\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)

Percentages

Definition

Definition Percentage
A percentage is a ratio out of \(100\).
\(\pourcent\) is read ``percent'', meaning ``out of a hundred.''
Example
As \(\textcolor{colorprop}{23}\) parts out of \(\textcolor{colordef}{100}\) are colored, this is \(\textcolor{colorprop}{23}\textcolor{colordef}{\pourcent} = \dfrac{\textcolor{colorprop}{23}}{\textcolor{colordef}{100}}\).

Percentage as a Number


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)$$\begin{aligned}\dfrac{\textcolor{colorprop}{3}}{\textcolor{colordef}{4}} &= \left( \dfrac{\textcolor{colorprop}{3}}{\textcolor{colordef}{4}} \times \textcolor{colordef}{100} \right)\textcolor{colordef}{\pourcent} \\&= \textcolor{colorprop}{75}\textcolor{colordef}{\pourcent} \quad \text{(compute } 3 \div 4 \times 100 = 75)\end{aligned}$$
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:$$\begin{aligned}0.05 &= 0.05 \times \textcolor{colordef}{100}\textcolor{colordef}{\pourcent} \\&= \textcolor{colorprop}{5}\textcolor{colordef}{\pourcent}\end{aligned}$$

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:$$\begin{aligned}\frac{\textcolor{colorprop}{x}}{\textcolor{colordef}{100}} &= \frac{\textcolor{colorprop}{12}}{\textcolor{colordef}{20}}\\\textcolor{colorprop}{x} &= \frac{\textcolor{colorprop}{12}}{\textcolor{colordef}{20}} \times \textcolor{colordef}{100} && \text{(multiplying both sides by }\textcolor{colordef}{100})\\\textcolor{colorprop}{x} &= \textcolor{colorprop}{60}\\\end{aligned}$$The percentage of girls in the class is \(\textcolor{colorprop}{60}\textcolor{colordef}{\pourcent}\). This means \(\textcolor{colorprop}{60}\) out of every \(\textcolor{colordef}{100}\) students are girls.

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

  • \(\begin{aligned}[t]\text{Percentage of women in Parliament A} &= \left(\frac{\textcolor{colorprop}{26}}{\textcolor{colordef}{50}} \times \textcolor{colordef}{100}\right)\textcolor{colordef}{\pourcent} \\&= \textcolor{colorprop}{52}\textcolor{colordef}{\pourcent}\end{aligned}\)
    In Parliament A, \(\textcolor{colorprop}{52}\) out of every \(\textcolor{colordef}{100}\) deputies are women.
  • \(\begin{aligned}[t]\text{Percentage of women in Parliament B} &= \left(\frac{\textcolor{colorprop}{30}}{\textcolor{colordef}{80}} \times \textcolor{colordef}{100}\right)\textcolor{colordef}{\pourcent} \\&= \textcolor{colorprop}{37.5}\textcolor{colordef}{\pourcent}\end{aligned}\)
    In Parliament B, \(\textcolor{colorprop}{37.5}\) out of every \(\textcolor{colordef}{100}\) deputies are women.
Conclusion: Since \(\textcolor{colorprop}{52}\textcolor{colordef}{\pourcent} > \textcolor{colorprop}{37.5}\textcolor{colordef}{\pourcent}\), women 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 \(\textcolor{colorprop}{14}\) girls in total.
Find the total number of students in the class.

  • Method 1 (using the formula)$$\begin{aligned}\textcolor{colordef}{\text{total students}} &= \dfrac{\textcolor{colorprop}{14}}{\textcolor{colorprop}{40}\textcolor{colordef}{\pourcent}} \\&= \dfrac{\textcolor{colorprop}{14}}{\left( \dfrac{\textcolor{colorprop}{40}}{\textcolor{colordef}{100}} \right)} \\&= \textcolor{colordef}{35} &&\text{(compute: } 14 \div (40 \div 100) = 35)\end{aligned}$$
  • 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
  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: $$ \text{New quantity} = \text{Original quantity} + \text{Increase} $$
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
  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: $$ \text{New quantity} = \text{Original quantity} - \text{Decrease} $$
Example
If the original price of a shirt is \(\dollar\)50 and it is decreased by 20\(\pourcent\), find the new price.

  • Calculate the decrease: $$ \begin{aligned} \text{Decrease} &= 20\pourcent \times \dollar 50 \\ &= \frac{20}{100} \times \dollar 50 \\ &= \dollar 10 \end{aligned} $$
  • Calculate the new price: $$ \text{New price} = \dollar 50 - \dollar 10 = \dollar 40 $$