Study CommonsRead · reason · practise

Home / Math / Divide Fractions: Meaning, Reciprocals, Mixed Numbers, Estimate, Check

Math · Fractions

Divide Fractions: Meaning, Reciprocals, Mixed Numbers, Estimate, Check

Move from fraction multiplication into division: interpret how-many-groups situations, use reciprocals with meaning, divide mixed numbers, estimate size and check by multiplication.

Grades 5–6 · 40 practice questions · Original material
Learning objectiveInterpret fraction division, divide whole numbers and fractions by fractions, convert mixed numbers when needed, explain why reciprocal multiplication works, estimate quotient size, and check division with multiplication.

The idea

Division can ask how many groups fit or how much is in each group. For 3 ÷ 1/2, the question is how many halves fit in 3, so the answer is 6.

Dividing by a fraction can be rewritten as multiplying by its reciprocal because dividing by a/b asks how many a/b-sized groups fit. Multiplying by b/a reverses the scaling.

Estimate first. Dividing a positive number by a proper fraction should make the result larger, while dividing by a number greater than 1 should make the result smaller.

Worked examples

Example 1 · How many groups?

Find 3 ÷ 1/2.

  1. Ask how many half-units fit inside 3 whole units.
  2. Each whole contains two halves, so 3 wholes contain 6 halves.
  3. Check: 6 × 1/2 = 3.

3 ÷ 1/2 = 6.

Example 2 · Fraction ÷ unit fraction

Find 3/4 ÷ 1/8.

  1. Rewrite fourths in eighths: 3/4 = 6/8.
  2. Six groups of 1/8 fit in 6/8.
  3. Check: 6 × 1/8 = 3/4.

3/4 ÷ 1/8 = 6.

Example 3 · Fraction ÷ fraction

Find 2/3 ÷ 4/5.

  1. Multiply by the reciprocal: 2/3 × 5/4.
  2. Simplify 10/12 to 5/6.
  3. Because the divisor is less than 1, a quotient a little larger than 2/3 is reasonable.

2/3 ÷ 4/5 = 5/6.

Example 4 · Mixed number

Find 2 1/4 ÷ 3/5.

  1. Convert 2 1/4 to 9/4.
  2. Multiply by the reciprocal of 3/5: 9/4 × 5/3.
  3. Simplify to 15/4 = 3 3/4.

2 1/4 ÷ 3/5 = 3 3/4.

Example 5 · Divide by a whole number

Find 4/5 ÷ 2.

  1. Write 2 as 2/1.
  2. Multiply 4/5 × 1/2 = 4/10.
  3. Simplify to 2/5.

4/5 ÷ 2 = 2/5.

Example 6 · Missing value

x ÷ 3/4 = 8. Find x.

  1. If x is split into groups of size 3/4, there are 8 groups.
  2. Multiply 8 × 3/4 to reconstruct the whole amount.
  3. x = 6; check: 6 ÷ 3/4 = 8.

x = 6.

Common mistakes

These are tempting because they use familiar operations. Check what the question is actually measuring.

Flipping the first fraction instead of the divisor.

Why this fails: Keep the dividend. Replace division by a/b with multiplication by b/a.

Assuming division always makes a number smaller.

Why this fails: Dividing by a proper fraction counts small groups, so the quotient can be larger than the dividend.

Using the reciprocal rule without checking meaning.

Why this fails: A quick group model or multiplication check makes the rule interpretable instead of a symbol trick.

Leaving mixed numbers unchanged when multiplying by a reciprocal.

Why this fails: Convert mixed numbers to improper fractions before multiplying.

Cancelling across addition or mixed-number notation.

Why this fails: Cancel only common factors in a multiplication expression, never terms joined by addition.

Foundation

Foundation practice

Build the meaning of how-many-groups division and simple reciprocal calculations.

Compute

1. 3/4 ÷ 1/4

Compute

2. 2/3 ÷ 1/3

Compute

3. 5 ÷ 1/2

Compute

4. 3 ÷ 3/4

Compute

5. 4/5 ÷ 2

Compute

6. 7/8 ÷ 7

Compute

7. 2/3 ÷ 4/5

Compute

8. 3/4 ÷ 2/3

Compute

9. 5/6 ÷ 5/12

Compute

10. 7/9 ÷ 14/27

Core

Core practice

Divide fractions and mixed numbers efficiently, then check by multiplication.

Compute

11. 4/7 ÷ 2/3

Compute

12. 5/8 ÷ 15/16

Compute

13. 9/10 ÷ 3/5

Compute

14. 11/12 ÷ 22/9

Mixed number

15. 1 1/2 ÷ 3/4

Mixed number

16. 2 1/4 ÷ 3/5

Mixed numbers

17. 3 1/3 ÷ 1 2/3

Mixed numbers

18. 4 1/2 ÷ 1 1/8

Mixed divisor

19. 2/5 ÷ 1 1/5

Mixed numbers

20. 1 3/4 ÷ 2 1/3

Think

Think practice

Estimate quotient size, analyse errors and interpret what division is asking.

Estimate + exact

21. Estimate 5/6 ÷ 1/4, then find the exact quotient.

Reasonableness

22. Without calculating first, should 3/8 ÷ 7/8 be less than or greater than 1?

Error analysis

23. A student says 2/3 ÷ 4/5 = 8/15. Explain the error.

Error analysis

24. A student says 3 ÷ 1/2 = 1 1/2 because division makes numbers smaller. What is wrong?

Missing value

25. x ÷ 3/4 = 8. Find x.

