Multiplying Fractions: GCSE Exam Questions

This guide explains how to tackle multiplying fractions questions in GCSE Maths, covering the method, common pitfalls, and worked examples that match exam style. You’ll learn the straightforward rule (multiply numerators, multiply denominators) and how to apply it to mixed numbers, algebraic fractions, and worded problems. By the end, you’ll be able to approach any multiplication of fractions question with confidence.

Formula

The rule for multiplying two fractions is:

\[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \]

Where:

• \( a \) and \( c \) are the numerators (top numbers)
• \( b \) and \( d \) are the denominators (bottom numbers), and neither can be zero

You do not need a common denominator when multiplying – that’s only for addition and subtraction. Multiply straight across, then simplify the result if possible.

Worked Example

Using the default values from the calculator: \( \frac{2}{3} \times \frac{4}{5} \)

1. Multiply the numerators: \( 2 \times 4 = 8 \)
2. Multiply the denominators: \( 3 \times 5 = 15 \)
3. Write as a fraction: \( \frac{8}{15} \)
4. Check for simplification: 8 and 15 have no common factors, so the answer is \( \frac{8}{15} \).

The calculator shows 0.53 (rounded to 2 decimal places), which is \( 8 \div 15 = 0.5333… \). In an exam, you’d leave it as the fraction \( \frac{8}{15} \).

How to Multiply Fractions Step by Step

Exam questions often ask for the answer “in its simplest form.” Follow these steps:

1. Write both fractions as proper or improper fractions (convert mixed numbers first).
2. Multiply the numerators together.
3. Multiply the denominators together.
4. Simplify the resulting fraction by dividing top and bottom by their highest common factor (HCF).
5. If the answer is an improper fraction (numerator larger than denominator), you may convert it to a mixed number – but check the question’s instructions.

For example, \( \frac{3}{4} \times \frac{2}{9} \):

• Numerators: \( 3 \times 2 = 6 \)
• Denominators: \( 4 \times 9 = 36 \)
• Simplify: \( \frac{6}{36} = \frac{1}{6} \) (divide by 6)

Multiplying Mixed Numbers and Whole Numbers

GCSE questions often include mixed numbers (e.g., \( 1\frac{1}{2} \)). Always convert them to improper fractions first:

• \( 1\frac{1}{2} = \frac{3}{2} \) (multiply the whole number by the denominator, add the numerator, keep the denominator)
• \( 2\frac{3}{4} = \frac{11}{4} \)

Then multiply as usual. For a whole number like 3, write it as \( \frac{3}{1} \). For example:

\( 3 \times \frac{2}{5} = \frac{3}{1} \times \frac{2}{5} = \frac{6}{5} = 1\frac{1}{5} \)

Common Mistakes and How to Avoid Them

Students often lose marks on fraction multiplication for these reasons:

• Finding a common denominator – not needed for multiplication. Only addition/subtraction requires that.
• Forgetting to simplify – always reduce your final answer. If you leave \( \frac{2}{8} \), you may lose a mark even though it’s mathematically correct.
• Mishandling mixed numbers – multiplying the whole number parts separately is wrong. Convert to improper fractions first.
• Sign errors – if a fraction is negative, apply the sign to the final answer. Two negatives make a positive.
• Not cancelling before multiplying – you can simplify diagonally (cross-cancel) to make numbers smaller, but it’s not required. For example, in \( \frac{2}{3} \times \frac{3}{4} \), cancel the 3s to get \( \frac{2}{1} \times \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \).

Worded Problems and Real-World Contexts

GCSE exam questions often wrap fraction multiplication in a scenario. For instance:

“A recipe needs \( \frac{2}{3} \) of a cup of sugar. You want to make \( \frac{1}{2} \) of the recipe. How much sugar do you need?”

Answer: \( \frac{2}{3} \times \frac{1}{2} = \frac{2}{6} = \frac{1}{3} \) cup.

Another common type: “What is \( \frac{3}{5} \) of 20?” This is the same as \( \frac{3}{5} \times 20 = \frac{3}{5} \times \frac{20}{1} = \frac{60}{5} = 12 \).

Always read the question to see whether the answer should be a fraction, a decimal, or a mixed number.

Algebraic Fractions

Higher tier GCSE papers include algebraic fractions. The rule is the same – multiply numerators and denominators, then simplify by cancelling common factors.

Example: \( \frac{x}{2} \times \frac{3}{x+1} = \frac{3x}{2(x+1)} \)

If you have \( \frac{x^2}{y} \times \frac{y^2}{x} \), cancel \( x \) and \( y \) before multiplying:

\( \frac{x}{1} \times \frac{y}{1} = xy \)

Be careful with factors – you can only cancel when a factor appears in both a numerator and a denominator, not when it’s added or subtracted.

Frequently Asked Questions

Do I need a common denominator to multiply fractions?
No. A common denominator is only needed for adding or subtracting fractions. For multiplication, just multiply straight across.

How do I multiply mixed numbers?
Convert each mixed number to an improper fraction first (whole number × denominator + numerator, over the denominator). Then multiply as usual and simplify.

What does “simplify your answer” mean?
It means reduce the fraction to its lowest terms by dividing the numerator and denominator by their highest common factor. For example, \( \frac{6}{8} \) simplifies to \( \frac{3}{4} \).

Can I cross-cancel before multiplying?
Yes, cross-cancelling (dividing a numerator and a denominator by a common factor) is a valid shortcut. It doesn’t change the answer, but it makes the numbers smaller.

What if my answer is an improper fraction?
In GCSE, you can leave it as an improper fraction unless the question asks for a mixed number. If it says “give your answer as a mixed number,” convert it.

Why do I keep getting the wrong answer?
Check that you haven’t added denominators, forgotten to convert mixed numbers, or missed a negative sign. Also ensure you simplify at the end.

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in a | First numerator | - | 2 | 0
in b | First denominator | - | 3 | 0
in c | Second numerator | - | 4 | 0
in d | Second denominator | - | 5 | 0
out product | Product | - | (a * c) / (b * d) | 2