Percentages

What is a Percentage?

Definition Percentage

A percentage is a ratio out of $100$.
The symbol $\pourcent$ means "percent," which comes from the phrase "per centum," meaning "out of one hundred."

Example

This grid has 100 squares. Since $\textcolor{colorprop}{23}$ out of $\textcolor{colordef}{100}$ squares are colored, we say that $\textcolor{colorprop}{23}\textcolor{colordef}{\pourcent}$ of the grid is colored.

$$\textcolor{colorprop}{23}\textcolor{colordef}{\pourcent} = \dfrac{\textcolor{colorprop}{23}}{\textcolor{colordef}{100}}$$

Converting Between Forms

Discover

Percentages, fractions, and decimals are three different ways to talk about the same value. For example, $\textcolor{colorprop}{50}\textcolor{colordef}{\pourcent}$, the fraction $\dfrac{\textcolor{colorprop}{1}}{\textcolor{colordef}{2}}$, and the decimal $0.5$ all mean "half." Being able to convert between these forms is a very useful skill.
Remark: A very important idea is that $100\pourcent = \dfrac{100}{100} = 1$.
Since $100\pourcent$ is just another way of writing the number $1$, you can multiply any number by $100\pourcent$ without changing its value. This is a very useful trick for converting decimals and fractions into percentages.

Method Percentage to Fraction

To convert a percentage to a fraction, write it as a fraction over 100, then simplify if possible.

Method Fraction to Percentage

To convert a fraction like $\dfrac{\textcolor{colorprop}{3}}{\textcolor{colordef}{4}}$ to a percentage, you have two common methods:

Method 1: Equivalent Fractions. Find an equivalent fraction with a denominator of 100.
Method 2: Multiply by $100\pourcent$. This works because multiplying by $100\pourcent$ is the same as multiplying by 1. $$ \begin{aligned} \dfrac{\textcolor{colorprop}{3}}{\textcolor{colordef}{4}} &= 0.75 \quad (\text{since } 3 \div 4 = 0.75) \\ &= 0.75 \times \textcolor{colordef}{100\pourcent} \\ &= \textcolor{colorprop}{75}\textcolor{colordef}{\pourcent} \end{aligned} $$

Method Percentage to Decimal

To convert a percentage to a decimal, divide by 100. A quick way to do this is to move the decimal point two places to the left.$$ \textcolor{colorprop}{45}\textcolor{colordef}{\pourcent} = \textcolor{colorprop}{45} \div \textcolor{colordef}{100} = 0.45 $$

Method Decimal to Percentage

To convert a decimal to a percentage, multiply by 100. A quick way to do this is to move the decimal point two places to the right and add the percent sign ($\pourcent$).$$ 0.68 = 0.68 \times 100\pourcent = \textcolor{colorprop}{68}\textcolor{colordef}{\pourcent} $$