Percent vs percentage points, and common percent mistakes
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The formulas for percent are simple, yet the answers often come out wrong. The cause is rarely the multiplication. It’s losing track of the base: “percent of what?” This guide sets out the three basic calculations, then works through the classic base-value mistakes with numbers.
The three basic percent calculations
Almost every percent question is one of these three.
| Question | Formula | Example |
|---|---|---|
| What is B% of A? | A × B ÷ 100 | 12% of 250 is 250 × 12 ÷ 100 = 30 |
| A is what percent of B? | A ÷ B × 100 | 30 is 30 ÷ 250 × 100 = 12% of 250 |
| By what percent did A change to become B? | (B − A) ÷ A × 100 | From 80 to 100 is (100 − 80) ÷ 80 × 100 = a 25% increase |
In all three, the number you divide by is the base. In the third, the base is A, the value before the change. If you divide the change from 80 to 100 by 100 instead, you get 20%, which answers a different question: “80 is 20% less than 100.”
Percent and percentage points
When you compare two rates, there are two ways to describe the change. Say 500 people were asked a question, and the number answering “yes” rose from 20 (4%) to 25 (5%).
- Percentage points are the difference between the two rates. 5 − 4 = 1, so the rate rose by 1 percentage point.
- Percent is the relative change. (5 − 4) ÷ 4 × 100 = 25, so it rose by 25%. Using the head counts gives the same answer: (25 − 20) ÷ 20 × 100 = 25%.
Both statements are correct, but they mean different things. Writing that the rate “rose by 1%” would be wrong, because 4% rising by 1% is 4 × 1.01 = 4.04%. Glossaries and style guides from statistical offices make this distinction: use percentage points for the difference between rates. When you read figures that are themselves percentages, such as interest rates, approval ratings or unemployment rates, check whether “rose” refers to the difference or to the rate of change.
Raising and then lowering by the same percent doesn’t bring you back
Raise 100 by 20% and you get 120. Lower 120 by 20% and you get 120 × 0.8 = 96, not 100. The 20% on the way up is 20% of 100 (20), while the 20% on the way down is 20% of 120 (24). The base is different.
In one step, 1.2 × 0.8 = 0.96, so lowering first and raising second gives the same 96, which is 4% below the original. To get from 120 back to 100, you need to lower it by 20 ÷ 120 × 100 = about 16.67%.
Stacked discounts multiply; they don’t add
A 30% discount followed by a further 20% discount is not 50% off. Starting from 200:
- 30% off: 200 × 0.7 = 140
- A further 20% off: 140 × 0.8 = 112
The share that remains is 0.7 × 0.8 = 0.56, so the overall discount is 44%. The second discount is taken from the already reduced price of 140, so it removes 20% of 140 (28), not 20% of 200 (40).
Repeated increases work the same way. Raise 100 by 10% twice and you get 100 × 1.1 × 1.1 = 121, a rise of 21%, not 20%. Compound interest follows this principle.
Working back to the price before a discount
If the price after a 30% discount is 84, the original price is 84 ÷ (1 − 0.3) = 84 ÷ 0.7 = 120. To check: 120 × 0.7 = 84.
The common mistake here is to add 30% back onto 84. That gives 84 × 1.3 = 109.2, which doesn’t match 120. The discount took away 30% of the original price, but adding back puts on 30% of the discounted price. To recover the original from a value that was reduced by a rate r, divide by (1 − r); from a value that was increased by r, divide by (1 + r).
Increases and decreases aren’t symmetrical
When 50 becomes 100, that’s (100 − 50) ÷ 50 × 100 = a 100% increase. When 100 becomes 50, that’s (50 − 100) ÷ 100 × 100 = −50%, a 50% decrease. The difference is 50 both times, but the bases are 50 and 100, so the percentages differ.
This is why a decrease can’t exceed 100% (as long as the value doesn’t go below 0), while an increase has no upper limit. It’s also why a value that has fallen by 50% has to rise by 100%, not 50%, to get back to where it was.
What to check before you calculate
- What is the base? “20% more” and “20% less” use different bases, so one isn’t the reverse of the other. Decide what you’re dividing by first.
- Difference or rate of change? If you’re comparing two rates, say whether you mean percentage points or percent.
- Did it change more than once? Don’t add the percentages. Multiply by (1 ± rate) for each step in turn.
- Are you working backwards? Don’t apply the same percent in the opposite direction. Divide by (1 ± rate).
Enter the numbers from this guide into the related tools below to confirm that you get the same results.