Percentage increase and decrease

Increasing or decreasing a quantity by a certain percentage is the most frequent manoeuvre in which we meet the symbol $\%$. The key idea is that increasing by $p\%$ is the same as multiplying by a single factor, and the same holds for a decrease.

Property — Percentage increase and decrease

• If a quantity $a$ increases by $p\%$: new value $= a + \dfrac{p}{100}\,a = \dfrac{100+p}{100}\,a$.
• If a quantity $a$ decreases by $p\%$: new value $= a - \dfrac{p}{100}\,a = \dfrac{100-p}{100}\,a$.

In both cases the initial value is multiplied by a single factor: $\frac{100+p}{100}$ for the increase, $\frac{100-p}{100}$ for the decrease. Reasoning in terms of multiplicative factors also makes immediate the cases where several changes follow one another.

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Topics: Percentages
Concepts: Percentage increase · Percentage · Percentage decrease
Skills: Calculating · Using formulae