Percentage Change vs Percentage Points vs Percentage Difference

A savings rate rises from 5 percent to 8 percent. Is that a 3 percent rise or a 60 percent rise? Both statements are true, and they mean different things. Most percentage confusion comes from not saying which comparison you are making, so it is worth being precise about three of them.

Percentage of a number

The most basic question: what is x percent of y? Multiply y by x and divide by 100. Fifteen percent of 80 is 15 × 80 ÷ 100 = 12. The reverse question, what percent is a part of a whole, divides the part by the whole and multiplies by 100: 12 is 15 percent of 80. The Percentage Calculator answers both and shows the working.

Percentage change

Percentage change compares a new value with the old one, measured against the old value.

Change = (new − old) ÷ old × 100.

• From 80 to 100: (100 − 80) ÷ 80 × 100 = +25 percent.
• From 100 to 80: (80 − 100) ÷ 100 × 100 = −20 percent.

The two results are different because the base is different. That asymmetry is the source of the classic trap: a price that rises 25 percent must fall 20 percent to return to where it started. Likewise, an investment that loses 50 percent needs a 100 percent gain to recover: from 100 to 50 is −50 percent, from 50 back to 100 is +100 percent.

You cannot calculate a percentage change from zero, because dividing by zero is undefined. The calculator says so instead of showing a meaningless number.

Percentage points

When the numbers are themselves percentages, the plain subtraction is measured in percentage points.

• Savings rate from 5 percent to 8 percent: a rise of 3 percentage points.
• The same change as a percentage change: (8 − 5) ÷ 5 × 100 = +60 percent.

If an interest rate moves from 4 percent to 5 percent, the loan costs 1 percentage point more, and the rate itself is 25 percent higher. News stories often mix these up, so read carefully whether a figure is points or percent. When you write about it, say which you mean.

Percentage difference

Percentage difference compares two values when neither is the original, such as two products, two cities or two sets of test results. It divides the gap by the average of the two.

Difference = |a − b| ÷ ((a + b) ÷ 2) × 100.

For 80 and 100: 20 ÷ 90 × 100 = 22.22 percent. Notice that this is neither +25 nor −20, and it is the same whichever number you put first. Use percentage difference when there is no before and after; use percentage change when there is.

Increasing and decreasing by a percentage

To raise a number by p percent, multiply it by 1 + p ÷ 100. To lower it, multiply by 1 − p ÷ 100. A 20 percent rise on 50 is 50 × 1.20 = 60, and a 20 percent fall is 50 × 0.80 = 40. The calculator has both modes. When you apply several changes, multiply the factors: a 25 percent rise followed by a 20 percent fall is 1.25 × 0.80 = 1.00, which is no change at all. Stacked discounts work the same way.

Finding the whole from a part

If 12 is 15 percent of some number, the number is 12 ÷ 0.15 = 80. This is how you recover a full price from a deposit or a total from a share.

A quick guide

Common mistakes

1. Treating a drop from 8 percent to 5 percent as a 3 percent drop. It is 3 percentage points, or a 37.5 percent fall.
2. Using the new value instead of the old one as the base for a change.
3. Subtracting the same percentage to undo an increase.
4. Averaging percentages of different-sized groups without weighting them.

The same care helps with everyday numbers. The Discount Calculator applies a percentage to a price, the Inflation Calculator shows a yearly percentage rise compounding over time, and the Salary Calculator lets you see how a percentage raise changes pay in every period.