Percentages, Ratios and Proportions
What you must understand
You need fluency with percentage increase and decrease, finding the original amount after a change, successive percentage changes, percentage points versus percent, and ratio sharing. A ratio a:b splits a total into a+b parts; each part equals total ÷ (a+b).
Method
For a percentage change, use multiplier form: an increase of 18% multiplies by 1.18; a decrease of 18% multiplies by 0.82. Successive changes multiply: two rises of 10% give 1.10 × 1.10 = 1.21, a 21% rise, not 20%. To reverse a change, divide by the multiplier: if a price after a 15% rise is €92, the original was 92 ÷ 1.15 = €80.
Traps
- Adding successive percentages instead of multiplying multipliers.
- Confusing percentage points with percent: a rate rising from 4% to 6% rises 2 percentage points but 50%.
- Reversing a change by subtracting the percentage from the final value rather than dividing.
- Ratio questions where the total given is not the whole but one part.
- Mixing up 'of' (a multiplication base) with 'more than'.
Worked example
A jacket is reduced by 20%, then the reduced price is reduced by a further 25%. What single discount is equivalent? Multipliers: 0.80 × 0.75 = 0.60, so the final price is 60% of the original - an equivalent single discount of 40%, not 45%.