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Glacius
FoundationsConcept reference

Dividing fractions

Division by a nonzero fraction is multiplication by that fraction’s reciprocal.

1 min read1 figureSources & further reading
On this page 8 sections
  1. Overview
  2. Dividing asks how many divisor-sized amounts fit
  3. The divisor must be nonzero
  4. Check your understanding
  5. Key takeaway
  6. Further questions
  7. Sources & further reading
  8. Concept connections

01Dividing asks how many divisor-sized amounts fit#

Dividing asks how many divisor-sized amounts fit. Three quarters contains three quarter-sized pieces, so 34÷14=3\dfrac{3}{4}\div\dfrac{1}{4}=3.

Three quarters contains three pieces of size one quarter, illustrating three quarters divided by one quarter equals three.Three quarters contains three pieces of size one quarter, illustrating three quarters divided by one quarter equals three.
Figure 1Three quarters contains three pieces of size one quarter, illustrating three quarters divided by one quarter equals three.
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The reciprocal of a nonzero fraction swaps its numerator and denominator. A fraction times its reciprocal equals 11. This makes the reciprocal undo multiplication by that fraction.

23×32=1\frac{2}{3}\times\frac{3}{2}=1

Keep the dividend and multiply by the divisor’s reciprocal. To divide three quarters by one half, multiply three quarters by two. The answer is one and a half half-sized pieces.

34÷12=34×21=32\frac{3}{4}\div\frac{1}{2}=\frac{3}{4}\times\frac{2}{1}=\frac{3}{2}
Check your reasoning

Calculate 23÷45\dfrac{2}{3}\div\dfrac{4}{5}.

  1. A815\dfrac{8}{15}
  2. B56\dfrac{5}{6}
  3. C65\dfrac{6}{5}
Show answer and explanation
56\dfrac{5}{6}

Keep 2/3 and multiply by 5/4 to get 10/12 = 5/6.

02The divisor must be nonzero#

The divisor must be nonzero. A fraction with a zero numerator has no reciprocal because flipping it would place zero in the denominator. A zero dividend is allowed if the divisor is nonzero.

Check your reasoning

Calculate 34÷14\dfrac{3}{4}\div\dfrac{1}{4}.

Show answer and explanation
3

There are three quarter-sized pieces in three quarters. Equivalently, multiply 3/4 by 4 to get 3.

03Check your understanding#

Check your reasoning

A ribbon is 34\dfrac{3}{4} m. How many 18\dfrac{1}{8} m pieces fit?

  1. A66
  2. B33
  3. C88
Show answer and explanation
66

Divide by 1/81/8: multiply 3/43/4 by 88 to get 66.

Key takeaway

Keep the first fraction and multiply by the reciprocal of the second.

  • Divide a fraction by a nonzero fraction

Further questions

Should I simplify the final fraction?
Equivalent forms have the same value. A simplified form is usually easier to read; the app accepts the equivalent choice shown.

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Dividing fractions.” Math behind ML. /learn/f-fraction-divide