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How do you calculate successive percentage changes?

A rigorous multiplier method, general formula, and verifiable example for avoiding the mistake of adding percentages with different bases.

Convert each decimal change $r_i$ into the multiplier $1+r_i$ , multiply the factors, then subtract $1$ : $r_{total}=(1+r_1)(1+r_2)\cdots(1+r_n)-1.$ A decrease of $p\%$ uses $r=-p/100$ .

From a percentage to a multiplier

If the initial value is $V$ , an increase of $p\%$ produces $V\left(1+\frac{p}{100}\right),$ while a decrease of $p\%$ produces $V\left(1-\frac{p}{100}\right).$ The multiplier combines the original value and the part added or removed. For example, $+20\%$ corresponds to $1.20$ , while $-20\%$ corresponds to $0.80$ .

The formula for two changes

Write the two rates as decimals $p$ and $q$ , positive for an increase and negative for a decrease. After both changes, the value is $V_f=V(1+p)(1+q).$ Relative to $V$ , the overall rate is therefore $(1+p)(1+q)-1=p+q+pq.$ The $pq$ term explains why simply adding the percentages is generally wrong: the second change acts on the value already produced by the first.

Example: +20% followed by −20%

Starting from $100$ , the increase gives $100\cdot1.20=120.$ The following decrease is $20\%$ of $120$ , which is $24$ , so $120\cdot0.80=96.$ A direct check uses the combined multiplier: $1.20\cdot0.80=0.96.$ The final value is $96\%$ of the initial value, so the net change is $-4\%$ , not zero.

How to reverse a change exactly

To undo a $20\%$ decrease, use the reciprocal of $0.80$ : $\frac{1}{0.80}=1.25.$ The required increase is therefore $25\%$ , because $0.80\cdot1.25=1$ . More generally, a nonzero multiplier $m$ is reversed by $1/m$ . For any number of successive changes, multiply every factor first and then subtract $1$ .

Frequently asked questions

Why can’t I simply add the percentages?

You can add them only when every percentage is measured from the same base. With successive changes, each percentage acts on the value produced by the preceding step.

Does the order of the changes affect the final result?

For percentage multipliers alone, with no rounding or other operations, the final result is unchanged because multiplication is commutative. The intermediate values do change.

What increase reverses a 20% decrease?

A 25% increase. The decrease leaves a multiplier of 0.80, whose reciprocal is 1/0.80=1.25.

Are percent and percentage points the same?

No. Moving, for example, from 20% to 25% is a rise of 5 percentage points but a relative increase of 25%.