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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 $$\displaystyle \frac{26}{178} = 0\Torange{.14}6 = \Torange{14.}6\%. $$

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.
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.

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