Percentages, percentage change and percentage points
8 min read · Updated 4 October 2026
Percentages are everywhere (discounts, tax, interest rates, test scores, survey results, growth figures), and most of us learned them at school. Yet a handful of mistakes come up again and again, in shops, in spreadsheets and in news headlines. Almost all of them come from one source: forgetting what the percentage is a percentage of.
This guide goes through the basic calculations, then the common traps, each with a worked example.
Percent means "per hundred"
A percentage is a fraction with 100 as the denominator. 25% means 25 per 100, or 0.25 as a decimal. To use a percentage in a calculation, divide it by 100 first.
There are three basic questions you can ask, and each has its own formula.
1. What is X% of Y? Multiply.
15% of 240 = 0.15 × 240 = 36
2. X is what percent of Y? Divide, then multiply by 100.
45 is what % of 180? 45 ÷ 180 = 0.25 → 25%
3. What is the percentage change from A to B? Find the difference, divide by the starting value.
change from 80 to 100: (100 − 80) ÷ 80 = 0.25 → +25%
change from 100 to 80: (80 − 100) ÷ 100 = −0.20 → −20%
Notice the second pair. Going from 80 to 100 is a 25% increase, but going back from 100 to 80 is a 20% decrease. The difference is the same 20 units in both cases, but the starting value (the base) is different. This asymmetry is behind most of the traps below.
The percentage calculator handles all three question types, so you can check any of these examples.
Trap 1: increases and decreases do not cancel out
A share price rises 50%, then falls 50%. Are you back where you started?
100 → +50% → 150 → −50% → 75
No: you have lost 25%. The fall was 50% of a bigger number. The same applies to salaries: a 10% pay cut followed by a 10% raise takes a salary of 50,000 to 45,000 and then to 49,500, still 1% below the original.
To undo a percentage change, you need to divide by the multiplier, not apply the opposite percentage. To recover from a 20% drop (a multiplier of 0.8), you need 1 ÷ 0.8 = 1.25, a 25% rise.
Trap 2: stacked discounts do not add up
A shop offers 20% off, plus an extra 10% off at the checkout. That is not 30% off.
price × 0.80 × 0.90 = price × 0.72
You pay 72% of the original price, so the total discount is 28%. On a 50.00 item, that is 36.00 rather than the 35.00 a true 30% discount would give. The order of the two discounts does not matter, because multiplication gives the same result either way.
The tip, tax and discount calculator applies stacked discounts in sequence and shows the effective total discount.
Trap 3: removing tax or a discount from a final price
A price including 20% VAT is 120. What was the price before tax?
The tempting answer is 120 − 20% = 96. That is wrong, because the 20% was calculated on the pre-tax price, not on 120. The correct method divides by the multiplier:
120 ÷ 1.20 = 100
Check it: 100 + 20% of 100 = 120. With a US sales tax of 8.25%, a receipt total of 54.13 gives 54.13 ÷ 1.0825 ≈ 50.00 before tax.
Discounts work the same way in reverse. If a jacket costs 60 after a 25% discount, the original price was 60 ÷ 0.75 = 80, not 60 × 1.25 = 75.
Trap 4: percentage points are not percent
When a percentage changes, there are two different ways to describe the change:
- Percentage points (pp) describe the simple difference between two percentages.
- Percent describes the change relative to the starting value.
If an interest rate rises from 4% to 5%:
- it has risen by 1 percentage point (5 − 4);
- it has risen by 25 percent ((5 − 4) ÷ 4).
Both statements are true, but they sound very different. "Rates up 25%" and "rates up one point" describe exactly the same change. A rise from 2% to 3% is also one percentage point, but it is a 50% relative increase.
The distinction matters most in news and health statistics. If a treatment reduces the risk of a condition from 2% to 1%, it is accurate to say it "halves the risk" (a 50% relative reduction), and also accurate to say it reduces the risk by 1 percentage point (an absolute reduction). The relative figure sounds more dramatic; the absolute figure tells you how many people are actually affected: one in a hundred. Good reporting gives both.
In finance, small changes are often given in basis points: one basis point is 0.01 percentage points. A rate cut of 25 basis points takes a rate from, say, 4.50% to 4.25%.
Trap 5: "more than" and "less than" use different bases
If product X costs 130 and product Y costs 100:
- X is 30% more than Y, because 30 ÷ 100 = 0.30;
- Y is 23.1% less than X, because 30 ÷ 130 ≈ 0.231.
