Percentage change: why the starting number matters

The short answer

Percentage change is the difference between the new and old values, divided by the old value, multiplied by 100. In symbols: ((new − old) ÷ old) × 100. For a change from 80 to 100, the difference is 20. Divide by the starting value, 80, to get 0.25, then multiply by 100: a 25% increase. The starting number is the reference point. This article uses non-negative starting quantities such as counts and prices, where that interpretation is straightforward.

Separate the amount of change from the percentage

A number can rise by 20 without rising by 20%. From 100 to 120, the change of 20 is 20% of the original 100. From 200 to 220, that same change is only 10% of the original 200. When reading a question, write down the starting value, the ending value and the difference separately. This makes it harder to accidentally divide by the ending number.

Work through a decrease

Suppose a quantity falls from 120 to 90. The difference is 90 − 120 = −30. Divide −30 by 120 to get −0.25, then multiply by 100 to get −25%. The minus sign tells you it is a decrease. In words, say a 25% decrease. You can check this by finding a quarter of 120: 30. Removing that quarter leaves 90.

Why equal percentage drops and rises do not cancel

Start at 100 and decrease it by 20%: you reach 80. A 20% increase from 80 adds 16, giving 96, not 100. The percentages use different starting values. To return from 80 to 100, you need to add 20, which is 20 ÷ 80 × 100 = 25%. Nothing mysterious happened: the reference number changed between the two steps. Always restart your calculation from the value at that step.

Percentage points answer a different question

If a rate goes from 10% to 15%, it rises by 5 percentage points. Relative to its starting value, the rate has increased by 50%, because (15 − 10) ÷ 10 × 100 = 50. Both descriptions can be correct, but they describe different things. Percentage points give the direct difference between two percentages. Percentage change measures that difference relative to the original rate.

What if the starting value is zero?

The usual percentage-change formula divides by the starting value, so it cannot be used when that value is zero. A change from zero to five is an increase of five units, but it has no finite percentage increase under this formula. Describe the actual quantities instead of reporting a made-up percentage. Negative starting values also need care: the sign of the formula can give a misleading everyday interpretation. State the values and context clearly.

A quick check before you finish

For a positive starting value, a new value twice as large represents a 100% increase, not a 200% increase. It is 200% of the original amount, but the extra part is 100%. A value cut in half represents a 50% decrease. These reference cases help you spot an answer that is the wrong size. Try the examples below, then use the percentage practice set to get comfortable finding percentages of numbers.

Check your answers

A quantity rises from 40 to 50. What is the percentage increase?

25%. The change is 10; 10 ÷ 40 × 100 = 25.

A quantity falls from 60 to 45. What is the percentage decrease?

25%. The drop is 15; 15 ÷ 60 × 100 = 25.

A rate changes from 4% to 6%. How many percentage points, and what relative increase?

2 percentage points; a 50% relative increase, since (6 − 4) ÷ 4 × 100 = 50.

Try the method yourself.

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