GCSE Maths Practice: fractions

Question 12 of 12

This question tests your ability to simplify fractions by dividing both numerator and denominator by their greatest common divisor (GCD). Simplifying helps express fractions in their lowest terms for easier comparison and calculation.

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\( \text{Simplify }\frac{16}{64}. \)

Choose one option:

To simplify any fraction, divide numerator and denominator by their GCD. If unsure, start with small prime numbers (2, 3, or 5) and keep dividing until no further reduction is possible.

Understanding Simplifying Fractions

In GCSE Maths, simplifying fractions is one of the most important skills in the Number topic. It means reducing a fraction to its lowest terms so that the numerator and denominator share no common factors except 1. This process makes calculations easier and shows the true relationship between parts and the whole.

Why Simplify Fractions?

Simplified fractions make mathematical results cleaner, easier to compare, and less prone to mistakes in later steps. For instance, when adding or multiplying fractions, it is much more efficient to work with smaller numbers. Simplifying also helps recognise equivalent fractions — different-looking fractions that represent the same value.

Step-by-Step Method

  1. Find the GCD (or HCF) – The greatest common divisor of the numerator and denominator is the largest number that divides both exactly.
  2. Divide both numbers by the GCD – This removes all shared factors.
  3. Check the result – Make sure there are no further common factors.

For example, with \(\frac{16}{64}\), both 16 and 64 are powers of 2. Since \(16=2^4\) and \(64=2^6\), dividing numerator and denominator by \(2^4\) gives \(\frac{1}{4}\).

Worked Examples

  • Example 1: Simplify \(\frac{6}{9}\). GCD = 3, so \(\frac{6}{9}=\frac{2}{3}\).
  • Example 2: Simplify \(\frac{15}{25}\). GCD = 5, giving \(\frac{3}{5}\).
  • Example 3: Simplify \(\frac{20}{100}\). GCD = 20, so \(\frac{1}{5}\).
  • Example 4 (using prime factors): \(\frac{18}{30}\). Prime factors: 18 = 2×3×3, 30 = 2×3×5. Common factors = 2×3 = 6, so \(\frac{18}{30}=\frac{3}{5}\).

Common Mistakes

  • Forgetting to simplify fully: Stopping after dividing by a small number, e.g. \(\frac{12}{16}\) → \(\frac{6}{8}\) instead of \(\frac{3}{4}\).
  • Incorrect division: Dividing only one part (numerator or denominator) changes the value of the fraction entirely.
  • Confusing GCD with LCM: The least common multiple (LCM) is used for adding fractions, not simplifying them.

Real-Life Applications

Fractions appear in cooking (recipes scaled up or down), in building and design (proportions of materials), and in data interpretation (percentages and ratios). For instance, if a recipe calls for 16 g of sugar out of 64 g of mixture, that’s one quarter sugar — exactly \(\frac{1}{4}\). Understanding simplified fractions allows quick estimation and accurate scaling.

FAQs

Q1: What if both numbers are odd and share no factors?
A1: Then the fraction is already in simplest form. For example, \(\frac{5}{9}\) cannot be simplified further.

Q2: Can decimals be simplified like fractions?
A2: You can convert decimals to fractions first, then simplify. For example, 0.25 = \(\frac{25}{100}\) = \(\frac{1}{4}\).

Q3: Is simplifying fractions required in exams?
A3: Yes. Many GCSE questions require final answers in their simplest form, and marks can be lost for not simplifying.

Study Tip

When stuck, break numbers into prime factors or use a quick mental check: if both are even, divide by 2; if both end in 0 or 5, divide by 5; if the digits sum to a multiple of 3, divide by 3. Repeat until no further simplification is possible. Practice with different examples to build fluency and accuracy.

Mastering the skill of simplifying fractions strengthens your understanding of ratio, proportion, and algebraic fractions — key foundations for higher-level GCSE Maths topics and beyond.