割合・百分率

To find $x\%$ of an amount, turn the percentage into a decimal (or fraction) by dividing by $100$, then multiply by the amount. For example $15\%$ of $80$ is $\frac{15}{100}\times 80=12$. To express one quantity as a percentage of another, compute $\frac{\text{part}}{\text{whole}}\times 100\%$. For instance $30$ out of $50$ is $\frac{30}{50}\times 100\%=60\%$.

To increase by $p\%$, multiply by the multiplier $\left(1+\frac{p}{100}\right)$; to decrease by $p\%$, multiply by $\left(1-\frac{p}{100}\right)$. For example, increasing $200$ by $20\%$ gives $200\times 1.2=240$. Percentage change is $\frac{\text{change}}{\text{original}}\times 100\%$. A change from $40$ to $50$ is $\frac{10}{40}\times 100\%=25\%$ increase.

A reverse percentage problem gives the value after a $p\%$ change and asks for the original. Divide by the multiplier that produced the change. For example, if $240$ is the amount after a $20\%$ increase, the original is $240÷1.2=200$. If $68$ is the price after a $15\%$ discount, the original is $68÷0.85=80$. The common trap is applying the percentage to the changed value instead of dividing.

Simple interest adds the same amount each year on the original principal: $I=\frac{P\times r\times t}{100}$. Compound interest earns interest on interest, so after $n$ years the total amount is $A=P{\left(1+\frac{r}{100}\right)}^n$. For example, with principal $1000$ at $5\%$ for $2$ years, simple interest is $100$, while compound gives $A=1000\times 1.05^2=1102.5$.