times\dfrac{\cancel{4}1}{1}=\dfrac{7}{2} = 3\dfrac{1}{2}[/latex]

Example H

[latex]5\div\dfrac{2}{3}=[/latex]

Step 1:

[latex]5\div\dfrac{2}{3}= \dfrac{5}{1}\div\dfrac{2}{3}=[/latex]

Step 2:

[latex]\dfrac{5}{1}\times\dfrac{3}{2}=[/latex]

Step 3 & 4:

[latex]\dfrac{5}{1}\times\dfrac{3}{2}=\dfrac{15}{2}=7\dfrac{1}{2}[/latex]

Example I

[latex]3\dfrac{1}{2}\div2\dfrac{3}{4}=[/latex]

Step 1:

[latex]3\dfrac{1}{2}\div2\dfrac{3}{4}=\dfrac{7}{2}\div\dfrac{11}{4}[/latex]

Step 2: 

[latex]\dfrac{7}{2}\times\dfrac{11}{4}=[/latex]

Step 3 & 4:

[latex]\dfrac{7}{\cancel{2}1}\times\dfrac{\cancel{4}2}{11}=\dfrac{14}{11}=1 \dfrac{3}{11}[/latex]

Exercise 2

Divide these fractions using the steps you have just learned.

Example:     [latex]\dfrac{4}{9}\div 4 = \dfrac{4}{9} \div \dfrac{4}{1} = \dfrac{4}{9} \times \dfrac{1}{4} = \dfrac{\cancel{4}1}{9} \times \dfrac{1}{\cancel{4}1} = \dfrac{1}{9}[/latex]

 

  1. [latex]\dfrac{7}{2}\div\dfrac{3}{5}=[/latex]
  2. [latex]\dfrac{5}{8}\div\dfrac{7}{16}=[/latex]
  3. [latex]\dfrac{2}{3}\div\dfrac{8}{9}=[/latex]
  4. [latex]\dfrac{1}{5}\div\dfrac{1}{2}=[/latex]
  5. [latex]\dfrac{5}{6}\div\dfrac{5}{3}=[/latex]
  6. [latex]\dfrac{1}{3}\div\dfrac{3}{8}=[/latex]
  7. [latex]\dfrac{6}{7}\div\dfrac{1}{6}=[/latex]

Exercise 2 Answers

  1. [latex]5\dfrac{5}{6}[/latex]
  2. [latex]1\dfrac{3}{7}[/latex]
  3. [latex]\dfrac{3}{4}[/latex]
  4. [latex]\dfrac{2}{5}[/latex]
  5. [latex]\dfrac{1}{2}[/latex]
  6. [latex]\dfrac{8}{9}[/latex]
  7. [latex]5\dfrac{1}{7}[/latex]

Exercise 3

If you need more practice, try a few more of these division questions. If you are not having any trouble, go on to Exercise 4, which has mixed numbers in it.

Divide these fractions using the steps you have just learned.

Example:     [latex]\dfrac{1}{2}\div\dfrac{1}{8}= \dfrac{1}{2} \times \dfrac{8}{1} = \dfrac{1}{\cancel{2}1} \times \dfrac{\cancel{8}4}{1} = \dfrac{4}{1} = 4[/latex]

 

  1. [latex]\dfrac{8}{9}\div\dfrac{3}{2}=[/latex]
  2. [latex]\dfrac{3}{4}\div\dfrac{3}{4}=[/latex]
  3. [latex]\dfrac{5}{6}\div\dfrac{3}{3}=[/latex]
  4. [latex]\dfrac{1}{3}\div\dfrac{3}{4}=[/latex]
  5. [latex]\dfrac{2}{3}\div\dfrac{1}{2}=[/latex]

Exercise 3 Answers

  1. [latex]\dfrac{16}{27}[/latex]
  2. 1
  3. [latex]1\dfrac{5}{6}[/latex]
  4. [latex]\dfrac{4}{9}[/latex]
  5. [latex]1\dfrac{1}{3}=\dfrac{4}{3}[/latex]

Exercise 4

More practice: You might want to save some of this exercise for review before a test.

  1. [latex]8\div\dfrac{1}{2}=[/latex]
  2. [latex]2\dfrac{2}{5}\div\dfrac{1}{8}=[/latex]
  3. [latex]\dfrac{1}{6}\div\dfrac{1}{5}=[/latex]
  4. [latex]\dfrac{1}{8}\div\dfrac{1}{5}=[/latex]
  5. [latex]\dfrac{3}{5}\div\dfrac{1}{4}=[/latex]
  6. [latex]2\dfrac{4}{5}\div\dfrac{1}{5}=[/latex]
  7. [latex]\dfrac{2}{5}\div\dfrac{1}{2}=[/latex]
  8. [latex]\dfrac{1}{4}\div\dfrac{2}{3}=[/latex]
  9. [latex]2\dfrac{3}{4}\div1\dfrac{7}{8}=[/latex]
  10. [latex]5\dfrac{1}{10}\div3\dfrac{3}{10}=[/latex]
  11. [latex]1\dfrac{5}{9}\div3\dfrac{1}{3}=[/latex]
  12. [latex]\dfrac{1}{2}\div\dfrac{3}{8}=[/latex]

Exercise 4 Answers

  1. 16
  2. [latex]19\dfrac{1}{5}[/latex]
  3. [latex]\dfrac{5}{6}[/latex]
  4. [latex]\dfrac{5}{8}[/latex]
  5. [latex]2\dfrac{2}{5}[/latex]
  6. 14
  7. [latex]\dfrac{4}{5}[/latex]
  8. [latex]\dfrac{3}{8}[/latex]
  9. [latex]1\dfrac{7}{15}[/latex]
  10. [latex]1\dfrac{6}{11}[/latex]
  11. [latex]\dfrac{7}{15}[/latex]
  12. [latex]1\dfrac{1}{3}[/latex]

Problems Which Use Division of Common Fractions

Look for word patterns and keywords in the division problems. Thinking about the problems using whole numbers instead of fractions may sometimes help you recognize the division pattern. Start your division equation with the dividend. The dividend is the total.

