1.4 Fractions, decimals, percentages and ratio
Percentages
Everything on percentages that can come up in the WJEC GCSE Mathematics and Numeracy, in all three units. Each skill below has the method, the mistake that costs the most marks, a worked example, and free practice marked the way WJEC marks it.
A worked example
Percentage of an amount · [2 marks]
A pair of trainers costs £110. Seren pays a 10% deposit. How much is the deposit?
- M1£110 × 0.1 (or find 10%, 5%, 1% and build up)
- A1£11.00
The exam skills (9)
Percentage of an amount
- Turn the percentage into a decimal: 5% = 0.05.
- Multiply the amount by it.
- Without a calculator, build it up: 10% (÷ 10), 5% (half of 10%), 1% (÷ 100).
Watch out: Giving money with one decimal place. Write £0.50, never £0.5.
Percentage increase
- Add the increase to 100%: up 5% means 105%.
- Write it as a multiplier: 1.05.
- Multiply the original amount by the multiplier.
Watch out: Working out the 5% and stopping there. The question wants the new amount.
Percentage decrease
- Take the decrease away from 100%: down 5% means 95%.
- Write it as a multiplier: 0.95.
- Multiply the original amount by the multiplier.
Watch out: Using 0.05 instead of 0.95. That gives the discount, not the sale price.
Reverse percentage · decrease
- The price you are given is AFTER the change, so it is 95% (or whatever is left).
- Write that as a multiplier: 0.95.
- Divide by the multiplier to get back to the original.
Watch out: Taking 5% of the sale price and adding it on. The 5% was of the original, so this is always wrong.
Reverse percentage · increase
- The new price is 100% + the rise, e.g. 105%.
- Write that as a multiplier: 1.05.
- Divide by the multiplier to undo the rise.
Watch out: Taking 5% off the new price. Divide by 1.05 instead.
Increase then decrease
- Write a multiplier for each change, e.g. 1.05 and 0.95.
- Multiply them together to get one combined multiplier.
- Divide the final amount by the combined multiplier.
Watch out: Thinking up 5% then down 5% cancels out. It does not: 1.05 × 0.95 = 0.9975.
Decrease then increase
- Write a multiplier for each change, e.g. 0.95 then 1.05.
- Multiply them together.
- Divide the final amount by the combined multiplier.
Watch out: Adding and subtracting the percentages. Multiply the multipliers instead.
Compound depreciation
- Losing 5% a year means a multiplier of 0.95 each year.
- For n years the multiplier is 0.95ⁿ.
- The value after n years ÷ 0.95ⁿ gives the value when new.
Watch out: Using 5% × n (simple). Depreciation compounds, so use a power.
Compound interest
- Interest of 2% a year means a multiplier of 1.02 each year.
- For n years it is 1.02ⁿ.
- Divide the final amount by 1.02ⁿ to find what was put in.
Watch out: Multiplying instead of dividing when the question gives you the final amount.
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