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Equivalent Fractions and Simplifying

Level 1 Fractions · Reviewed by The NumThink editorial team · Last reviewed

Short version: Multiply or divide the top and the bottom by the same number and the value does not change: 3/4 = 9/12 (×3), and 18/24 = 3/4 (÷6).

This page is part of the NumThink Fractions course. Read why the method works, watch it run on worked examples, then drill it in the practice box below until it feels automatic.

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Why it works

Multiplying top and bottom by 3 cuts every part into 3 smaller parts: there are 3 times as many parts, and you take 3 times as many — the same amount. Three quarters of a pizza is exactly nine twelfths of it.

Dividing top and bottom by a common factor does the reverse and gives a simpler form. Divide by the greatest common factor to reach the simplest form: dividing the top and bottom of 18/24 by 6 gives 3/4.

The steps

  1. Find the scale factor. 3/4 = ?/12: the bottom went from 4 to 12, which is ×3.
  2. Do the same to the top. 3 × 3 = 9, so 3/4 = 9/12.
  3. To simplify, divide both by a common factor. 18/24: both divide by 6, giving 3/4. Repeat until no common factor is left.

Worked examples

Easy 1/2 = ?/10

Show the thinking

2 × 5 = 10, so the scale factor is 5

1 × 5 = 5

So 1/2 = 5/10.

Answer: 5

Medium 3/4 = ?/20

Show the thinking

4 × 5 = 20

3 × 5 = 15

So 3/4 = 15/20.

Answer: 15

Hard 18/24 = ?/4

Show the thinking

24 ÷ 6 = 4, so divide by 6

18 ÷ 6 = 3

So 18/24 = 3/4.

Answer: 3

When to use it

  • Before adding fractions with different denominators: 1/2 + 1/4 becomes 2/4 + 1/4.
  • To simplify an answer, or to compare two fractions: 2/3 and 3/4 become 8/12 and 9/12, so 3/4 is bigger.

When NOT to use it

  • When the new denominator is not a multiple of the old one, simplify first (2/4 = 1/2 = 3/6); if the fraction is already in simplest form (3/4 = ?/10), there is no whole-number answer.
  • When you really need a percentage — go straight there instead (3/4 is 75%).

Common mistakes

Adding instead of multiplying. Turning 1/2 into 2/3 by adding 1 to the top and the bottom. Fix: Half is not two thirds. Only multiplying or dividing both by the same number keeps the value.
Changing only one number. Writing 3/4 = 3/12. Fix: Whatever you do to the bottom (×3), do to the top: 3/4 = 9/12.
Check yourself: Cross-multiply: 3/4 = 9/12 because 3 × 12 = 36 and 4 × 9 = 36 ✓.