Both are correct. The base is whatever comes after "than". When comparing prices, salaries or populations, be clear which one you are using as the reference.
Trap 6: markup is not margin
Businesses use two different percentages to describe profit on a sale, and they are easy to confuse.
- Markup is profit as a percentage of cost.
- Margin is profit as a percentage of selling price.
An item that costs 60 and sells for 90 makes a profit of 30:
markup = 30 ÷ 60 = 50%
margin = 30 ÷ 90 ≈ 33.3%
The confusion is expensive. If you need a 40% margin and mistakenly add a 40% markup to a cost of 60, you charge 84 and get a margin of only 24 ÷ 84 ≈ 28.6%. The correct price for a 40% margin is cost ÷ (1 − 0.40) = 60 ÷ 0.60 = 100.
| Markup | Equivalent margin |
|---|---|
| 25% | 20% |
| 50% | 33.3% |
| 100% | 50% |
| 200% | 66.7% |
Margin can never reach 100% (that would mean the item cost nothing), while markup has no upper limit. The margin and markup calculator converts between the two and finds the selling price for a target margin.
Trap 7: averaging percentages
Two online shops report their conversion rates:
| Shop | Visitors | Orders | Conversion rate |
|---|---|---|---|
| A | 1,000 | 100 | 10% |
| B | 10 | 5 | 50% |
The simple average of the two rates is 30%. But across both shops, 105 orders came from 1,010 visitors, so the true combined rate is 105 ÷ 1,010 ≈ 10.4%. Shop B's tiny number of visitors should barely move the result.
When combining percentages from groups of different sizes, go back to the underlying counts: add up the parts and the totals separately, then divide. This is a weighted average, and it is what you almost always want.
Trap 8: growth over several periods
If something grows 10% a year for three years, it has not grown 30% in total, because each year's growth is calculated on a bigger base:
1.10 × 1.10 × 1.10 = 1.331 → +33.1%
Going the other way, if sales rose from 200 to 288 over two years (a 44% total increase), the average yearly growth is not 22%. It is the rate that, applied twice, gives a factor of 1.44:
√1.44 = 1.20 → 20% per year
Check it: 200 × 1.20 = 240, and 240 × 1.20 = 288. This is called the compound annual growth rate. For more than two years, use the nth root: for a total factor F over n years, the yearly rate is F^(1/n) − 1.
Trap 9: big percentages from small numbers
"Complaints up 100%" sounds alarming. It could mean complaints went from 2 to 4. A percentage change from a very small base can be huge while the actual change is trivial. When a percentage looks striking, look for the underlying numbers.
Percentage change is also meaningless when the starting value is zero or negative. A company whose profit goes from −20 to +10 has improved by 30, but no percentage describes that sensibly.
Everyday examples
Tipping. An 18% tip on a bill of 64.50 is 0.18 × 64.50 = 11.61. In places where tax is added to the bill, tips are often calculated on the amount before tax; check local custom.
Test scores. 42 correct out of 56 questions is 42 ÷ 56 = 0.75, or 75%.
Battery or storage. If 48 GB of a 64 GB phone is used, that is 75% used and 25% free. If you delete 12 GB, usage drops to 36 GB, or 56.25%: a fall of 18.75 percentage points, or 25% in relative terms.
Price increases. A subscription rising from 8.99 to 10.99 is an increase of 2.00 ÷ 8.99 ≈ 22.2%, not 20%.
A checklist before you trust a percentage
- What is the base? Every percentage is "of" something. Identify it.
- Is this a change in percent or in percentage points?
- Are percentages being added or averaged? If so, check whether they should be multiplied or weighted instead.
- What are the underlying numbers? Large percentages from small counts can mislead.
- Is a reversal involved? Undoing a percentage means dividing by the multiplier.
FAQ
How do I add a percentage to a number? Multiply by 1 plus the percentage as a decimal. Adding 15% to 80 is 80 × 1.15 = 92.
How do I subtract a percentage? Multiply by 1 minus the percentage. 80 minus 15% is 80 × 0.85 = 68.
Can a percentage be more than 100%? Yes, when comparing to a base. Sales that triple have risen by 200%, and a price of 250 is 250% of a price of 100. A share of a whole, such as the fraction of a pizza eaten, cannot exceed 100%.
What is the difference between a percentage and a percentile? A percentage is a proportion. A percentile is a position in a ranked list: scoring in the 90th percentile means you scored higher than about 90% of people, regardless of what your score was as a percentage.