Think in multipliers
"Increase by 20%" means multiply by 1.2. "Decrease by 15%" means multiply by 0.85. Turning every change into a multiplier makes long questions short.
Fractions worth memorising
% | Fraction | % | Fraction |
|---|---|---|---|
50% | 1/2 | 12.5% | 1/8 |
33⅓% | 1/3 | 11.11% | 1/9 |
25% | 1/4 | 10% | 1/10 |
20% | 1/5 | 9.09% | 1/11 |
16⅔% | 1/6 | 8⅓% | 1/12 |
14.28% | 1/7 | 6.25% | 1/16 |
25% of 30% of 800 = 1/4 × 3/10 × 800 = 60.
Percentage change
Percentage change = (new − old) ÷ old × 100.
Successive changes
Multiply the multipliers. For two changes of a% and b%, the net change is a + b + ab/100 (use negative numbers for decreases).
+10% then +20%: 10 + 20 + 2 = 32% increase.
+20% then −20%: 20 − 20 − 4 = 4% decrease. Equal rises and falls never cancel.
More than and less than
If A is r% more than B, then B is less than A by r / (100 + r) × 100%.
Example. A earns 25% more than B, so B earns 25/125 × 100 = 20% less than A.
Consumption and expenditure
Expenditure = price × consumption. If the price rises by r%, consumption must fall by r / (100 + r) × 100% to keep spending the same.
Example. Sugar becomes 25% dearer, so cut consumption by 25/125 = 20%.
Growth and depreciation
A quantity growing r% a year becomes P(1 + r/100)ⁿ after n years.
Example. A town of 50,000 growing at 10% a year has 50,000 × 1.1 × 1.1 = 60,500 people after 2 years.
Shortcuts and tricks
The standard way is to take 100 as the starting value and work step by step. These shortcuts give the same answer in one line.
Percentages as fractions
Example: 37.5% of 640.
Standard way: 640 × 37.5 ÷ 100 = 240.
Shortcut: 37.5% = 3/8, so 640 ÷ 8 × 3 = 80 × 3 = 240.
Successive changes: a + b + ab/100
Example: a value rises 30% and then falls 10%.
Standard way: 100 → 130 → 130 − 13 = 117, a 17% rise.
Shortcut: 30 + (−10) + (30 × −10)/100 = 30 − 10 − 3 = 17%.
Equal rise and fall
Example: a price goes up 10% and then down 10%.
Standard way: 100 → 110 → 110 − 11 = 99, a 1% fall.
Shortcut: an equal rise and fall always ends in a loss of x²/100 = 100/100 = 1%.
"More than" turned round
Example: A is 50% more than B. By what percentage is B less than A?
Standard way: B = 100 and A = 150, so B is 50/150 = 33⅓% less.
Shortcut: 50% = 1/2 more, so B is 1/(2 + 1) = 1/3 = 33⅓% less.
Work back from a part
Example: 35% of a number is 84. Find the number.
Standard way: 84 ÷ 0.35 = 240.
Shortcut: 35% is 7 lots of 5%, so 5% = 84 ÷ 7 = 12 and 100% = 20 × 12 = 240.
Exam patterns
Pass marks: a student needs 40% and fails by 20 marks after scoring 180, so 40% of the maximum is 200 and the maximum is 500.
Elections (two candidates): if the winner gets 60%, the loser gets 40%, and the 20% gap equals the winning margin.
Chains: if x is 20% of y and y is 50% of z, then x = 0.2 × 0.5 × z = 10% of z.