Introduction

Percentages are very common in everyday life. For example, they are used in retail, finance, and sports! In this lesson, we will learn how to calculate the percentage of a whole number in real-world situations.

For example, suppose Daniel and Luisa traveled 200 kilometers in their car. If Luisa drove 40 \% of the way, how many kilometers did she drive?

To solve this problem, we can use the following formula: \textrm{part} = \textrm{percentage} \times \textrm{whole}

In this case, the whole is the number of kilometers traveled, which is 200 kilometers. The percentage is 40 \%, and the part is the distance Luisa drove.

As usual, we can use two methods to solve this problem:

Method 1

First, we convert the percentage to a decimal: 40 \% = 40 \div 100 = 0.4

Applying our formula, we have \begin{align*} \textrm{part} &= \textrm{percentage} \times \textrm{whole} \\[5pt] &= 40 \% \times 200 \\[5pt] &= 0.4 \times 200 \\[5pt] &= 80. \end{align*}

So Luisa drove 80 kilometers.

Method 2

First, we convert the percentage to a fraction: 40 \% = \dfrac{40}{100} = \dfrac{40 \div 20}{100 \div 20} = \dfrac{2}{5}

Applying the formula, we get \begin{align*} \textrm{part} &= \textrm{percentage} \times \textrm{whole} \\[5pt] &= 40\% \times 200 \\[5pt] &= \dfrac{2}{5} \times 200 \\[5pt] &= \dfrac{400}{5} \\[5pt] &= 80. \end{align*}

Again, we find that Luisa drove 80 kilometers.

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Example: Finding Part of a Number Given a Whole and a Percentage

In a garden, there are 40 flowers, 30 \% of which are roses. How many roses are there in the garden?

EXPLANATION

To solve this problem, we can use the following formula: \textrm{part}= \textrm{percentage}\times \textrm{whole}

In our case, the whole is the number of flowers in the garden, which is 40 flowers. The percentage is 30 \%, and the part is the number of roses in the garden.

Method 1

First, we convert the percentage to a decimal: 30 \% = 30 \div 100 = 0.3

Applying our formula, we get \begin{align*} \textrm{part} &= \textrm{percentage}\times \textrm{whole} \\[5pt] &= 30 \% \times 40 \\[5pt] &= 0.3 \times 40 \\[5pt] & = 12. \end{align*}

So, there are 12 roses in the garden.

Method 2

First, we convert the percentage to a fraction: 30 \% = \dfrac{30}{100} = \dfrac{30 \div 10}{100 \div 10} = \dfrac{3}{10}

Applying our formula, we get \begin{align*} \textrm{part}&= \textrm{percentage}\times \textrm{whole} \\[5pt] &= 30 \% \times 40 \\[5pt] &= \dfrac{3}{10} \times 40 \\[5pt] & = 3 \times \dfrac{40}{10}\\[5pt] & = 3 \times 4\\[5pt] & = 12. \end{align*}

So, there are 12 roses in the garden.

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Practice: Finding Part of a Number Given a Whole and a Percentage

Esther has $16$ balloons. If $25 \%$ of the balloons are yellow, how many yellow balloons does Esther have?

a
 $5$
b
 $4$
c
 $3$
d
 $6$
e
 $8$
Practice: Finding Part of a Number Given a Whole and a Percentage

The original price of a toy robot in a department store was $\$75.$ The store manager decides to reduce the price by $12\%.$ How much of a discount has the manager given?

a
 $\$12$
b
 $\$8$
c
 $\$9$
d
 $\$15$
e
 $\$6$
Example: Finding Part of a Number Given a Whole and a Decimal Percentage

Dora works 800 hours per year. She rests for 12.5 \% ​​of her total working hours. How much time does Dora rest per year at work?

EXPLANATION

To solve this problem, we can use the following formula: \textrm{part}= \textrm{percentage}\times \textrm{whole}

In our case, the whole is the total number of hours Dora works yearly, which is 800 hours. The percentage is 12.5 \%, and the part is the number of hours she rests.

