How to calculate percentage change — and the mistakes everyone makes

Prices go up, salaries change, a score improves from one test to the next. Percentage change turns those differences into a single comparable number — and it is also one of the easiest calculations to get subtly wrong.

The formula

percentage change = (new − old) ÷ old × 100

Subtract the old value from the new one, divide by the old value, and multiply by 100. A positive answer is an increase; a negative one is a decrease.

• A price rising from 80 to 100: (100 − 80) ÷ 80 × 100 = +25%.
• A price falling from 100 to 80: (80 − 100) ÷ 100 × 100 = −20%.
• Rent rising from 1,200 to 1,290: 90 ÷ 1,200 × 100 = +7.5%.

The Percentage Calculator has a percentage change mode that shows this working for any two numbers.

Mistake 1: dividing by the wrong number

The two examples above use the same gap of 20, yet one is +25% and the other −20%. That is not a contradiction. A percentage change is always measured against where you started, and the starting points differ. Dividing by the new value instead of the old one is the most common error, and it makes increases look smaller and decreases look bigger than they are.

Mistake 2: expecting a rise and a fall to cancel out

A 10% rise followed by a 10% fall does not bring you back to where you started. 100 rises to 110, and 10% of 110 is 11, so it falls to 99 — a 1% loss overall. A 50% fall needs a 100% rise to recover. Successive changes multiply rather than add:

overall change = (1 + first change) × (1 + second change) − 1

The same applies to discounts. A 25% markup followed by a 25% discount leaves an item at 93.75% of its original price, and two stacked discounts of 20% and 10% add up to 28% off, not 30%. The Discount Calculator works out stacked discounts exactly.

Mistake 3: mixing up percent and percentage points

When the thing being measured is already a percentage — an interest rate, a tax rate, an exam pass rate — there are two ways to describe a change, and they give very different numbers. If a mortgage rate moves from 4% to 6%:

• it has risen by 2 percentage points (6 − 4);
• it has risen by 50 percent ((6 − 4) ÷ 4 × 100).

Both are correct; they answer different questions. News reports sometimes use whichever sounds more dramatic, so it pays to notice which one is meant.

Mistake 4: averaging yearly changes

If something grows from 100 to 150 over five years, the total change is 50%, but the average yearly growth is not 10%. Because growth compounds, the steady yearly rate that gets from 100 to 150 in five years is about 8.45%. The formula for this compound annual growth rate is:

yearly rate = (final ÷ start)^(1 ÷ years) − 1

For 100 to 150 over five years: 1.5 raised to the power 0.2, minus 1, which is 0.0845, or 8.45% a year.

Working backwards

To find an original value from a changed one, divide rather than subtract. If a price is 126 after a 5% increase, the original was 126 ÷ 1.05 = 120 — not 126 minus 5% of 126, which gives 119.70. The same idea removes tax from a price: see how to add or remove VAT.

When percentage change does not work

• Starting from zero. Dividing by zero is undefined, so going from 0 sales to 10 cannot be expressed as a percentage increase. Give the actual numbers instead.
• Negative starting values. A loss of −50 improving to a profit of 25 is hard to express meaningfully as a percentage. Again, the actual figures are clearer.
• Tiny bases. Going from 2 to 6 is a 200% increase, which sounds dramatic but is only four more. Quote the numbers alongside the percentage.

Quick reference

For marks and grades specifically, see how to turn a test score into a percentage and a letter grade.