Dividing Fractions
Dividing Fractions is a key number topic at GCSE Maths. This Foundation and Higher worksheet gives you exam-style questions on dividing fractions, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. To divide by a fraction, multiply by its reciprocal — flip and multiply.
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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
Topic overview
Dividing by a fraction is done by multiplying by its reciprocal. You turn the second fraction upside down and change the division to a multiplication.
The reason is worth understanding rather than memorising. Asking how many halves fit into three is asking \(3 \div \frac{1}{2}\), and the answer is \(6\) — dividing by a fraction smaller than one makes the answer larger, which is exactly what multiplying by its reciprocal does.
Once flipped, the method is identical to multiplying fractions, so you can cancel before multiplying in the same way. The most common error is flipping the wrong fraction: it is always the one you are dividing by, never the first.
Revision notes
Flipping the second fraction
Keep the first fraction, change the sign to multiplication, and invert the second. Some students remember this as keep, change, flip.
So \(\frac{2}{3} \div \frac{4}{5}\) becomes \(\frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}\).
Dividing by a whole number
Write the whole number over \(1\), then flip it. Dividing by \(4\) is the same as multiplying by \(\frac{1}{4}\).
So \(\frac{3}{5} \div 4 = \frac{3}{5} \times \frac{1}{4} = \frac{3}{20}\).
Mixed numbers
Convert every mixed number to an improper fraction before flipping anything, or the inversion will be wrong.
For \(2\frac{1}{2} \div 1\frac{1}{4}\), write \(\frac{5}{2} \div \frac{5}{4}\), then \(\frac{5}{2} \times \frac{4}{5} = 2\).
Key points
- Dividing by a fraction means multiplying by its reciprocal.
- Keep the first fraction, change the sign, flip the second.
- Only the second fraction is inverted.
- Write a whole number over 1 before flipping.
- Convert mixed numbers to improper fractions first.
- Dividing by a fraction below 1 gives a larger answer.
Worked examples
Example 1
Work out \(\frac{3}{4} \div \frac{2}{5}\).
Working
Example 2
Work out \(\frac{5}{6} \div 10\).
Working
Example 3
Work out \(3\frac{1}{3} \div 1\frac{1}{9}\).
Working
Common mistakes
Flipping the first fraction instead of the second.
Only the fraction you are dividing by is inverted. Flipping the first gives the reciprocal of the correct answer.
Flipping a mixed number before converting it.
2½ inverted is meaningless. Convert to 5/2 first, then flip to 2/5.
Changing the sign but forgetting to flip.
Both steps are needed. Multiplying without inverting gives a completely different answer.
Assuming division always makes numbers smaller.
Dividing by a fraction below 1 increases the value, which is correct.
Exam tips
- Write keep, change, flip as three separate steps in your working.
- Convert mixed numbers before doing anything else.
- Cancel after flipping, in the same way as ordinary multiplication.
- Sense-check: dividing by a fraction under 1 should give a bigger answer.
Key terms
- Reciprocal
- The fraction turned upside down, so the reciprocal of 3/4 is 4/3.
- Divisor
- The number you are dividing by.
- Improper fraction
- A fraction whose numerator exceeds its denominator.
- Invert
- To turn a fraction upside down.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.