The three basic formulas
Almost every percentage question is one of three: find a part, find a rate, or find a change. Each is one line.
Percent of a number
A percentage is a fraction with 100 on the bottom. 15% is 15 ÷ 100, or 0.15. To take a percentage of a number, turn the percentage into that decimal and multiply:
- Part = whole × percent ÷ 100
- 15% of 80 = 80 × 0.15 = 12
- 7.5% of 240 = 240 × 0.075 = 18
One trick makes this easier: you can swap the numbers. 8% of 50 is the same as 50% of 8, which is 4 and needs no calculator. The percentage calculator covers this form and the next two, with the formula shown for each answer.
What percent X is of Y
This is the reverse question. You have a part and a whole and want the rate. Divide the part by the whole, then multiply by 100 to get out of decimal form:
- Percent = part ÷ whole × 100
- 30 out of 120 = 30 ÷ 120 × 100 = 25%
- 45 out of 60 = 45 ÷ 60 × 100 = 75%
The order of the division matters. 120 ÷ 30 gives 4, which tells you the whole is four times the part, a true statement that is not the answer. If your result is above 100% and that surprises you, the two numbers are probably swapped.
Percent change
To say how much something grew or shrank, compare the difference with where you started:
- Change = (new − old) ÷ old × 100
- 80 to 100: (100 − 80) ÷ 80 × 100 = +25%
- 100 to 80: (80 − 100) ÷ 100 × 100 = −20%
Notice that the same two numbers give 25% one way and 20% the other. The base is always the old value, and the base is smaller going up than coming down. That asymmetry is behind a lot of confusion about prices, salaries and investments, and the percentage change calculator exists mostly so you never have to guess which number goes underneath.
A related pair of questions comes up in pricing. Markup is the increase as a share of cost; margin is the same dollar amount as a share of the selling price. Buy at 80 and sell at 100 and the markup is 25% while the margin is 20%: again the same numbers, different base. The markup calculator converts between the two.
Percentage points vs percent
When both numbers are already percentages, there are two honest ways to describe the gap, and they are far apart. Say a savings rate moves from 4% to 5%.
- In percentage points: 5 − 4 = 1 point. Plain subtraction.
- In percent: (5 − 4) ÷ 4 × 100 = a 25% increase. Relative to the start.
Both are correct. "The rate rose 1 point" and "the rate rose 25 percent" describe the same move, and a headline can pick whichever sounds bigger or smaller. When you read a change in a rate, an approval figure, a tax bracket or a market share, check which of the two is meant before you react. When you write one, say "points" if you subtracted.
Reverse percentages
You know the result after a percentage was applied and want the number before. The temptation is to take the percentage off the result, and that is wrong, because the percentage was applied to the original, smaller number, not to the result.
Take a receipt of $119 that includes 19% tax. Taking 19% off 119 gives 96.39, which is not the pre-tax price. The total is 119% of the original, so divide by 1.19 instead: 119 ÷ 1.19 = 100. The tax was 19, and 19 ÷ 100 checks out at 19%.
- Original = result ÷ (1 + percent ÷ 100) after an increase
- Original = result ÷ (1 − percent ÷ 100) after a decrease
- A jacket costs $36 after 20% off. Original price: 36 ÷ 0.8 = $45.
The discount calculator runs the second case in both directions, from the list price to the sale price and back, and finds the percent off from any two prices.
Ten worked examples
| Question | Calculation | Answer |
|---|---|---|
| 15% of 80 | 80 × 0.15 | 12 |
| 7.5% of 240 | 240 × 0.075 | 18 |
| 30 is what percent of 120? | 30 ÷ 120 × 100 | 25% |
| 45 is what percent of 60? | 45 ÷ 60 × 100 | 75% |
| Change from 80 to 100 | (100 − 80) ÷ 80 × 100 | +25% |
| Change from 100 to 80 | (80 − 100) ÷ 100 × 100 | −20% |
| 20% off $45 | 45 × (1 − 0.20) | $36.00 |
| 25% more than 200 | 200 × 1.25 | 250 |
| Price before 19% tax, total $119 | 119 ÷ 1.19 | $100.00 |
| Two discounts of 20% on $100 | 100 × 0.8 × 0.8 | $64.00, not $60 |
Mental math shortcuts
- Start from 10%. Move the decimal point one place left. 10% of 64 is 6.4. Halve it for 5% (3.2), double it for 20% (12.8), add the two for 15% (9.6). A 15% tip on a $64 bill is $9.60 without a phone.
- 1% for odd rates. Two places left gives 1%: 1% of 250 is 2.5, so 7% is 17.5 and 23% is 57.5.
- Swap the numbers. 4% of 75 is awkward; 75% of 4 is 3.
- Work with what remains. 30% off means you pay 70%. 0.7 × 45 is 31.5, one multiplication instead of a subtraction after it.
- Increase by a multiplier. Up 25% is × 1.25; up 8% is × 1.08. 200 × 1.25 = 250.
- Chain multipliers. Up 10% then down 10% is 1.1 × 0.9 = 0.99, a 1% loss overall.
Mistakes that cost money
- Adding percentages that have different bases. A 5% raise this year and 5% next year is not 10%; it is 1.05 × 1.05 = 10.25%. Over 10 years the gap between adding and multiplying is the gap between 50% and 63%.
- Adding stacked discounts. 20% off plus a further 20% off is 36% off, not 40%: 0.8 × 0.8 = 0.64, so you pay 64% of the price. Two 25% discounts take off 43.75%, not 50%.
- Taking the percentage off the total. The tax, tip or markup was added to the smaller original number, so divide by the multiplier to get back, as in the reverse section above.
- Choosing the wrong base for a change. A stock that falls from 100 to 50 lost 50%. It needs a 100% gain, not 50%, to get back.
- Rounding too early. 7.5% of 240 is 18; rounding the rate to 8% first gives 19.2.
Frequently asked questions
What is the fastest way to work out a percentage in my head?
Find 10% by moving the decimal point one place left, then build from there. 10% of 64 is 6.4, so 5% is 3.2, 20% is 12.8 and 15% is 6.4 + 3.2 = 9.6. For 1% move the point two places.
Is 50% off then 20% off the same as 70% off?
No. The 20% comes off the already halved price, so the combined discount is 1 − 0.5 × 0.8 = 60%. A $100 item ends at $40, not $30. Discounts multiply, they never add.
Why is a 20% loss not undone by a 20% gain?
Because the second percentage works on a smaller base. 100 falls 20% to 80; a 20% gain on 80 is 16, which brings you to 96, not 100. To get back to 100 from 80 you need 20 ÷ 80 = 25%. The percentage change calculator shows the exact recovery figure for any drop.
What is the difference between percent and percentage points?
Percentage points measure the gap between two percentages by subtraction; percent measures it relative to the starting value. An interest rate that goes from 4% to 5% rises by 1 percentage point and by 25 percent. Use points when both numbers are already percentages and you want the plain difference.
How do I calculate a percentage of a percentage?
Multiply the two as decimals. 30% of 50% is 0.3 × 0.5 = 0.15, so 15%. Stacked discounts are the same arithmetic: two 25% discounts leave 0.75 × 0.75 = 0.5625 of the price, so they take off 43.75%, not 50%.