How Percentages Really Work: Six Everyday Calculations Explained

A percentage is just a fraction with 100 underneath. "35%" means 35 out of every 100, or 0.35. Nearly every percentage problem you will meet is a rearrangement of one relationship: part = percent × whole. Once you see that, the six situations below stop feeling like six different skills.

1. Finding a percentage of a number

Turn the percent into a decimal by dividing by 100, then multiply. To find 15% of 240, compute 0.15 × 240 = 36. A mental shortcut: find 10% by moving the decimal one place (24), find 5% as half of that (12), and add them: 24 + 12 = 36.

2. Finding what percent one number is of another

This is the same equation solved for the percent: divide the part by the whole and multiply by 100. If you answered 42 questions correctly out of 60, your score is 42 ÷ 60 = 0.70, which is 70%. The most common error is dividing the wrong way round; the whole always goes on the bottom.

3. Working backwards: finding the original number

This is the one people most often get stuck on. Suppose you know that 18 is 12% of some unknown total. Rearrange the same relationship: whole = part ÷ percent. So the total is 18 ÷ 0.12 = 150.

This is how you work out a population from a survey result ("12% of respondents is 18 people, so about 150 people were surveyed"), a pre-tax price from a tax amount, or a full salary from a bonus expressed as a share. Whenever you know a piece and what share of the whole that piece represents, divide.

4. Percent change

Percent change compares a new value with an old one: (new − old) ÷ old × 100. If a monthly electricity bill goes from 80 to 92, the change is (92 − 80) ÷ 80 = 0.15, an increase of 15%. A negative result means a decrease. Note that the old value is always the base, not the new one, which is why the order of the numbers matters.

Notice that going up 15% and then back down to where you started is a drop of only about 13%. The two directions use different bases, so they are not mirror images.

5. Increasing or decreasing a number by a percentage

To raise a value by 8%, multiply by 1.08. To cut it by 8%, multiply by 0.92. Folding the change into a single multiplier is faster than calculating the percentage and then adding or subtracting it, and it makes chains of changes easy.

6. Why +20% then −20% does not cancel out

Start at 100. Add 20% and you have 120. Now remove 20% of that: 20% of 120 is 24, leaving 96. You ended 4% below where you began. The multipliers show why: 1.20 × 0.80 = 0.96. Percentages applied one after another multiply, they do not add.

This matters in real decisions. A stock that falls 50% needs to rise 100% to recover. A shop that marks a price up 25% and later offers "25% off" does not return to the original price: 1.25 × 0.75 = 0.9375, so the final price is 6.25% below where it started.

Percentage points versus percent

If an interest rate rises from 4% to 5%, it has risen by one percentage point, but by 25% in relative terms (1 ÷ 4). News stories and ads often blur this on purpose or by accident. When you see a headline about a change in a percentage, ask which of the two it means, and what the starting figure was. A "50% increase in risk" from 2 in 100,000 to 3 in 100,000 is very different from a 50-point jump.

Common mistakes to avoid

• Using the wrong base. Percent change is always measured against the starting value.
• Adding successive percentages. Two discounts of 20% and 10% are not a 30% discount; they multiply to 0.80 × 0.90 = 0.72, a 28% discount.
• Forgetting to convert. Typing 15 instead of 0.15 into a calculator multiplies by fifteen instead of fifteen hundredths.
• Rounding too early. Keep full precision through the calculation and round only the final answer.

If you want to check any of the examples above, or try your own numbers, the calculator linked below handles all of these cases, including working backwards from a part and its percentage to the original total.

Try it yourself: the Percentage Calculator, including the "Find the total" tab →