A price tag offers 20% off, followed by another 10%. That sounds like 30% in total. Yet a hypothetical price of 1000 units becomes 720: a 28% reduction. The second 10% is calculated from a different amount. A percentage always means a share of a stated base. The base is the quantity treated as 100% in that calculation.
Definitions of fractional numbers are in the first lesson on number sets. Here we will focus on what we are comparing.
A recipe as a proportion
Four equal servings need 300 g of grain. For six servings we need 6/4 = 3/2 times as much: 300 ⋅ 3/2 = 450 g. A ratio is the result of dividing one quantity by another; here 300 g / 4 servings = 75 g per serving. We preserve that ratio:
(300)/(4) = (450)/(6) = 75.
An equality of two ratios is called a proportion. Both numerators here are grams and both denominators are numbers of servings, in the same order on either side. The cross-product check gives 300 ⋅ 6 = 450 ⋅ 4 = 1800. This works because both denominators are nonzero: we can multiply the equality by their product. Definition and conditions in OpenStax, §6.5.
Not everything in a kitchen scales the same way: keeping the mass per serving constant does not prove that cooking time must also be multiplied by 3/2.
For fraction arithmetic, the size of each part matters. To add 3/4 and 1/2 of a litre, express both amounts in quarters: 3/4 + 2/4 = 5/4 litres. Do not add the denominators: they name the size of the parts. To find half of 3/4 of a litre, multiply: 1/2 ⋅ 3/4 = 3/8 of a litre. Dividing by 3/4 means multiplying by its reciprocal, 4/3; zero has no reciprocal.
A percentage names hundredths of a base
1% = 1/100, so 20% = 20/100 = 1/5 = 0.2. These are different ways to write the same share. OpenStax, §6.1: converting percentages, fractions and decimals.
| What we want | Example with a base of 1000 |
|---|---|
| A share from a percentage | 20% of 1000: 1000 · 0.2 = 200 |
| The whole from a known share | If 200 is 20%, the whole is 200 / 0.2 = 1000 |
| The percentage from a part and a whole | 200 out of 1000: 200 / 1000 · 100% = 20% |
The share in the second row is nonzero; the base in the third row is nonzero. Percentages can exceed 100%: 150% of a recipe means one and a half times its original quantities.
Why two discounts do not add
After a 20% discount, 80% of the original price remains: 1000 ⋅ 0.8 = 800. The second discount removes 10% of 800, which is 80 rather than 100. That leaves 800 ⋅ 0.9 = 720.
We removed 280 units from the original thousand: (1000 − 720)/1000 = 0.28 = 28%. We could instead multiply the retained fractions directly: 0.8 ⋅ 0.9 = 0.72. Adding 20% and 10% would work if both discounts were calculated separately from the original 1000. That is a different condition, which needs checking.
Percentages and percentage points
Suppose a category’s share changes from 10% to 12%. The difference is 2 percentage points: 12 − 10 = 2. The relative increase in the share itself is 20%: (12 − 10)/10 ⋅ 100% = 20%. These answer different questions. The official ONS explanation defines a percentage point as the difference between percentages.
To calculate a percentage change, divide the difference by the old value. If that value is zero, this calculation is undefined: going from 0 to 5 is not a finite percentage increase relative to zero.
Branch · CheckThis extra example of a changing base can be skipped.
Why plus 20% does not undo minus 20%
After a reduction from 1000 to 800, a 20% increase gives only 800 ⋅ 1.2 = 960. Returning to 1000 requires adding 200 relative to the current 800: 200/800 = 0.25 = 25%. The inverse of multiplying by 0.8 is dividing by 0.8, or multiplying by 1.25.
Before your next percentage calculation, add the words “of what?” In the recipe the base is four servings; in the second discount it is already 800; in the relative change of a share it is the previous 10%. Choosing the base determines the right operation.