GCSE Maths Practice: fractions

Understanding Multiplying Fractions

Multiplying fractions is one of the simplest operations in GCSE Maths once you know the rule: multiply the numerators together and multiply the denominators together. The result often needs simplifying to its lowest terms. Mastering this skill is essential not only for exams but also for later topics such as algebraic fractions, ratios, and probability.

Why the Rule Works

Each fraction represents a part of a whole. When you multiply two fractions, you are finding a part of a part — so the fraction becomes smaller. For example, half of a half is a quarter because you are taking half of the already-halved whole. The same logic applies when numerators and denominators are multiplied: the product shows how many parts out of a new total the result covers.

Step-by-Step Method

1. Multiply numerators: Multiply the top numbers together.
2. Multiply denominators: Multiply the bottom numbers together.
3. Simplify the result: Divide by the greatest common divisor (GCD) if possible.
4. Check for cancellation early: If any numerator and denominator share a factor, cancel it before multiplying to make calculations easier.

For this question, \(\frac{1}{5}\times\frac{5}{6}=\frac{5}{30}\). Dividing top and bottom by 5 gives \(\frac{1}{6}\).

Worked Examples

• Example 1: \(\frac{3}{4}\times\frac{2}{5}=\frac{6}{20}=\frac{3}{10}\).
• Example 2: \(\frac{1}{2}\times\frac{3}{8}=\frac{3}{16}\) (already simplified).
• Example 3: \(\frac{2}{3}\times\frac{9}{10}=\frac{18}{30}=\frac{3}{5}\).
• Example 4 (using cancellation): \(\frac{4}{9}\times\frac{3}{8}\). Cancel the 3 and 9 to get \(\frac{4}{3}\times\frac{1}{8}=\frac{4}{24}=\frac{1}{6}\).

Common Mistakes to Avoid

• Adding instead of multiplying: \(\frac{1}{2}\times\frac{1}{3}\neq\frac{2}{5}\). You must multiply, not add.
• Forgetting to simplify: Always check if the fraction can be reduced to lowest terms.
• Mixing up numerator and denominator order: Keep multiplication consistent — top with top, bottom with bottom.
• Not cancelling early: Cancelling before multiplying avoids large numbers and reduces errors.

Real-Life Applications

Fraction multiplication is used in everyday life. For example, if a recipe needs one-fifth of a 5/6 cup of sugar, the total sugar used is \(\frac{1}{5}\times\frac{5}{6}=\frac{1}{6}\) of a cup. In finance, finding interest over fractions of a year uses the same rule, and in probability, multiplying fractions finds combined event chances (e.g., the probability of two independent events both happening).

FAQs

Q1: Why can we cancel before multiplying?
A1: Because multiplication is commutative — the order doesn’t matter — so cancelling common factors first gives the same final answer but with smaller numbers.

Q2: What happens when a whole number is multiplied by a fraction?
A2: Write the whole number as a fraction over 1. For example, \(3\times\frac{1}{2}=\frac{3}{2}\).

Q3: Does the answer always get smaller?
A3: Usually yes, but if one factor is greater than 1 (an improper fraction), the result can be larger — e.g., \(\frac{3}{2}\times\frac{4}{3}=2\).

Study Tip

Always look for numbers that can be simplified before multiplying. This reduces working time in exams. Remember the sequence “top times top, bottom times bottom.” Practise with real examples involving recipes, scaling, and percentages to strengthen your confidence. Fluency with fraction multiplication supports success in GCSE Maths and more advanced algebra topics later on.