The idea that makes all of them quick: the multiplier
Turn every percentage change into a single number to multiply by. It saves keystrokes on a calculator that has no brackets and works through sums in the order you key them.
| Change | Multiply by |
|---|---|
| Find 36% of something | 0.36 |
| Increase by 15% | 1.15 |
| Decrease by 15% | 0.85 |
| Increase by 4% | 1.04 |
| Decrease by 30% | 0.7 |
1. A percentage of an amount
Multiply by the percentage as a decimal. 36% of 250 is 250 × 0.36 = 90.
2. Percentage change
Change divided by the original value, times 100. The original is where you started, whichever number is larger.
Original example
A clinic saw 460 patients in March and 630 in April. By what percentage did the number rise, to the nearest whole number?
Answer: 37%
Change: 630 − 460 = 170. Divide by the original: 170 ÷ 460 = 0.3696, which is 37%.
Dividing by the new figure instead (170 ÷ 630 = 27%) is the commonest mistake, and 27% will be one of the options.
Quicker still: new ÷ old, then read off the change. 630 ÷ 460 = 1.3696, so a rise of 37%. A result below 1 is a fall: 48 ÷ 64 = 0.75 is a fall of 25%.
3. Reverse percentages
You are given the value after a change and asked for the value before it. Divide by the multiplier. Do not take the percentage off the new value.
Original example
After a 15% price rise, a monthly bus pass costs $391. What did it cost before the rise?
Answer: $340
The new price is the old price × 1.15. So the old price is 391 ÷ 1.15 = $340.
Taking 15% off $391 gives $332.35, which is wrong: 15% of the new price is more than 15% of the old one. Check by going forwards: 340 × 1.15 = 391.
The same works for a fall. If a jacket costs $84 after 30% off, it cost 84 ÷ 0.7 = $120 before.
4. Percentage points
When a rate moves from 12% to 15%, it has risen by 3 percentage points. As a percentage change it has risen by 3 ÷ 12 = 25%. Questions use the exact wording, and both figures are usually among the options, so read which is asked for.
5. Repeated change
Multiply the multipliers. Do not add the percentages.
- Up 20% two years running: 1.2 × 1.2 = 1.44, a rise of 44%, not 40%. $250 becomes $360.
- Up 10% then down 10%: 1.1 × 0.9 = 0.99, a fall of 1%, not back where you started.
- Up 25% then down 20%: 1.25 × 0.8 = 1, exactly back where you started.
Which number is the base?
Before you touch the calculator, decide what the percentage is a percentage of. "What percentage of the total", "by what percentage did it rise" and "what percentage more than last year" each divide by a different number. Almost every percentage mistake is the right arithmetic on the wrong base.
Do these in your head
| Percentage | Shortcut |
|---|---|
| 50% | Halve it |
| 25% | Halve it twice |
| 10% | Move the decimal point one place left |
| 5% | Half of 10% |
| 1% | Move the decimal point two places left |
| 15% | 10% plus half of that |
| 20% | Double 10% |
Estimating first also protects you from keying errors. If 37% of roughly 600 comes out as 2,220, you know at once that something slipped.
On the UCAT calculator
- Key the multiplier, not the % key. 250 × 0.36 is two fewer things to go wrong.
- The calculator has no brackets and no order of operations. Do the subtraction first, then the division: 630 − 460 = 170, then ÷ 460.
- Use the number pad. Calculator and keyboard shortcuts explains the memory keys.
Practise them
Percentages are worth learning until they are automatic, because they sit inside other question types: interest, tax bands, discounts and charts. Try free Quantitative Reasoning questions, or read the full Quantitative Reasoning guide.
Official sources
Dates, fees and entry rules change every year. Always confirm them on the official page before you act on them.