How to calculate percentages properly

Four related questions people mix up constantly, and one trap that costs real money.

Most percentage confusion comes from not being precise about which of several different questions is being asked.

The four questions

• What is 15% of 240? Multiply: 36.
• What percentage is 36 of 240? Divide and multiply by 100: 15%.
• 240 increased by 15%? Multiply by 1.15: 276.
• 276 is 15% more than what? Divide by 1.15: 240.

The fourth is the reverse percentage, and it is the one people get wrong.

The trap: a rise and fall do not cancel

A price rises 20%, then falls 20%. It is not back where it started — it is 4% below. The rise was 20% of the original; the fall was 20% of the larger figure.

The same applies to investments, and there it is worse: lose 50% and you need a 100% gain to recover. This asymmetry is why percentage losses hurt more than the equivalent percentage gains help.

Percentage points

If a rate goes from 4% to 6%, that is a rise of 2 percentage points, and a 50% increase. Both are correct and they describe the same change. Reporting picks whichever sounds more dramatic, so it is worth checking which is meant.

Two useful tricks

Percentages commute: 8% of 25 is the same as 25% of 8, which is 2. When one direction looks awkward, reverse it.

Build from 10%: shift the decimal for 10%, halve it for 5%, shift twice for 1%. Most percentages are a short sum of those.

Frequently asked questions

If a price rises 20% then falls 20%, is it back to the start?

No, it is 4% lower. The increase applied to the original and the decrease applied to the larger figure, so they do not cancel.

What is the difference between percent and percentage points?

Moving from 4% to 6% is a rise of two percentage points and a 50% increase. Both are accurate; they answer different questions.

How do I find the original price before a discount?

Divide by one minus the discount as a decimal. A 25% discount means dividing by 0.75, not adding 25% back.