How to Calculate Percentages

What a Percentage Means

"Per cent" means "per hundred". 15% is 15 parts of every 100, or the fraction 15/100 = 0.15. To use a percentage in a calculation, divide it by 100 first; to turn a decimal into a percentage, multiply it by 100. Every calculation below is a rearrangement of one relation: part = base × percent ÷ 100.

The Five Calculations

The percentage calculator has one card for each of these, plus separate cards for discounts and markups. It works with exact fractions, so 0.1% of 0.3 is exactly 0.0003, and only rounds the figure it shows.

Percent Change: Divide by the Starting Value

A change is measured against where you started. A price that rises from 80 to 100 has gone up by 20, which is 25% of 80. If it then falls from 100 back to 80, the fall is still 20, but now it is 20% of 100. That is why a rise and a fall of the same amount are different percentages.

When the starting value is negative, as with a loss that grows from −10 to −20, the calculator divides by the size of the starting value, so the result reads as a 100% decrease rather than a confusing +100%. A change from zero cannot be expressed as a percentage at all, because it would mean dividing by zero.

Discounts and the Price Before a Sale

A 25% discount leaves 75% of the price: 80 × 0.75 = 60. Going back is where mistakes happen. If an item costs 60 after 25% off, adding 25% of 60 gives 75, which is wrong, because the 25% was taken from the original price, not from 60. Divide instead: 60 ÷ 0.75 = 80.

The same applies to tax and markups. A price of 120 that includes 20% tax had a net price of 120 ÷ 1.2 = 100; the tax is 20, which is one sixth of the gross price, not 20% of it. The VAT guide covers this in detail.

Two Changes in a Row

Percentages applied one after another multiply rather than add. Raising 100 by 10% gives 110; lowering 110 by 10% gives 99, not 100. Two discounts of 20% and 10% give 100 × 0.8 × 0.9 = 72, a total discount of 28%, not 30%. To undo an increase of X%, divide by (1 + X ÷ 100) instead of subtracting X%.

Percent and Percentage Points

When a rate itself changes, there are two ways to describe it. An interest rate that goes from 4% to 5% has risen by 1 percentage point, the plain difference between the two figures. As a change of the rate itself, it has risen by 25%, because 1 is a quarter of 4. News reports and loan offers usually mean percentage points; check which one is meant before comparing.

Mental Shortcuts

• 10% is the number divided by 10; 5% is half of that; 1% is the number divided by 100.
• X% of Y equals Y% of X: 8% of 50 is the same as 50% of 8, which is 4.
• A 50% loss needs a 100% gain to recover; a 20% loss needs a 25% gain.

Frequently Asked Questions