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Mathematics 05801.4

Ratio, proportion and percentage

Sharing in a ratio, direct and inverse proportion, and percentage change.

Learning objectives

What you need to be able to do

Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.

  • 1.4.1Divide a quantity in a given ratio and use direct and inverse proportion.
  • 1.4.2Calculate percentage increase, decrease, reverse percentages and compound interest.

6 minute read

Ratio, proportion and percentage

Sharing in a ratio

To divide a quantity in a ratio, add the parts to find the total number of shares, divide the quantity by that total to find one share, then multiply out. Share £120 in the ratio 2 : 3. Total parts = 5, so one part = £24, giving £48 and £72. Always check the parts add back to the original.

Direct and inverse proportion

  • Direct (y ∝ x, so y = kx): as one increases, the other increases in the same ratio.
  • Inverse (y ∝ 1/x, so y = k/x): as one increases, the other decreases — more workers, less time.

Method: substitute the known pair to find k, then use k with the new value.

Percentages — the multiplier method

Using multipliers is faster and far less error-prone than finding a percentage and adding it separately:

  • Increase by 15% → multiply by 1.15.
  • Decrease by 15% → multiply by 0.85.
  • Compound interest over n years → multiply by the multiplier n times: Final = P × (multiplier)ⁿ.

Reverse percentages — the classic trap

If a price after a 20% increase is £60, you must divide by 1.2 to get the original £50. Taking 20% off £60 gives £48, which is wrong, because the 20% was calculated on the smaller original, not on £60. The test: if the question gives you the value after the change and asks for the value before, you divide.

Think of it like this

A percentage change is a scale factor, not a fixed amount. Undoing it means dividing by that factor — just as the way back from a photo enlarged 3× is to divide by 3, not to subtract what was added.

Worked examples

Method, step by step

A jacket costs £84 after a 40% reduction in a sale. Find the original price.

  1. 1A 40% reduction means the sale price is 60% of the original, so the multiplier is 0.6.
  2. 2Sale price = original × 0.6, so original = 84 ÷ 0.6.
  3. 3original = £140.
  4. 4Check: 140 × 0.6 = 84 ✓

£140

£2000 is invested at 5% compound interest per year. Find the value after 3 years.

  1. 1The multiplier for a 5% increase is 1.05.
  2. 2For compound interest over 3 years, apply it three times: 2000 × 1.05³.
  3. 31.05³ = 1.157625, so 2000 × 1.157625 = 2315.25.

£2315.25

Common misconceptions

  • Doing reverse percentages by subtracting. If £60 is after a 20% rise, the original is 60 ÷ 1.2 = £50, not 60 − 20% = £48.
  • Thinking a 10% rise followed by a 10% fall returns to the start. 1.1 × 0.9 = 0.99, a 1% overall loss.
  • Treating compound interest as simple interest. Compound multiplies repeatedly, so it grows faster each year.

In the exam

  • Write the multiplier before calculating anything. It makes multi-stage percentage questions almost mechanical.
  • For "find the original amount" questions, divide by the multiplier — and check your answer by applying the change forwards.