You don’t always start with a whole pizza or a whole hundred dollar bill. You must also be able to find $30$ % of $$250$, or $25$ % of 700 grams of sugar, as but two examples.

You have already seen that calculating with percentages is the same as using fractions. A fraction is a part of a whole. This whole could be 700 grams of sugar, for example, and we want to know how much $25$ % $\phantom{\rule{-0.17em}{0ex}}\left(\frac{25}{100}\right)$ of that is.

Rule

When a part is $n\phantom{\rule{0.17em}{0ex}}\text{\%}$ of a whole, you have that

$$\text{part}=\frac{n}{100}\times \text{whole}=\frac{n\times \text{whole}}{100}$$ |

This is the most important rule about percentages. The rule tells you that you can swap the percent sign (%) for the fraction $\frac{1}{100}$.

Example 1

**Find out how much $\text{}25\text{}\phantom{\rule{0.17em}{0ex}}\text{\%}$ of 700 grams of minced meat is **

You write “g” for “grams” in the calculation: $$\begin{array}{llll}\hfill 25\phantom{\rule{0.17em}{0ex}}\text{\%}\times 700\phantom{\rule{0.17em}{0ex}}\text{g}& =\frac{25\times 700\phantom{\rule{0.17em}{0ex}}\text{g}}{100}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{25\times 7\phantom{\rule{0.17em}{0ex}}\text{g}}{1}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =175\phantom{\rule{0.17em}{0ex}}\text{g}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$

When you calculate with percentages, it’s often concerning money. For that reason, here’s an example with money:

Example 2

**Find $\text{}20\text{}\phantom{\rule{0.17em}{0ex}}\text{\%}$ of $\text{\$}\text{}2500\text{}$ **

$$\begin{array}{llll}\hfill \text{}20\text{\%of\$}2500\text{}& =\frac{20\times \text{\$}2500}{100}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{20\times \text{\$}25}{1}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\text{\$}500\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$ You can also rely on the fact that $20\phantom{\rule{0.17em}{0ex}}\text{\%}=\frac{1}{5}$ and write

$$\text{}20\text{\%of\$}2500\text{}=\frac{1}{5}\times \text{\$}2500=\text{\$}500.$$ |

Think About This

You can draw your own picture of this problem to help you understand. Use hundred dollar bills, or something else of your choice.

Percentages are also used to study the ratio between a part and a whole. The fraction $\frac{3}{4}$ can be the ratio between three croissants and a whole pack of four croissants. Here, you need to remember that

$$\frac{1}{4}=0.25=25\phantom{\rule{0.17em}{0ex}}\text{\%}$$ |

This means that one croissant is $25$ %, and three croissants are

$$3\times 25\phantom{\rule{0.17em}{0ex}}\text{\%}=75\phantom{\rule{0.17em}{0ex}}\text{\%}$$ |

of all the croissants in the bag.

Rule

That a $\text{part}=\frac{n}{100}\times \text{whole}$ is the same as

$$\frac{\text{part}}{\text{whole}}=\frac{n}{100}$$ |

That means that the ratio between a part and the whole is $n\phantom{\rule{0.17em}{0ex}}\text{\%}$.

Remember that all fractions express a ratio.

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