Arithmetic
Percentages formulas
At a glance
r% = r / 100 x% of y = y% of x % change = (new − old) / old × 100 +r% → × (1 + r/100) −r% → × (1 − r/100) two changes: a + b + ab/100 r% more → r/(100 + r) less r% less → r/(100 − r) more 1/6 = 16.67% 1/7 = 14.29% 1/8 = 12.5% 1/9 = 11.11% 1/11 = 9.09%
Fractions to know by heart
| Fraction | Percent | Handy multiples |
|---|---|---|
| 1/2 | 50% | — |
| 1/3 | 33.33% | 2/3 = 66.67% |
| 1/4 | 25% | 3/4 = 75% |
| 1/5 | 20% | 2/5 = 40%, 3/5 = 60%, 4/5 = 80% |
| 1/6 | 16.67% | 5/6 = 83.33% |
| 1/7 | 14.29% | 2/7 = 28.57%, 3/7 = 42.86% |
| 1/8 | 12.5% | 3/8 = 37.5%, 5/8 = 62.5%, 7/8 = 87.5% |
| 1/9 | 11.11% | 2/9 = 22.22%, 4/9 = 44.44% |
| 1/10 | 10% | 3/10 = 30%, 7/10 = 70% |
| 1/11 | 9.09% | 2/11 = 18.18%, 3/11 = 27.27% |
| 1/12 | 8.33% | 5/12 = 41.67%, 7/12 = 58.33% |
| 1/13 | 7.69% | 2/13 = 15.38% |
| 1/14 | 7.14% | 3/14 = 21.43% |
| 1/15 | 6.67% | 2/15 = 13.33%, 4/15 = 26.67% |
1/7 is 14.2857…%, and every k/7 uses the same six repeating digits 142857 in a rotated order.
Shortcut: Build k/n from 1/n: 3/8 = 3 × 12.5% = 37.5%, and 5/6 = 5 × 16.67% = 83.33%.
Finding a percent of a number
Shortcut: x% of y is always equal to y% of x, because both are xy / 100. So 16% of 25 = 25% of 16 = 4.
Percentage change: always against the base
% change = (new − old) / old × 100
Multiplying factors
+20% → × 1.2 = × 6/5 −15% → × 0.85 +12.5% → × 1.125 = × 9/8 −25% → × 0.75 = × 3/4
Successive changes
net = a + b + ab/100 +20% then −20%: 20 − 20 − 400/100 = −4%
More than, less than: switching the base
A is r% more than B → B is r / (100 + r) × 100% less than A A is r% less than B → B is r / (100 − r) × 100% more than A
Price × quantity: keeping spending fixed
Shortcut: Use fractions: +25% is × 5/4, so the quantity must be × 4/5, a 20% cut. +20% is × 6/5, so × 5/6, a 16.67% cut.
Formulas stick when you use them
The Percentages lesson has worked problems and quick checks for every formula here. Free to start.
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