GCSE Maths Practice: fractions

Question 7 of 12

This question tests your understanding of multiplying fractions and simplifying the result. Multiply the numerators and denominators, then simplify to the lowest terms.

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\( \text{Calculate }\frac{3}{4}\times\frac{5}{6}.\\\text{Give your answer in simplest form.} \)

Choose one option:

Multiply numerators and denominators, cancel any common factors early, and simplify the final answer. Early cancellation saves time and avoids big numbers.

Understanding How to Multiply Fractions

Multiplying fractions is a straightforward process once you remember one golden rule: multiply the numerators together and multiply the denominators together. This gives the correct product, which you then simplify if possible. Simplification ensures the fraction is written in its lowest terms so that it’s easy to understand and compare.

Why This Works

Each fraction represents a portion of a whole. When you multiply fractions, you’re finding a portion of a portion — a smaller share. For instance, half of three-quarters means taking half of something already divided into four parts. Mathematically, \(\frac{1}{2}\times\frac{3}{4}=\frac{3}{8}\), because each new part is smaller than before. This same reasoning applies to any pair of fractions.

Step-by-Step Method

  1. Multiply the numerators: Multiply the top numbers together.
  2. Multiply the denominators: Multiply the bottom numbers together.
  3. Simplify the fraction: Divide numerator and denominator by their greatest common divisor (GCD).
  4. Optional: Cancel any common factors before multiplying to make calculations easier.

For this question, \(\frac{3}{4}\times\frac{5}{6}\): multiply 3×5 = 15, and 4×6 = 24, giving \(\frac{15}{24}\). Divide both by 3 to get \(\frac{5}{8}\).

Worked Examples

  • Example 1: \(\frac{2}{5}\times\frac{3}{4}=\frac{6}{20}=\frac{3}{10}\).
  • Example 2: \(\frac{4}{9}\times\frac{3}{8}=\frac{12}{72}=\frac{1}{6}\).
  • Example 3: \(\frac{7}{10}\times\frac{5}{14}=\frac{35}{140}=\frac{1}{4}\).
  • Example 4 (cancel early): \(\frac{3}{4}\times\frac{5}{6}\). You can cancel 3 and 6 → \(\frac{1}{4}\times\frac{5}{2}=\frac{5}{8}\).

Common Mistakes to Avoid

  • Adding instead of multiplying: \(\frac{1}{2}\times\frac{1}{3}\neq\frac{2}{5}\). Always multiply the numerators and denominators.
  • Forgetting to simplify: Write your final answer in simplest form for full marks in GCSE exams.
  • Multiplying across diagonally: This is only valid when dividing fractions, not multiplying them.
  • Skipping cancellation: Cancelling early avoids working with unnecessarily large numbers.

Real-Life Applications

Fraction multiplication is used in many practical contexts. For example, if a recipe calls for three-quarters of a cup of sugar, but you want only five-sixths of the recipe, the total sugar you need is \(\frac{3}{4}\times\frac{5}{6}=\frac{5}{8}\) of a cup. In finance, multiplying fractions helps when calculating discounts, interest rates, or proportional reductions. It’s also essential in geometry, where scaling shapes or finding areas of fractions of regions relies on this skill.

FAQs

Q1: Do I need common denominators to multiply fractions?
A1: No, that’s only required for addition and subtraction. You can multiply directly.

Q2: Can the product ever be larger than both fractions?
A2: Yes — if both fractions are greater than 1, or if one is improper, the result can exceed 1.

Q3: What’s the quickest way to simplify?

A3: Look for common factors before multiplying — for example, cancel a 3 in the numerator and a 6 in the denominator immediately.

Study Tip

Remember: “top times top, bottom times bottom.” Check if any numbers can cancel before multiplying to save time. With consistent practice, this becomes second nature, helping you tackle GCSE fraction, ratio, and algebraic fraction problems quickly and confidently.

Understanding how to multiply fractions accurately builds a strong foundation for percentages, scaling, and algebraic manipulation across GCSE Maths topics.