GCSE Revision

Ratio

Simplifying ratios

A compares amounts, such as 2 : 3. Simplify it like a fraction: divide every part by the HCF.

  1. Write the parts in the same units.
  2. Turn decimals or fractions into whole numbers by multiplying every part by the same number.
  3. Divide every part by the HCF.

Example: simplify 24 : 36.

  1. The HCF of 24 and 36 is 12
  2. 2 : 3

Harder examples:

  1. 45 cm : 2 m = 45 : 200 = 9 : 40
  2. 1.5 : 2.5 = 15 : 25 = 3 : 5 (× 10, then ÷ 5)
  3. ½ : ⅓ = 3 : 2 (multiply both by 6): 3 : 2

Common mistake

Change to the same units first: 45 cm : 2 m is not 45 : 2.

Key terms

Ratios and fractions

A ratio splits a whole into . Add the parts to find the total number of parts; each share is that many parts out of the total.

  1. Add the numbers in the ratio.
  2. Each amount is its number of parts over the total.

Example: the ratio of boys to girls in a class is 3 : 5. What fraction of the class are girls? There are 32 students; how many are girls?

  1. Total parts: 3 + 5 = 8
  2. Girls: 5/8 of the class
  3. 5/8 of 32 = 32 ÷ 8 × 5 = 20 girls

The fraction for boys is 3/8, not 3/5: the denominator is the total, not the other part.

Key terms

Sharing in a ratio

  1. Add the parts to find the total number of parts.
  2. Divide the amount by the total parts to find one part.
  3. Multiply by each number in the ratio.

Example: share £72 in the ratio 2 : 3 : 4.

  1. Total parts: 2 + 3 + 4 = 9
  2. One part: £72 ÷ 9 = £8
  3. 2 × 8 = 16, 3 × 8 = 24, 4 × 8 = 32
  4. £16, £24 and £32

Check: 16 + 24 + 32 = 72.

Key terms

Using one part or the difference

If you know one share, or the difference between two shares, find what one part is worth first.

Example: red and blue counters are in the ratio 2 : 5. There are 14 red counters. How many are blue?

  1. 2 parts = 14, so 1 part = 7
  2. Blue: 5 × 7 = 35

Example: Amy and Ben share money in the ratio 3 : 7. Ben gets £28 more than Amy. How much is shared?

  1. The difference is 7 − 3 = 4 parts, so 4 parts = £28
  2. 1 part = £7
  3. Total: 10 parts = £70

Check: Amy gets £21, Ben gets £49, and 49 − 21 = 28.

Combining ratios (Higher tier only)

To combine a : b and b : c into a : b : c, make the shared part (b) the same in both.

  1. Find the LCM of the two numbers for b.
  2. Multiply each ratio so b equals the LCM.
  3. Write the three parts together.

Example: a : b = 2 : 3 and b : c = 4 : 5. Find a : b : c.

  1. The LCM of 3 and 4 is 12
  2. 2 : 3 = 8 : 12 (× 4) and 4 : 5 = 12 : 15 (× 3)
  3. a : b : c = 8 : 12 : 15

Ratio problems with algebra (Higher tier only)

When a ratio changes, use a letter for one part and form an equation.

  1. Write each amount using x for one part.
  2. Write the new amounts after the change.
  3. Use the new ratio to form an equation and solve.

Example: Ali and Ben share sweets in the ratio 3 : 5. Ben gives Ali 6 sweets, and now they have the same number. How many sweets are there?

  1. Ali has 3x and Ben has 5x
  2. After: Ali has 3x + 6 and Ben has 5x − 6
  3. 3x + 6 = 5x − 6, so 12 = 2x and x = 6
  4. Total: 8x = 48 sweets

Check: Ali 18 + 6 = 24 and Ben 30 − 6 = 24.

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