Method 1

First, we convert the percentage to a decimal: 12.5 \% = 12.5 \div 100 = 0.125

Applying our formula, we get \begin{align*} \textrm{part} &= \textrm{percentage}\times \textrm{whole} \\[5pt] &= 12.5 \% \times 800 \\[5pt] &= 0.125 \times 800 \\[5pt] & = 100. \end{align*}

Therefore, Dora rests for 100 hours.

Method 2

First, we convert the percentage to a fraction: 12.5 \% = \dfrac{12.5}{100}

Applying our formula, we get \begin{align*} \textrm{part} &= \textrm{percentage}\times \textrm{whole} \\[5pt] &= 12.5 \% \times 800 \\[5pt] &= \dfrac{12.5}{100} \times 800 \\[5pt] &= 12.5 \times \dfrac{800}{100} \\[5pt] &=12.5\times 8 \\[5pt] & = 100. \end{align*}

Therefore, Dora rests for 100 hours.

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Practice: Finding Part of a Number Given a Whole and a Decimal Percentage

A concrete mix weighs $600$ kilograms. Of this amount, $19.5 \%$ represents the weight of the crushed stone in the mix. How much crushed stone is there in the concrete mix?

a
 $128$ kilograms
b
 $130$ kilograms
c
 $112$ kilograms
d
 $121$ kilograms
e
 $117$ kilograms
Practice: Finding Part of a Number Given a Whole and a Decimal Percentage

Margaret has $ \$1,500$ saved. Of this amount, she decides to spend $22.4 \%$ to buy some supplies for her farm. How much money does Margret plan to spend?

a
 $\$ 344$
b
 $\$ 320$
c
 $\$ 328$
d
 $\$ 354$
e
 $\$ 336$
Example: Finding Part of a Number Given a Whole and a Percentage Greater Than 100%

Marcus has a tank with a capacity of 1,500 liters. He decides to buy a second tank whose capacity is 130 \% of the capacity of the first tank. What is the capacity of the second tank?

EXPLANATION

To solve this problem, we can use the following formula: \textrm{part}= \textrm{percentage}\times \textrm{whole}

In our case, the whole is the capacity of the first tank, which is 1,500 liters. The percentage is 130 \%, and the part is the capacity of the second tank.

Method 1

First, we convert the percentage to a decimal: 130 \% = 130 \div 100 = 1.3

Applying our formula, we get \begin{align*} \textrm{part} &= \textrm{percentage}\times \textrm{whole} \\[5pt] &= 130 \% \times 1,500 \\[5pt] &= 1.3 \times 1,500 \\[5pt] &= 1,950. \end{align*}

Therefore, the second tank has a capacity of 1,950 liters.

Method 2

First, we convert the percentage to a fraction: 130 \% = \dfrac{130}{100}

Applying our formula, we get \begin{align*} \textrm{part}&= \textrm{percentage}\times \textrm{whole} \\[5pt] &= 130 \% \times 1,500 \\[5pt] &= \dfrac{130}{100} \times 1,500 \\[5pt] & = 130 \times \dfrac{1,500}{100}\\[5pt] & = 130 \times 15 \\[5pt] & = 1,950. \end{align*}

Therefore, the second tank has a capacity of 1,950 liters.

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Practice: Finding Part of a Number Given a Whole and a Percentage Greater Than 100%

Anna, a dressmaker, bought $15$ meters of fabric in a particular month to make children's pants. The following month, she decided to buy $160 \%$ of the amount of fabric she bought the previous month. How much fabric did Anna buy on her second purchase?

a
 $24$ meters
b
 $22$ meters
c
 $18$ meters
d
 $21$ meters
e
 $28$ meters
Practice: Finding Part of a Number Given a Whole and a Percentage Greater Than 100%

In his first season in the youth soccer league, Fernando scored $8$ goals. In his second season, he scored $250 \%$ of the number of goals he made in the first season. How many goals did Fernando score in his second season?

a
 $15$ goals
b
 $24$ goals
c
 $22$ goals
d
 $18$ goals
e
 $20$ goals
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