# Understanding percentages

## In a nutshell

The symbol that represents percent is $\%$. Per cent means "out of a hundred", and $100\%$ represents the whole amount. Percentages can be written as fractions with a denominator of $100$, or can be converted into decimals.

**Curiosity: **The word "percent" comes from the language Latin, which was spoken in ancient Rome. "Per" means "each" or "every", and "cent" means "one hundred". The word *cent**imetre comes from the same root word - there are $100$ centimetres in a metre!*

## Relating percentages and fractions

Percentages can be written as fractions by placing the percentage number over a denominator of $100$. The means that the percentage ($\%$) sign is effectively another way of writing $\dfrac{}{100}$. Any percentage can be converted to a fraction with a denominator of $100.$

#### Procedure

1. | Recognise the percentage sign $(\%)$ and recall that this means "out of one hundred". |

2. | Write out a fraction with a denominator of $100.$ |

3. | Write the number in front of the percentage as the numerator of the fraction. |

##### Example 1

*Write *$15\%$* as a fraction.*

*Recognise the *$\%$* and remember this relates to a fraction out of *$100.$

$\%= \dfrac{}{100}$

*Write the number in front of the percentage as the numerator of the fraction.*

*$15\%= \underline{\dfrac{15}{100}}$*

**Note: ***As you may have spotted, this fraction can be simplified, as *$\dfrac{15}{100} = \dfrac{3}{20}$*. Make sure to fully simplify the fraction if the question asks you to.*

##### Example 2

*Are* $40\%$* and *$\dfrac{20}{50}$ *equivalent?*

*Convert $40 \%$* *into a fraction out of $100.$*

*$40 \% = \dfrac{40}{100}$*

*Expand the other fraction so that it has a denominator of $100$, to check whether the fractions are equivalent.*

$\dfrac{20}{50}\times \dfrac{2}{2}=\dfrac{40}{100}$

*Therefore:*

$\underline{\text{Yes}},$ $40 \%$ *and* $\dfrac{20}{50}$ *are* *equivalent*.

## Relating percentages and decimals

To change a percentage into a decimal, you divide by $100.$ To change a decimal into a percentage ($\%$), you apply the *inverse operation* and multiply by $100.$

##### Example 3

*Change $42 \%$ into a decimal.*

*Divide the number in front of the percentage sign by $100$.*

*$42 \% \longrightarrow 42 \div 100 = 0.42$*

*Therefore:*

*$42\% = \underline{0.42}$*