Percentages

A percentage is a ratio out of \(100\).
\(\pourcent\) reads "percent," which is short for the Latin term "per centum," meaning "by the hundred."$$\textcolor{colorprop}{x}\textcolor{colordef}{\pourcent} = \frac{\textcolor{colorprop}{x}}{\textcolor{colordef}{100}}$$

As \(\textcolor{colorprop}{23}\) parts out of \(\textcolor{colordef}{100}\) are colored, this is \(\textcolor{colorprop}{23}\textcolor{colordef}{\pourcent} = \dfrac{\textcolor{colorprop}{23}}{\textcolor{colordef}{100}}\).

By definition, the percentage \(\textcolor{colorprop}{5}\textcolor{colordef}{\pourcent}\) is equal to the fraction \(\dfrac{\textcolor{colorprop}{5}}{\textcolor{colordef}{100}}\). Also, the fraction \(\dfrac{\textcolor{colorprop}{5}}{\textcolor{colordef}{100}}\) is equal to the decimal number \(0.05\), since \(5\div 100=0.05\). It is important to be able to convert between these different representations of a percentage.

To convert the percentage \(\textcolor{colorprop}{5}\textcolor{colordef}{\pourcent}\) into a fraction, use the definition:$$\textcolor{colorprop}{5}\textcolor{colordef}{\pourcent} = \dfrac{\textcolor{colorprop}{5}}{\textcolor{colordef}{100}}$$

To convert the fraction \(\dfrac{\textcolor{colorprop}{3}}{\textcolor{colordef}{4}}\) into a percentage:

• Method 1 (Equivalent Fractions)
• Method 2 (Formula) $$\begin{aligned} \dfrac{\textcolor{colorprop}{3}}{\textcolor{colordef}{4}} &= \left( \dfrac{\textcolor{colorprop}{3}}{\textcolor{colordef}{4}} \times \textcolor{colordef}{100} \right)\textcolor{colordef}{\pourcent} \\ &= \textcolor{colorprop}{75}\textcolor{colordef}{\pourcent} \quad \text{(compute } 3 \div 4 \times 100 = 75) \end{aligned}$$

To convert the percentage \(\textcolor{colorprop}{5}\textcolor{colordef}{\pourcent}\) into a decimal:$$\begin{aligned}\textcolor{colorprop}{5}\textcolor{colordef}{\pourcent} &= \dfrac{\textcolor{colorprop}{5}}{\textcolor{colordef}{100}} \\&= \textcolor{colorprop}{5} \div \textcolor{colordef}{100} \\&= 0.05\end{aligned}$$

To convert the decimal \(0.05\) into a percentage:$$\begin{aligned}0.05 &= 0.05 \times \textcolor{colordef}{100}\textcolor{colordef}{\pourcent} \\&= \textcolor{colorprop}{5}\textcolor{colordef}{\pourcent}\end{aligned}$$

We use percentages to compare a part with a whole.
For example, in a class with \(\textcolor{colordef}{20}\) students, there are \(\textcolor{colorprop}{12}\) girls. To find the percentage of girls in the class, we find an equivalent fraction with a denominator of 100:$$\begin{aligned}\frac{\textcolor{colorprop}{x}}{\textcolor{colordef}{100}} &= \frac{\textcolor{colorprop}{12}}{\textcolor{colordef}{20}}\\\textcolor{colorprop}{x} &= \frac{\textcolor{colorprop}{12}}{\textcolor{colordef}{20}} \times \textcolor{colordef}{100} && \text{(multiplying both sides by }\textcolor{colordef}{100})\\\textcolor{colorprop}{x} &= \textcolor{colorprop}{60}\\\end{aligned}$$There are \(\textcolor{colorprop}{60}\textcolor{colordef}{\pourcent}\) girls in the class. In this class, for every \(\textcolor{colordef}{100}\) students, there would be \(\textcolor{colorprop}{60}\) girls in proportion.

To calculate the percentage of a ratio of a part to the whole, use the following formula:$$\textcolor{colorprop}{x}\textcolor{colordef}{\pourcent}= \left(\frac{\textcolor{colorprop}{\text{part}}}{\textcolor{colordef}{\text{whole}}} \times \textcolor{colordef}{100}\right)\textcolor{colordef}{\pourcent}$$

You took a math quiz and answered \(\textcolor{colorprop}{21}\) questions correctly out of \(\textcolor{colordef}{24}\) questions total.
Calculate the percentage of correct answers.

In Parliament A of Country A, there are \(\textcolor{colorprop}{26}\) girls out of \(\textcolor{colordef}{50}\) deputies. In Parliament B of Country B, there are \(\textcolor{colorprop}{30}\) girls out of \(\textcolor{colordef}{80}\) deputies. Hugo says, "Since there are more girls in Parliament B, girls are better represented in this parliament." Do you agree with that statement?

• Step 1: Calculate the percentage for each group.
• Step 2: Compare the percentages and conclude.

To calculate a part in a part-to-whole relationship, multiply the percentage by the whole:$$\textcolor{colorprop}{\text{part}} =\textcolor{colorprop}{x}\textcolor{colordef}{\pourcent} \times \textcolor{colordef}{\text{whole}}$$

In a school with \(\textcolor{colordef}{200}\) students, \(\textcolor{colorprop}{60}\textcolor{colordef}{\pourcent}\) are girls.
Calculate the number of girls in the school.

To calculate the whole in a part-to-whole relationship, divide the part by the percentage (as a decimal):$$\textcolor{colordef}{\text{whole}} = \dfrac{\textcolor{colorprop}{\text{part}}}{ \textcolor{colorprop}{x}\textcolor{colordef}{\pourcent} }$$

In a class, \(\textcolor{colorprop}{40}\textcolor{colordef}{\pourcent}\) of the students are girls, and there are \(14\) girls in total.
Find the total number of students in the class.

In everyday life, there are many situations where quantities are either increased or decreased by a certain percentage. For example:

• A store offers a \(20\pourcent\) discount on all items during a sale.
• A worker receives a salary increase of \(7\pourcent\).
• A person on a diet reduces their weight by \(10\pourcent\).

• A percentage increase occurs when a quantity is raised by a certain percentage.
• A percentage decrease occurs when a quantity is reduced by a certain percentage.

1. Calculate the increase: $$ \begin{aligned} \text{Increase} &= \text{Percentage of the original quantity} \\ &= \text{Percentage} \times \text{Original quantity} \end{aligned} $$
2. Calculate the new quantity: $$ \text{New quantity} = \text{Original quantity} + \text{Increase} $$

If the original price of a shirt is \(\dollar 50\) and it is increased by \(20\pourcent\), find the new price.

1. Calculate the decrease: $$ \begin{aligned} \text{Decrease} &= \text{Percentage of the original quantity} \\ &= \text{Percentage} \times \text{Original quantity} \end{aligned} $$
2. Calculate the new quantity: $$ \text{New quantity} = \text{Original quantity} - \text{Decrease} $$

If the original price of a shirt is \(\dollar \) 50 and it is decreased by \(20\pourcent\), find the new price.