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§ Fractions

Dividing Fractions

§ Fractions

Dividing Fractions

CCSS.6.NS3 min read

Dividing fractions involves finding how many times one fraction fits into another fraction. The standard method requires converting division into multiplication by using the reciprocal of the divisor. This operation appears in Year 6 of the UK National Curriculum and forms a foundation for algebraic manipulation in later mathematics.

§ 01

Why it matters

Dividing fractions appears frequently in real-world calculations involving rates, recipes, and measurements. A baker dividing 34 kg of flour into portions of 18 kg each needs to calculate 34 ÷ 18 = 6 portions. Construction workers dividing a 23 metre plank into 16 metre sections use the same principle. In GCSE mathematics, fraction division connects to solving equations with fractional coefficients and working with rational expressions. Engineering applications include calculating gear ratios, where 58 ÷ 14 = 2.5 represents the relationship between rotating components. Medical dosage calculations often require dividing fractional amounts, such as determining how many 13 mg doses fit into a 23 mg tablet. The reciprocal method also provides groundwork for understanding division of algebraic fractions in A-level mathematics.

§ 02

How to solve dividing fractions

Dividing Fractions

  • Keep the first fraction.
  • Flip the second fraction (reciprocal).
  • Multiply. Simplify.

Example: 23 ÷ 45 → 23 × 54 = 1012 = 56.

§ 03

Worked examples

Beginner§ 01

12 / 14 = _______

Answer: 2

  1. Invert and multiply → 12 x 41 = 42 — Flip the second fraction, then multiply across.
  2. Simplify → 2 — Reduce to lowest terms.
  3. Verify → 2 ✓ — Answer.
Easy§ 02

A rope is 45 m long. You cut it into pieces 15 m each. How many pieces?

Answer: 4

  1. Invert and multiply → 45 x 51 = 205 — Cutting into equal pieces is division. Flip the second fraction, then multiply across.
  2. Simplify → 4 — Reduce to lowest terms.
  3. Verify → 4 ✓ — Answer.
Medium§ 03

23 / 15 = _______

Answer: 3 13

  1. Invert and multiply → 23 x 51 = 103 — Flip the second fraction, then multiply across.
  2. Simplify → 3 13 — Reduce to lowest terms.
  3. Verify → 3 13 ✓ — Answer.
§ 04

Common mistakes

  • Dividing numerators and denominators separately, writing 2/3 ÷ 4/5 = (2÷4)/(3÷5) = 1/2 ÷ 3/5 instead of the correct answer 5/6
  • Forgetting to flip the second fraction, calculating 1/2 ÷ 1/4 as 1/2 × 1/4 = 1/8 instead of 1/2 × 4/1 = 2
  • Converting division to multiplication but flipping the wrong fraction, writing 3/4 ÷ 2/5 = 4/3 × 2/5 = 8/15 instead of 3/4 × 5/2 = 15/8
§ 05

Frequently asked questions

Why do we flip the second fraction when dividing?
Division by a fraction equals multiplication by its reciprocal. Asking 'how many 1/4s fit into 1/2' is equivalent to asking '1/2 times what equals 1?' The answer is 4/1, so 1/2 ÷ 1/4 = 1/2 × 4/1 = 2.
How do you divide a fraction by a whole number?
Convert the whole number to a fraction by placing it over 1, then apply the standard method. For example, 3/4 ÷ 2 becomes 3/4 ÷ 2/1, which equals 3/4 × 1/2 = 3/8.
What is the reciprocal of a fraction?
The reciprocal swaps the numerator and denominator. The reciprocal of 3/5 is 5/3, and the reciprocal of 2/7 is 7/2. Multiplying any fraction by its reciprocal always equals 1.
How do you check if your answer is correct?
Multiply your answer by the divisor to get the original dividend. If 2/3 ÷ 1/5 = 10/3, then 10/3 × 1/5 should equal 2/3. Calculate: 10/15 = 2/3 ✓
Can the answer to dividing fractions be larger than both fractions?
Yes, when dividing by a fraction less than 1, the answer grows larger. Since 1/4 ÷ 1/8 = 2, dividing by 1/8 doubles the original value. This happens because division by fractions less than 1 creates more groups.
§ 06

See also

FractionNumeratorDenominator
§ 06

Where to next?

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