Percentages Explained: Five Calculations You Use Every Day
Clear, worked examples of percent of a number, percentage change, discounts, tips and stacked percentages, with common mistakes to avoid.
What a percentage is
'Percent' means 'per hundred'. Fifteen percent is fifteen out of every hundred, or 0.15 as a decimal. Almost every percentage problem is one of three questions about three numbers: a part, a whole and a rate. Know two of them and you can find the third.
1. Finding a percentage of a number
Multiply the number by the percentage as a decimal. 20 percent of 150 is 150 × 0.20 = 30. A mental shortcut: find 10 percent by moving the decimal point one place (15), then double it for 20 percent, or halve it for 5 percent. The Percentage Calculator does this in its first row.
2. What percent is X of Y?
Divide the part by the whole and multiply by 100. If you scored 42 out of 60, that is 42 ÷ 60 = 0.7, or 70 percent. If the whole is zero the question has no answer, and the calculator will say so.
3. Percentage change
Subtract the old value from the new one, divide by the old value, and multiply by 100. A price rising from 80 to 100 is (100 − 80) ÷ 80 = 25 percent up. Notice the trap: going back from 100 to 80 is (80 − 100) ÷ 100 = 20 percent down. Increases and decreases are measured against different starting points, which is why a stock that falls 50 percent must rise 100 percent to break even.
4. Discounts
A sale price is the original price multiplied by (1 − discount). A 25 percent discount on 200 leaves 200 × 0.75 = 150, and you save 50. Stacked discounts do not add: 20 percent off followed by a further 10 percent off multiplies to 0.8 × 0.9 = 0.72, so the total reduction is 28 percent, not 30. Retailers sometimes use this in 'extra' coupon offers, so compute the final price with the Discount Calculator instead of adding percentages together.
5. Tips and splitting a bill
A tip is a percentage of the bill, usually calculated before or after tax depending on custom. A 15 percent tip on 80 is 12, giving a total of 92. Divide by the number of people for each share: 92 ÷ 4 = 23. The Tip Calculator does the three steps at once. Tipping norms vary widely by country, and a service charge may already be on the receipt, so check before adding a tip.
Common mistakes
Mixing up percentage points and percent: moving from 4 to 6 percent interest is a rise of 2 percentage points but a 50 percent increase. Applying a percentage to the wrong base. Adding successive percentages. Rounding too early in a multi-step calculation. When something important depends on the answer, calculate it twice by different routes.
Percentages in everyday decisions
Interest rates: a loan at 8 percent a year on 1,000 costs about 80 in the first year. Nutrition labels: '% daily value' compares one serving with a reference daily amount, so 20 percent or more is high, and 5 percent or less is low. Grades: a test worth 30 percent of the course, on which you score 80 percent, adds 24 percentage points to your final mark. Tax and VAT: to add 15 percent, multiply by 1.15; to remove it from a price that includes it, divide by 1.15, not subtract 15 percent.
Percentages in data and the news
Headlines often use relative changes. If a risk rises from 1 in 1,000 to 2 in 1,000, it 'doubles' (up 100 percent), but the absolute change is only 0.1 percentage points. Both are true; the second gives context. Ask what the starting number was, and what the group size is. A '50 percent increase' from 2 to 3 cases is very different from 50 percent of 20,000.
Watch for percentages of different bases being compared, for example a rise in the percentage of a shrinking group. Averages of percentages are also not the same as the overall percentage, unless the groups are the same size.
A quick estimation toolkit
10 percent: move the decimal one place. 1 percent: move it two places. 5 percent: half of 10 percent. 15 percent: 10 percent plus 5 percent. 25 percent: divide by 4. 50 percent: halve. 75 percent: three quarters. These shortcuts let you check any answer from the calculators before you rely on it.