Percent means 'per hundred': x% = x/100. Percentage is used in almost every arithmetic topic, including profit and loss, interest, mixtures and data interpretation, so quick and accurate handling of percentages saves a lot of time.
Basic Formulas
| What is asked | Formula | Example |
|---|---|---|
| x% of y | x × y / 100 | 15% of 240 = 36 |
| A as a percentage of B | (A / B) × 100 | 20 of 80 = 25% |
| Percentage change | (New − Old) / Old × 100 | 200 → 250 is +25% |
| Value after r% increase / decrease | Old × (100 ± r) / 100 | 800 + 15% = 920 |
| Original before r% increase / decrease | New × 100 / (100 ± r) | 500 after +25% → 400 |
| x% of y = y% of x | Swap when it is easier | 4% of 75 = 75% of 4 = 3 |
| Fraction | Percent | Fraction | Percent |
|---|---|---|---|
| 1/2 | 50% | 1/8 | 12.5% |
| 1/3 | 33⅓% | 1/9 | 11⅑% |
| 1/4 | 25% | 1/11 | 9 1/11 % |
| 1/5 | 20% | 1/12 | 8⅓% |
| 1/6 | 16⅔% | 1/16 | 6.25% |
| 1/7 | 14 2/7 % | 1/20 | 5% |
More Than / Less Than
Changing the base
If A is x% more than B, then B is [x / (100 + x)] × 100 % less than A. If A is x% less than B, then B is [x / (100 − x)] × 100 % more than A.
A is 25% more than B → B is 25/125 × 100 = 20% less than A. The two percentages differ because the base changes.
Price, quantity and expenditure
If price rises by r%, consumption must fall by [r / (100 + r)] × 100 % to keep expenditure the same; if price falls by r%, consumption can rise by [r / (100 − r)] × 100 %
Petrol up 25% → cut use by 20%. Wheat down 20% → use can go up by 25%.
Successive Percentage Change
Two changes a% and b%
a + b + (a × b) / 100
Use a minus sign for a decrease. +20% then −10% gives 20 − 10 − 2 = +8%. An equal rise and fall (±x%) always gives a net loss of x²/100 %: ±20% gives −4%.
- Three or more changes: multiply the factors. +10%, +20%, −25% → 1.1 × 1.2 × 0.75 = 0.99, a net fall of 1%.
- Successive discounts of 20% and 25% equal a single discount of 40% (0.8 × 0.75 = 0.6).
- Area of a square or circle: a 10% increase in side or radius gives 10 + 10 + 1 = 21% more area.
Population and Depreciation
Compound growth or decay
After n years: P × (1 ± r/100)ⁿ; n years ago: P ÷ (1 ± r/100)ⁿ
50,000 growing 4% a year → 50,000 × 1.0816 = 54,080 after 2 years. A machine worth Rs 72,900 now, losing 10% a year, was worth 72,900 ÷ 0.81 = Rs 90,000 two years ago. With different yearly rates, multiply each year's factor.
Typical Word Problems
- Income, expenditure, savings: take income as 100 and work out each part. If someone spends 75% and income rises 20% while expenditure rises 10%, savings go from 25 to 120 − 82.5 = 37.5, a 50% rise.
- Elections: margin of victory = (winner% − loser%) of valid votes. If 55% wins by 2800, then 10% of votes = 2800 and the total is 28,000. Subtract invalid votes first when they are given.
- Pass marks: if 60% fails by 15 and 80% passes by 35, then 20% of the maximum = 50, so maximum = 250 and pass mark = 165.
- Sets: failed in at least one = failed in A + failed in B − failed in both. With 30%, 20% and 10%, 40% failed in at least one, so 60% passed both.
Speed Tricks
Break any percentage into 10%, 5% and 1%: 17.5% of 400 = 40 + 20 + 10 = 70. For reverse percentage use fractions: 125% = 5/4, so a value of 500 after a 25% rise was 500 × 4/5 = 400.
Common Mistakes
Using the new value as the base for percentage change; assuming a 20% rise followed by a 20% fall gives no change (it gives a 4% fall); adding successive percentages directly instead of using a + b + ab/100.