Missing divisor

26. 5/6 ÷ x = 5/9. Find x.

Compare

27. Which is greater: 3/4 ÷ 1/2 or 3/4 × 1/2?

Compare

28. Is 2/3 ÷ 4/3 greater than or less than 2/3?

Measurement

29. A 2 1/2 m ribbon is cut into pieces 1/4 m long. How many pieces are made?

Generalize

30. Why can dividing a positive number by a proper fraction make the answer larger?

Challenge

Challenge practice

Use fraction division in equations, measurements and multi-step reasoning.

Compute

31. 7/8 ÷ 14/15

Compute

32. 12/13 ÷ 8/39

Mixed numbers

33. 3 3/4 ÷ 1 1/2

Mixed numbers

34. 5 1/4 ÷ 2 1/3

Equation

35. x ÷ 2/3 = 12. Find x.

Equation

36. 2 1/4 ÷ x = 3. Find x.

Recipe

37. There are 3 1/2 cups of soup. How many 3/4-cup servings is that?

Measurement

38. A 5 1/4 m rope is cut into 7 equal pieces. How long is each piece?

Multi-step

39. A 6 kg bag of flour has 2/3 used. The remainder is packed into 1/4 kg bags. How many bags are filled?

Generalize

40. Explain why a/b ÷ c/d equals a/b × d/c when c and d are nonzero.

Answer key

  1. 1. 3
    Three fourths contains three groups of one fourth.
  2. 2. 2
    Two thirds contains two groups of one third.
  3. 3. 10
    Each whole contains two halves, so 5 wholes contain 10 halves.
  4. 4. 4
    3 × 4/3 = 4.
  5. 5. 2/5
    4/5 × 1/2 = 4/10 = 2/5.
  6. 6. 1/8
    7/8 × 1/7 = 1/8.
  7. 7. 5/6
    2/3 × 5/4 = 10/12 = 5/6.
  8. 8. 1 1/8
    3/4 × 3/2 = 9/8 = 1 1/8.
  9. 9. 2
    5/6 × 12/5 = 2.
  10. 10. 1 1/2
    7/9 × 27/14 = 3/2.
  11. 11. 6/7
    4/7 × 3/2 = 12/14 = 6/7.
  12. 12. 2/3
    5/8 × 16/15 = 2/3.
  13. 13. 1 1/2
    9/10 × 5/3 = 3/2.
  14. 14. 3/8
    11/12 × 9/22 = 3/8.
  15. 15. 2
    3/2 × 4/3 = 2.
  16. 16. 3 3/4
    9/4 × 5/3 = 15/4.
  17. 17. 2
    10/3 ÷ 5/3 = 2.
  18. 18. 4
    9/2 ÷ 9/8 = 4.
  19. 19. 1/3
    2/5 ÷ 6/5 = 1/3.
  20. 20. 3/4
    7/4 ÷ 7/3 = 3/4.
  21. 21. Estimate: about 4; exact: 3 1/3.
    5/6 is close to 1. Exact: 5/6 × 4 = 10/3.
  22. 22. Less than 1.
    The dividend is smaller than the divisor; exact quotient is 3/7.
  23. 23. The correct quotient is 5/6.
    8/15 is the product. Division uses the reciprocal of the divisor.
  24. 24. 3 ÷ 1/2 = 6.
    The question asks how many halves fit in 3.
  25. 25. x = 6.
    Rebuild the dividend: 8 × 3/4 = 6.
  26. 26. x = 1 1/2.
    x = (5/6) ÷ (5/9) = 3/2.
  27. 27. 3/4 ÷ 1/2.
    The quotient is 1 1/2; the product is 3/8.
  28. 28. Less than 2/3.
    Dividing by a number greater than 1 makes a positive quantity smaller; exact quotient is 1/2.
  29. 29. 10 pieces.
    2 1/2 ÷ 1/4 = 10.
  30. 30. Because the divisor names a group smaller than one whole, so many small groups can fit inside the original amount.
    For example, six half-units fit inside 3.
  31. 31. 15/16
    7/8 × 15/14 = 15/16.
  32. 32. 4 1/2
    12/13 × 39/8 = 9/2.
  33. 33. 2 1/2
    15/4 × 2/3 = 5/2.
  34. 34. 2 1/4
    21/4 × 3/7 = 9/4.
  35. 35. x = 8.
    Multiply 12 × 2/3 to reconstruct x.
  36. 36. x = 3/4.
    9/4 ÷ x = 3, so x = 9/4 ÷ 3 = 3/4.
  37. 37. 4 2/3 servings.
    7/2 ÷ 3/4 = 14/3.
  38. 38. 3/4 m.
    21/4 ÷ 7 = 3/4.
  39. 39. 8 bags.
    2/3 of 6 is 4 kg, leaving 2 kg; 2 ÷ 1/4 = 8.
  40. 40. Dividing by c/d asks how many c/d-sized groups fit; multiplying by d/c reverses that scaling.
    The reciprocal is the multiplicative inverse, so multiplying by d/c undoes multiplication by c/d.

Related practice

Multiply FractionsAdd & Subtract FractionsDivision: Estimate & Check