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Percentage

What is a percentage?

A percentage is one-hundredth of a number. It uses the symbol $\%$. A percentage is a fraction where the denominator is $\Tblue{100}$.

$\Tred{50}\% = \frac{\Tred{50}}{\Tblue{100}} = \frac{1}{2} = 0.5,\quad \Tred{40}\% = \frac{\Tred{40}}{\Tblue{100}} = \frac{2}{5} = 0.4.$

To express a number as a percentage, write it in decimal form and multiply it by $\Tblue{100}$.

$0\Torange{.43}2 = \Tblue{100}\times 0\Torange{.43}2\% = \Torange{43.}2\%,\quad \frac{2}{7} = 0\Torange{.28}6 = \Torange{28.}6\%$

Percentages are often used to represent a fraction of a quantity.

Half a cake is the same as $50\%$ of the cake.

$25\%$ of $247$ is $\Torange{25}\%\times 247 = 0\Torange{.25}\times 247 = 61.75$

$26$ is $14.6\%$ of $178$ because $\displaystyle \frac{26}{178} = 0\Torange{.14}6 = \Torange{14.}6\%.$

Percentage change

Percentages are used to tell how a quantity will increase or decrease.

• Percentage change from an old number $\Tblue{O}$ to a new one $\Tgreen{N}$ is $\Tred{p} = \frac{\Tgreen{N}- \Tblue{O}}{\Tblue{O}} = \frac{\Tgreen{N}}{\Tblue{O}} - 1$

The price of a shirt went from $\Tblue{\11}$ to $\Tgreen{\12}$. It increased by $\frac{\Tgreen{12}- \Tblue{11}}{\Tblue{11}} = \frac{1}{\Tblue{11}} =\Tred{0.09} = \Tred{9\%}.$

• To know a value after the change $p$, multiply by $1+p$ $\Tgreen{N} = \Tblue{O} \times (1+\Tred{p})$

A population of $\Tblue{35}$ million increased by $\Tred{2\%}$. It is now $\Tblue{35}\cdot (1+\Tred{2\%}) = \Tblue{35} \times 1\Tred{.02} = \Tgreen{35.7}$

• To find the value before the change, divide by $1+p$. $\Tblue{O} =\Tgreen{N} \div (1+\Tred{p})$

After a $\Tred{20\%}$ discount, the price of a toy is $\Tgreen{\ 8}$. Before, it was $\frac{\Tgreen{8}}{1\Tred{-20\%}}=\frac{\Tgreen{8}}{1\Tred{-0.2}} = \frac{\Tgreen{8}}{0.8} = \Tblue{\ 10}$

Discounts in sales are often described using percentages. The new price is sometimes not given, so you need to know how to work it out.

Repeated percentage change

A change in percentage can be repeated several times.

The money I invest in a savings account increases by $4\%$ each year due to interest payment.

For a yearly percentage change of $\Tred{p\%}$, the new value $\Tgreen{N}$ after $\Torange{n}$ years from an old value $\Tblue{O}$ is $\Tgreen{N} = \Tblue{O} \times (1+\Tred{\frac{p}{100}})^\Torange{n}.$

I invest $\Tblue{1000}$ in a bank account with an interest rate of $\Tred{4\%}$. After $\Torange{3}$ years, I will have a balance of $\Tblue{1000} \times (1 + \Tred{4\%})^\Torange{3} = \Tblue{1000} \times 1\Tred{.04}^\Torange{3} = \Tblue{1000}\times 1.125 = \Tgreen{1125}.$

The population in this bacterial colony doubles every generation, an increase of 100%.