Percentages, without the guesswork
Updated 1 November 2026 · 6 minute read
Almost every percentage question is one of four shapes: a part of a whole, one number compared with another, a change from one value to another, or a value adjusted by a percentage. The wording changes — discount, tax, increase, margin, interest — but the arithmetic does not. This guide gives the formula for each shape, works an example, and collects the mistakes that cause most wrong answers.
1. Percent of a number
“What is 17% of 1,200?” A percent is a fraction with 100 in the denominator, so 17% is 17 ÷ 100 = 0.17.
part = whole × percent ÷ 1001,200 × 17 ÷ 100 = 1,200 × 0.17 = 204. The same answer comes from 1,200 × 17 = 20,400, then ÷ 100.
This is the shape behind discounts and taxes: a shop takes 17% off a 1,200 price, so the reduction is 204 and you pay 996. See the third shape below for the direct route.
2. One number as a percent of another
“45 is what percent of 200?” Now the whole is known and the percent is unknown. Divide the part by the whole and multiply by 100.
percent = part ÷ whole × 10045 ÷ 200 = 0.225, and 0.225 × 100 = 22.5%. This is the shape used for marks, market share, completion rates and expense ratios.
Order matters. “45 as a percent of 200” is 22.5%, but “200 as a percent of 45” is 444.4%. The whole is always the number after “of”.
3. Percentage change
“A price rises from 400 to 500. By what percent?” Subtract, then divide by the starting value.
change % = (new − old) ÷ old × 100(500 − 400) ÷ 400 = 0.25, so the price rose 25%. A fall from 500 to 400 is (400 − 500) ÷ 500 = −20%, not −25%. The same move is 25% up or 20% down, because the base changed.
This asymmetry is not a trick. It is why an investment that loses 50% needs a 100% gain to recover: falling from 100 to 50 is −50%, but 50 to 100 is +100%.
4. Adding or subtracting a percentage
To raise a value by a percentage, multiply by (1 + percent ÷ 100). To lower it, multiply by (1 − percent ÷ 100).
new = old × (1 ± percent ÷ 100)A 1,000 invoice with 17% tax: 1,000 × 1.17 = 1,170. A 1,000 price with a 17% discount: 1,000 × 0.83 = 830. Notice there is no separate “find 17%, then add” step — multiplying by 1.17 does both at once.
Adding and then subtracting the same percentage does not return the original. 1,000 × 1.17 = 1,170, then 1,170 × 0.83 = 971.10, which is 2.89% below where you started. The second percentage was taken from a larger base.
Percentage points are not percent
If an interest rate moves from 4% to 5%, it has risen by 1 percentage point — but by 25% in relative terms. Both statements are correct and they answer different questions. News reports often mix them, deliberately or not. When someone says “rates rose 1%”, check whether they mean 4% to 5% or 4% to 4.04%.
Reversing a percentage
A jacket costs 800 after a 20% discount. What was the original price? Subtracting 20% of 800 is wrong — the discount was taken from the original, not from the sale price. Undo the multiplication instead:
original = final ÷ (1 − discount ÷ 100) = 800 ÷ 0.8 = 1,000The same logic applies to tax-inclusive prices. If a receipt shows 1,170 including 17% tax, the pre-tax amount is 1,170 ÷ 1.17 = 1,000 and the tax is 170. Multiplying 1,170 by 0.17 would give 198.90, which is too much, because 17% was charged on 1,000, not on 1,170.
Worked examples
| Question | Working | Answer |
|---|---|---|
| 17% of 1,200 | 1,200 × 0.17 | 204 |
| 45 as a percent of 200 | 45 ÷ 200 × 100 | 22.5% |
| Change from 400 to 500 | (500 − 400) ÷ 400 × 100 | +25% |
| Change from 500 to 400 | (400 − 500) ÷ 500 × 100 | −20% |
| 1,000 plus 17% | 1,000 × 1.17 | 1,170 |
| 1,000 minus 17% | 1,000 × 0.83 | 830 |
| 1,170 including 17% tax — pre-tax | 1,170 ÷ 1.17 | 1,000 |
| 800 after a 20% discount — original | 800 ÷ 0.80 | 1,000 |
The mistakes that cause wrong answers
- Using the wrong base. Percentage change always divides by the starting value; a discount is always applied to the original price.
- Doing two percentage steps in the wrong order. Apply the tax after the discount if the tax is charged on the discounted amount — which is how most invoices work.
- Confusing points with percent. A move from 20% to 25% is 5 points and 25% relative.
- Reversing a percentage by applying it again. If the final price is known, divide; do not subtract the same percentage.
- Rounding too early. Keep full precision until the end. Converting 17% to 0.2 before multiplying can change the result by hundreds on a large invoice.
- Adding percentages of different wholes. 10% of 500 plus 5% of 200 is not 15% of 700. Add the parts, then divide by the combined whole if you need one figure.
What the calculator does
The percentage calculator offers the four shapes above as separate modes, each with its own labelled inputs, so there is nothing to rearrange by hand. It also shows the formula it used and a short reference table for common percentages of your number. All of it runs in the page: no values are submitted anywhere.
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