Multiplying and Dividing Fractions –

Multiplying and dividing fractions appears on every GCSE Maths paper across AQA, Edexcel, and OCR. Once you know the two core rules — multiply straight across, and Keep-Change-Flip for division — these questions become some of the easiest marks on the exam.

What Is Multiplying and Dividing Fractions?

Multiplying fractions is simpler than adding them because you do not need a common denominator. You multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. The key skill is to simplify — either at the end or by cross-cancelling before you multiply.

Dividing fractions uses the reciprocal. Instead of dividing by a fraction, you multiply by its reciprocal (the fraction flipped upside down). This is often remembered as Keep-Change-Flip: keep the first fraction, change ÷ to ×, flip the second fraction.

When mixed numbers are involved, always convert them to improper fractions before carrying out the operation, then convert back if the question requires a mixed number.

Key Formulas

Step-by-Step Method

1. If the question involves mixed numbers, convert each to an improper fraction.
2. For multiplication: multiply the numerators together and the denominators together.
3. For division: keep the first fraction, change ÷ to ×, flip the second fraction, then multiply.
4. Simplify by cross-cancelling before multiplying or by dividing the final answer by the HCF.
5. Convert back to a mixed number if the question asks for one.

Worked Example 1 — Foundation Level

Question: Work out 3/4 × 2/5. Give your answer as a fraction in its simplest form.

Working:

Step 1 — Multiply the numerators: 3 × 2 = 6.

Step 2 — Multiply the denominators: 4 × 5 = 20.

Step 3 — The result is 6/20. Simplify by dividing both by 2: 6/20 = 3/10.

Answer: 3/10

Worked Example 2 — Higher Level

Question: Work out 2 1/3 ÷ 1 3/4. Give your answer as a mixed number in its simplest form.

Working:

Step 1 — Convert to improper fractions: 2 1/3 = 7/3 and 1 3/4 = 7/4.

Step 2 — Keep-Change-Flip: 7/3 ÷ 7/4 = 7/3 × 4/7.

Step 3 — Cross-cancel: the 7 in the numerator and the 7 in the denominator cancel to give 1/3 × 4/1.

Step 4 — Multiply: 1 × 4 = 4 and 3 × 1 = 3, so the answer is 4/3.

Step 5 — Convert: 4/3 = 1 1/3.

Answer: 1 1/3

Worked Example 3 — Exam Style

Question: A piece of wood is 3 1/2 metres long. Emma cuts off 2/7 of the total length. What length does she cut off? (3 marks)

Working:

Step 1 — Convert: 3 1/2 = 7/2.

Step 2 — Find 2/7 of 7/2: multiply 2/7 × 7/2.

Step 3 — Cross-cancel: the 2s cancel and the 7s cancel, giving 1/1 × 1/1 = 1.

Answer: 1 metre

Common Mistakes

• Finding a common denominator when multiplying. You do not need a common denominator for multiplication — just multiply straight across.
• Flipping the wrong fraction when dividing. Only flip the second fraction (the divisor). Keep the first fraction exactly as it is.
• Forgetting to convert mixed numbers first. Multiplying or dividing with mixed numbers directly leads to errors. Always convert to improper fractions before operating.

Exam Tips

• Cross-cancelling before you multiply keeps the numbers small and reduces the risk of arithmetic errors. Look for any common factor between a numerator and a denominator.
• If the question says "Give your answer as a mixed number," you will lose the final mark if you leave it as an improper fraction.
• On non-calculator papers, show the cross-cancelling step — it earns method marks.

Practice Questions

Q1 (Foundation): Work out 5/8 × 4/9. Give your answer in its simplest form.

Q2 (Foundation): Work out 3/5 ÷ 2/3.

Q3 (Higher): Work out 1 2/5 × 2 1/3. Give your answer as a mixed number in its simplest form.

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Related Topics

• Fractions
• Algebraic Fractions
• Adding and Subtracting Fractions

Summary

• To multiply fractions, multiply numerators together and denominators together.
• To divide fractions, use Keep-Change-Flip: keep the first, change ÷ to ×, flip the second.
• Cross-cancel common factors before multiplying to keep numbers small.
• Always convert mixed numbers to improper fractions before calculating.
• Simplify your final answer and convert to a mixed number if required.