Equivalent Fractions and Simplifying –

Equivalent fractions and simplifying are essential skills that underpin almost every fractions question on GCSE Maths papers. You need to recognise when two fractions represent the same value and reduce fractions to their simplest form quickly and accurately.

What Are Equivalent Fractions?

Two fractions are equivalent if they represent the same proportion of a whole. You create equivalent fractions by multiplying or dividing both the numerator and the denominator by the same non-zero number.

For example, 2/3 and 8/12 are equivalent because 2 × 4 = 8 and 3 × 4 = 12. The value has not changed — only the way it is written.

Simplifying (also called cancelling down) means writing a fraction in its lowest terms. You do this by dividing the numerator and denominator by their highest common factor (HCF). A fraction is fully simplified when the HCF of the numerator and denominator is 1.

Key Formulas

Step-by-Step Method

1. Find the HCF of the numerator and the denominator.
2. Divide both the numerator and the denominator by the HCF.
3. Check that no common factor remains — the fraction is now in its simplest form.

To compare two fractions, either convert them to a common denominator or cross-multiply and compare the products.

Worked Example 1 — Foundation Level

Question: Simplify 18/24.

Working:

Step 1 — Find the HCF of 18 and 24. Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. HCF = 6.

Step 2 — Divide both by 6: 18 ÷ 6 = 3, 24 ÷ 6 = 4.

Step 3 — 3 and 4 share no common factor other than 1.

Answer: 3/4

Worked Example 2 — Higher Level

Question: Which is larger: 5/8 or 7/11? Show your working.

Working:

Cross-multiply: 5 × 11 = 55 and 7 × 8 = 56. Since 55 < 56, we have 5/8 < 7/11.

Alternatively, convert to a common denominator of 88: 5/8 = 55/88 and 7/11 = 56/88. Since 55 < 56, 7/11 is larger.

Answer: 7/11 is larger.

Worked Example 3 — Exam Style

Question: Write 42/60 in its simplest form. Show that it is equivalent to 7/10.

Working:

Step 1 — HCF of 42 and 60: 42 = 2 × 3 × 7 and 60 = 2² × 3 × 5, so HCF = 2 × 3 = 6.

Step 2 — 42 ÷ 6 = 7, 60 ÷ 6 = 10.

Step 3 — 7 and 10 share no common factor, so 7/10 is fully simplified. Since 7 × 6 = 42 and 10 × 6 = 60, the fractions are equivalent.

Answer: 42/60 = 7/10

Common Mistakes

• Dividing by a common factor that is not the HCF. For example, simplifying 24/36 by dividing by 2 to get 12/18, then stopping. Always check if you can simplify further, or find the HCF first.
• Multiplying numerator and denominator by different numbers. Both must be multiplied or divided by the same value to keep the fraction equivalent.
• Confusing cross-multiplication direction. When comparing a/b and c/d, compute ad and bc. If ad > bc then a/b is larger.

Exam Tips

• Always give your final answer in its simplest form unless told otherwise.
• If the numbers are large, use prime factorisation to find the HCF efficiently.
• When comparing fractions on a non-calculator paper, cross-multiplication is faster than finding a common denominator.

Practice Questions

Q1 (Foundation): Simplify 15/25.

Q2 (Foundation): Write three fractions equivalent to 3/7.

Q3 (Higher): Which is larger, 9/13 or 11/16? Show your working.

Practise equivalent fractions and simplifying questions with instant feedback — completely free on GCSEMathsAI.

Related Topics

• Fractions
• Adding and Subtracting Fractions
• Prime Factorisation, HCF and LCM

Summary

• Equivalent fractions have the same value but different numerators and denominators.
• To create equivalent fractions, multiply or divide both parts by the same number.
• Simplify a fraction by dividing numerator and denominator by their HCF.
• To compare fractions, use cross-multiplication or convert to a common denominator.