Keywords that point to division:

  • separated split cut shared
  • what is cost per…? unit pricing
  • what is distance per…? average (speed, cost, weight, time)

Exercise 5

Solve the following word problems.

  1. Every fall 3 friends get together to make antipasto. Last year, they filled [latex]4\tfrac{1}{2}[/latex] ice cream buckets with antipasto and then shared it equally. How many buckets of antipasto did each person get?
  2. A pick-up truck load of split wood is [latex]\tfrac{1}{2}[/latex] cord of wood. If you shared a full truckload of wood with a neighbour, how much of a cord of firewood would you each get?
  3. The distance from Trail, BC to Vancouver, BC is 640 km via the Crowsnest Highway. The trip can be made in [latex]7\tfrac{1}{2}[/latex] hours in good weather. What average speed must be maintained?
  4. The sweater that Janet is knitting has a complicated pattern. It takes her [latex]3\tfrac{3}{4}[/latex] hours to finish 15 rows. How long does each row take?
  5. Marian had [latex]\tfrac{12}{3}[/latex] lemon pies left, which she wanted to share equally amongst 10 people. How much of a pie will each person be given?
  6. Jack wants to cut his piece of trim for his square windows into 4 equal parts. The trim is [latex]2\tfrac{2}{5}[/latex] metres long. What will the measurement be of each piece?
  7. Tony is sewing 3 identical pairs of pants for his son’s dance performance. He bought metres of material. He uses up all of the material; how much material was used for each pair of pants?
  8. Joy has a [latex]7\tfrac{1}{4}[/latex] m long stick. She needs to split it into [latex]\tfrac{1}{3}[/latex] m pieces. How many pieces can she get? (Remember: your answer will be given with the unit of ‘pieces’ not metres!)

Exercise 5 Answers

  1. [latex]1\tfrac{1}{2}[/latex] buckets
  2. [latex]\tfrac{1}{4}[/latex] cord
  3. [latex]85\tfrac{1}{3}[/latex]km/h (85.3 km/h)
  4. [latex]\tfrac{1}{4}[/latex] hour or 15 minutes
  5. [latex]\tfrac{1}{6}[/latex] pie
  6. Each piece is [latex]\tfrac{3}{5}[/latex] metres long.
  7. He uses [latex]\tfrac{7}{9}[/latex] metre for each pair.
  8. She will get 21 pieces.

3.8: Practice Questions

  1. Divide and be sure the answers are in the lowest terms.
    1. [latex]\dfrac{3}{4}\div\dfrac{1}{4}=[/latex]
    2. [latex]\dfrac{1}{4}\div1\dfrac{1}{4}=[/latex]
    3. [latex]\dfrac{5}{8}\div\dfrac{15}{16}=[/latex]
    4. [latex]6\div\dfrac{7}{9}=[/latex]
    5. [latex]\dfrac{5}{11}\div11=[/latex]
    6. [latex]9\dfrac{3}{4}\div2=[/latex]
    7. [latex]3\div\dfrac{1}{3}=[/latex]
    8. [latex]3\dfrac{3}{7}\div2\dfrac{5}{14}=[/latex]

     

  2. Solve the following word problem.
    1. Joe is a school janitor. It takes him [latex]\tfrac{3}{4}[/latex] of an hour to clean one classroom. How many classrooms does he clean in his [latex]7\tfrac{1}{2}[/latex] hour shift?

3.8: Practice Answers

  1. Divide and be sure the answers are in the lowest terms.
    1. 3
    2. [latex]\dfrac{5}{14}[/latex]
    3. [latex]\dfrac{2}{3}[/latex]
    4. [latex]7\dfrac{5}{7}[/latex]
    5. [latex]\dfrac{5}{121}[/latex]
    6. [latex]4\dfrac{7}{8}[/latex]
    7. [latex]\dfrac{9}{13}[/latex]
    8. [latex]1\dfrac{5}{11}[/latex]

     

  2. Solve the following word problem.
    1. 10 classrooms

Attribution

All figures in this chapter are from Topic B: Dividing Common Fractions in Adult Literacy Fundamental Mathematics: Book 5 – 2nd Edition by Liz Girard, Wendy Tagami, and Leanne Caillier-Smith, via BCcampus.

This chapter has been adapted from Topic B: Dividing Common Fractions in Adult Literacy Fundamental Mathematics: Book 5 – 2nd Edition (BCcampus) by Liz Girard, Wendy Tagami, and Leanne Caillier-Smith (2023), which is under a CC BY 4.0 license.

License

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Intermediate PreAlgebra: Building Success Copyright © 2024 by Kim Tamblyn, TRU Open Press is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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