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# Percent of a Number

## Use multiplication and proportions to find the percent of a number.

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Percent of a Number

Taylor wants to buy a new pair of shoes. She goes to the shoe store and discovers a shelf on which all of the shoes are 60% off. She finds a pair of shoes on the shelf that are ticketed at $50. “I wonder how much money I will get back if I pay for the shoes with a$50 bill,” Taylor muses. At the checkout register, how much change will Taylor get back?

In this concept, you will learn to find the percent of a number using fraction multiplication.

### Finding the Percent of a Number

To work with percents you have to understand how they relate to parts and fractions.

The table below shows the fractional equivalents for common percents.

 5% 10% 20% 25% 30% 40% 50% 60% 70% 75% 80% 90% 120\begin{align*}\frac{1}{20}\end{align*} 110\begin{align*}\frac{1}{10}\end{align*} 15\begin{align*}\frac{1}{5} \end{align*} 14\begin{align*}\frac{1}{4}\end{align*} 310\begin{align*}\frac{3}{10}\end{align*} 25\begin{align*}\frac{2}{5}\end{align*} 12\begin{align*}\frac{1}{2}\end{align*} 35\begin{align*}\frac{3}{5}\end{align*} \begin{align*}\frac{7}{10}\end{align*} \begin{align*}\frac{3}{4}\end{align*} \begin{align*}\frac{4}{5}\end{align*} \begin{align*}\frac{9}{10}\end{align*}

The word “of” in a percent problem means to multiply. If you know the fractional equivalents for common percents, you can use this information to find the percent of a number by multiplying the fraction by that number. If you want to find a part of a whole using a percent, you use multiplication to solve.

Let’s look at an example.

Find 40% of 45.

First, 40% of 45 means \begin{align*}40\% \times 45\end{align*}.

Next, look at the chart. The fraction \begin{align*}\frac{2}{5}\end{align*} is equivalent to 40%.

Then perform the multiplication, simplifying as you go.

\begin{align*}40 \% \times 45 = \frac{2}{5} \times 45 = \frac{2}{\underset{1}{\cancel{5}}} \times \frac{\overset{9}{\cancel{45}}}{1} = \frac{18}{1} = 18\end{align*}

The answer is that 40% of 45 is 18.

Now, let’s look at another example using an alternate way to solve.

What is 18% of 50?

First, to figure this out, you can change the percent to a fraction and then create a proportion.

\begin{align*}\frac{18}{100} = \frac{x}{50}\end{align*}

Next, you can cross multiply and solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} 100 x &=& 900 \\ x &=& 9 \end{array}\end{align*}

The answer is 18% of 50 is 9.

### Examples

#### Example 1

Earlier, you were given a problem about Taylor and her shoes.

They were ticketed at $50 but there was a 60% off sale on them. Taylor wanted to know how much money she would get back if she paid for the shoes with a$50 bill.

First, 60% of 50 means \begin{align*}60\% \times 50\end{align*}.

Next, change the percent to a fraction in simplest form.

\begin{align*}60 \% = \frac{60}{100} = \frac{60 \div 20}{100 \div 20} = \frac{3}{5}\end{align*}

Then perform the multiplication, simplifying as you go.

\begin{align*}60 \% \times 50 = \frac{3}{5} \times 50 = \frac{3}{\underset{1}{\cancel{5}}} \times \frac{\overset{10}{\cancel{50}}}{1} = \frac{30}{1} = 30\end{align*}

The answer is that the shoes are marked down by $30. If Taylor pays with a$50 bill she will get \$30 back.

#### Example 2

Change the percent to a fraction in simplest form.

Find 85% of 20.

First, 85% of 20 means \begin{align*}85\% \times 20\end{align*}.

Next, change the percent to a fraction in simplest form.

\begin{align*}85 \% = \frac{85}{100} = \frac{85 \div 5}{100 \div 5} = \frac{17}{20} \end{align*}

Then perform the multiplication, simplifying as you go.

\begin{align*}85 \% \times 20 = \frac{17}{20} \times 20= \frac{17}{\underset{1}{\cancel{20}}} \times \frac{\overset{1}{\cancel{20}}}{1} = \frac{17}{1} = 17\end{align*}

The answer is that 85% of 20 is 17.

#### Example 3

What is 10% of 50?

First, 10% of 50 means \begin{align*}10\% \times 50\end{align*}.

Next, look at the chart. The fraction \begin{align*}\frac{1}{10}\end{align*} is equivalent to 10%.

Then perform the multiplication, simplifying as you go.

\begin{align*}10 \% \times 50 = \frac{1}{10} \times 50 = \frac{1}{\underset{1}{\cancel{10}}} \times \frac{\overset{1}{\cancel{50}}}{1} = \frac{5}{1} = 5 \end{align*}

The answer is that 10% of 50 is 5.

#### Example 4

What is 25% of 80?

First, 25% of 80 means \begin{align*}25\% \times 80\end{align*}.

Next, look at the chart. The fraction \begin{align*}\frac{1}{4}\end{align*} is equivalent to 25%.

Then perform the multiplication, simplifying as you go.

\begin{align*}25 \% \times 80 = \frac{1}{4} \times 80 = \frac{1}{\underset{1}{\cancel{4}}} \times \frac{\overset{20}{\cancel{80}}}{1} = \frac{20}{1} = 20\end{align*}

The answer is that 25% of 80 is 20.

#### Example 5

What is 22% of 100?

First, 22% of 100 means \begin{align*}22\% \times 100\end{align*}.

Next, change the percent to a fraction in simplest form.

\begin{align*}22 \% = \frac{22}{100}= \frac{22 \div 2}{100 \div 2} = \frac{11}{50}\end{align*}

Then perform the multiplication, simplifying as you go.

\begin{align*}22 \% \times 100 = \frac{11}{50} \times 100 = \frac{11}{\underset{1}{\cancel{50}}} \times \frac{\overset{2}{\cancel{100}}}{1} = \frac{22}{1} = 22\end{align*}

The answer is that 22% of 100 is 22.

### Review

Use fraction multiplication to find each percent of the number.

1. 10% of 25
2. 20% of 30
3. 25% of 80
4. 30% of 90
5. 75% of 200
6. 8% of 10
7. 10% of 100
8. 19% of 20
9. 15% of 30
10. 12% of 30
11. 15% of 45
12. 25% of 85
13. 45% of 60
14. 50% of 200
15. 55% of 300

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### Vocabulary Language: English

improper fraction

An improper fraction is a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator.