Percentages

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.

Method Converting a percentage to a fraction

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

Method Converting a fraction to a percentage

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 prove the formula:$$\begin{aligned}\textcolor{colorprop}{x}\textcolor{colordef}{\pourcent} &= \dfrac{\textcolor{colorprop}{3}}{\textcolor{colordef}{4}} \\\frac{\textcolor{colorprop}{x}}{\textcolor{colordef}{100}} &= \dfrac{\textcolor{colorprop}{3}}{\textcolor{colordef}{4}} \\\textcolor{colorprop}{x} &= \dfrac{\textcolor{colorprop}{3}}{\textcolor{colordef}{4}} \times \textcolor{colordef}{100} \quad \text{(multiplying both sides by } \textcolor{colordef}{100}) \\&= \textcolor{colorprop}{75}\end{aligned}$$

Method Converting a percentage to a decimal

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

Method Converting a decimal to a percentage

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

To prove the formula:$$\begin{aligned}0.05 &= 0.05 \times \frac{\textcolor{colordef}{100}}{\textcolor{colordef}{100}}\\&=0.05 \times \textcolor{colordef}{100}\textcolor{colordef}{\pourcent} \\\end{aligned}$$

Ratio to Percentage

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.

Method Ratio to Percentage

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

Example

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.

The $\textcolor{colorprop}{\text{part}}$ is the number of correct answers: $\textcolor{colorprop}{21}$.
The $\textcolor{colordef}{\text{whole}}$ is the total number of questions: $\textcolor{colordef}{24}$.
$\begin{aligned}[t]\text{Percentage of correct answers} &= \left(\frac{\textcolor{colorprop}{\text{part}}}{\textcolor{colordef}{\text{whole}}} \times \textcolor{colordef}{100}\right)\textcolor{colordef}{\pourcent}\\&= \left(\frac{\textcolor{colorprop}{21}}{\textcolor{colordef}{24}} \times \textcolor{colordef}{100}\right)\textcolor{colordef}{\pourcent}\\ &= \textcolor{colorprop}{87.5}\textcolor{colordef}{\pourcent} \quad \text{(compute: } 21 \div 24 \times 100 {=}87.5)\end{aligned}$

Comparing Percentages

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?

$\begin{aligned}[t]\text{Percentage of girls in Parliament A} &= \left(\frac{\textcolor{colorprop}{26}}{\textcolor{colordef}{50}} \times \textcolor{colordef}{100}\right)\textcolor{colorprop}{\pourcent} \\&= \textcolor{colorprop}{52}\textcolor{colordef}{\pourcent}\end{aligned}$
In Parliament A, for every $\textcolor{colordef}{100}$ deputies, there would be $\textcolor{colorprop}{52}$ girls in proportion.
$\begin{aligned}[t]\text{Percentage of girls in Parliament B} &= \left(\frac{\textcolor{colorprop}{30}}{\textcolor{colordef}{80}} \times \textcolor{colordef}{100}\right)\textcolor{colorprop}{\pourcent} \\&= \textcolor{colorprop}{37.5}\textcolor{colordef}{\pourcent}\end{aligned}$
In Parliament B, for every $\textcolor{colordef}{100}$ deputies, there would be $\textcolor{colorprop}{37.5}$ girls in proportion.

Conclusion: Since $\textcolor{colorprop}{52}\textcolor{colordef}{\pourcent} \gt \textcolor{colorprop}{37.5}\textcolor{colordef}{\pourcent}$, girls are better represented in Parliament A.

Method Comparing Percentages

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

Formula to Find a Part Using Percentages

Method Finding a Part Using Percentages

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

$$\begin{aligned}\frac{\textcolor{colorprop}{\text{part}}}{\textcolor{colordef}{\text{whole}}} &= \textcolor{colorprop}{x}\textcolor{colordef}{\pourcent} \\\textcolor{colorprop}{\text{part}} &= \textcolor{colorprop}{x}\textcolor{colordef}{\pourcent} \times \textcolor{colordef}{\text{whole}} &&\text{(multiplying both sides by }\textcolor{colordef}{\text{whole}})\end{aligned}$$

Example

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

Method 1 (using the formula)$$\begin{aligned}[t]\textcolor{colorprop}{\text{number of girls}} &= \textcolor{colorprop}{60}\textcolor{colordef}{\pourcent} \text{ of } \textcolor{colordef}{200\text{ students}} \\&= \textcolor{colorprop}{60}\textcolor{colordef}{\pourcent} \times \textcolor{colordef}{200} \\&= \frac{\textcolor{colorprop}{60}}{\textcolor{colordef}{100}} \times \textcolor{colordef}{200} \\&= \textcolor{colorprop}{120} & \text{(compute: }60{\div} 100 {\times} 200 {=}120)\end{aligned}$$
Method 2 (cross-multiplication)$$\begin{aligned}\frac{\textcolor{colorprop}{60}}{\textcolor{colordef}{100}} &= \frac{\textcolor{colorprop}{\text{number of girls}}}{\textcolor{colordef}{200}} \\\textcolor{colordef}{100} \times \textcolor{colorprop}{\text{number of girls}} &= \textcolor{colorprop}{60} \times \textcolor{colordef}{200} &&\text{(cross-multiplication)} \\\textcolor{colorprop}{\text{number of girls}} &= \frac{\textcolor{colorprop}{60} \times \textcolor{colordef}{200}}{\textcolor{colordef}{100}} &&\text{(dividing both sides by }\textcolor{colordef}{100}) \\\textcolor{colorprop}{\text{number of girls}} &= \textcolor{colorprop}{120} &&\text{(compute: }(60 \times 200)\div 100 {=}120)\end{aligned}$$

Formula to Find the Whole Using Percentages

Method Finding the Whole Using Percentages

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

$$\begin{aligned}\frac{\textcolor{colorprop}{\text{part}}}{\textcolor{colordef}{\text{whole}}} &= \textcolor{colorprop}{x}\textcolor{colordef}{\pourcent}\\\textcolor{colorprop}{\text{part}} &= \textcolor{colorprop}{x}\textcolor{colordef}{\pourcent} \times \textcolor{colordef}{\text{whole}} \\\textcolor{colordef}{\text{whole}} &= \dfrac{\textcolor{colorprop}{\text{part}}}{\textcolor{colorprop}{x}\textcolor{colordef}{\pourcent} } &&\text{(dividing both sides by } \textcolor{colorprop}{x}\textcolor{colordef}{\pourcent})\end{aligned}$$

Example

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.

Method 1 (using the formula)$$\begin{aligned}\textcolor{colordef}{\text{total students}} &= \dfrac{\textcolor{colorprop}{14}}{ \textcolor{colorprop}{40}\textcolor{colordef}{\pourcent} } \\&= \dfrac{\textcolor{colorprop}{14}}{ \left( \dfrac{\textcolor{colorprop}{40}}{\textcolor{colordef}{100}} \right) } \\&= \textcolor{colordef}{35} &&\text{(compute: }14 \div (14\div 100)=35)\end{aligned}$$
Method 2 (cross-multiplication)$$\begin{aligned}\frac{\textcolor{colorprop}{40}}{\textcolor{colordef}{100}} &= \frac{\textcolor{colorprop}{14}}{\textcolor{colordef}{\text{total students}}} \\\textcolor{colorprop}{40} \times \textcolor{colordef}{\text{total students}} &= \textcolor{colordef}{100} \times \textcolor{colorprop}{14} &&\text{(cross-multiplication)} \\\textcolor{colordef}{\text{total students}} &= \frac{\textcolor{colordef}{100} \times \textcolor{colorprop}{14}}{\textcolor{colorprop}{40}} &&\text{(dividing both sides by }\textcolor{colorprop}{40}) \\\textcolor{colordef}{\text{total students}} &= \textcolor{colordef}{35} &&\text{(compute: }(100 \times 14) \div 40 =35)\end{aligned}$$

Percentage Increase or Decrease

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 $10\pourcent$.

Definition Percentage Increase or Decrease

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.

Method Calculating the New Quantity for a Percentage Increase in Two Steps

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

Example

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

Calculate the increase: $$ \begin{aligned} \text{Increase} &= 20\pourcent \times \dollar 50 \\ &= \frac{20}{100} \times \dollar 50 \\ &= \dollar 10 \end{aligned} $$
Calculate the new price: $$ \text{New price} = \dollar 50 + \dollar 10 = \dollar 60 $$

Method Calculating the New Quantity for a Percentage Decrease in Two Steps

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

Example

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