The Number System

Dividing Fractions by Fractions

Dividing a fraction by a fraction using visual models and equations.

Intermediate15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Dividing by a Fraction That Isn't a Unit Fraction

In Grade 5, dividing by a unit fraction like 1/4 asked how many fourths fit into a whole number. Here's a harder question to predict before reading on: how many servings of 3/4 cup fit into a pot holding 6 cups of soup? This time the fraction being divided by, 3/4, isn't a single unit piece — it's 3 of them together.

The same 'how many fit' reasoning still applies, just with an extra step. First figure out how many fourths fit into 6 cups — 6 × 4 = 24 fourths. But each serving uses 3 fourths at a time, not just 1, so group those 24 fourths into groups of 3: 24 ÷ 3 = 8 servings.

Notice that finding how many fourths fit (multiplying by 4) and then grouping them by 3 (dividing by 3) is exactly the same as multiplying by 4/3 — the reciprocal of 3/4. This is why dividing by any fraction, not just a unit fraction, can always be found by multiplying by that fraction's reciprocal.

Worked Example — Dividing a Whole Number by a Fraction

Divide 6 ÷ 3/4 using the reciprocal method. Multiply 6 by the reciprocal of 3/4, which is 4/3: 6 × 4/3 = 24/3 = 8. This matches the servings found by reasoning directly: 8 servings.

Worked Example — Dividing a Fraction by a Fraction

Divide 2/3 ÷ 5/6. Multiply by the reciprocal of the divisor: 2/3 × 6/5 = 12/15, which simplifies to 4/5. So 2/3 ÷ 5/6 = 4/5.

\dfrac{2}{3} \div \dfrac{5}{6} = \dfrac{2}{3} \times \dfrac{6}{5} = \dfrac{12}{15} = \dfrac{4}{5}

Tip

Before calculating, predict whether the answer should be bigger or smaller than the number being divided — dividing by a fraction smaller than 1 always gives a bigger result, the same rule from Grade 5's unit fraction division.

Common Mistakes

Taking the reciprocal of the wrong fraction, such as flipping the first fraction instead of the divisor.
Only the divisor — the fraction after the division sign — gets flipped into its reciprocal. The first fraction stays exactly as written.
Forgetting to simplify the final fraction, leaving an answer like 12/15 instead of reducing it to 4/5.
Always check whether the numerator and denominator of the final answer share a common factor, and simplify if they do.

Key Takeaways

Dividing by any fraction, not just a unit fraction, can be found by multiplying by that fraction's reciprocal.
This works because dividing by a fraction is really 'how many of that fraction fit,' which combines multiplying and grouping.
Dividing by a fraction smaller than 1 always gives a result larger than the original number.

Summary

Dividing fractions by fractions extends Grade 5's reciprocal strategy to any two fractions. The next lesson moves to a different kind of number entirely — numbers that represent quantities in the opposite direction from what you've measured so far.

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