1.4: Understand Percent

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Learning Objectives

After completing this section, you should be able to:

Use the definition of percent
Convert percents to fractions and decimals
Convert decimals and fractions to percents

Use the Definition of Percent

How many cents are in one dollar? There are $100$ cents in a dollar. How many years are in a century? There are $100$ years in a century. Does this give you a clue about what the word “percent” means? It is really two words, “per cent,” and means per one hundred. A percent is a ratio whose denominator is $100 .$ We use the percent symbol $%,$ to show percent.

Percent

A percent is a ratio whose denominator is $100 .$

According to data from the American Association of Community Colleges (2015), out of 100 community college students, 57 are female. This is demonstrated visually in Figure 1.4.1. Out of the 100 squares on the grid, 57 are shaded, which we write as the ratio $\frac{57}{100} .$

The figure shows a hundred flat with 57 units shaded. — Figure 1.4.1 Among every $100$ community college students, $57$ are female.

Similarly, $25%$ means a ratio of $\frac{25}{100}, 3%$ means a ratio of $\frac{3}{100}$ and $100%$ means a ratio of $\frac{100}{100} .$ In words, "one hundred percent" means the total $100%$ is $\frac{100}{100},$ and since $\frac{100}{100} = 1,$ we see that $100%$ means $1$ whole.

Example 1.4.1

According to the Public Policy Institute of California $(2010), 44%$ of parents of public school children would like their youngest child to earn a graduate degree. Write this percent as a ratio.

Answer

The amount we want to convert is 44%.	$44%$
Write the percent as a ratio. Remember that percent means per 100.	$\frac{44}{100}$

Your Turn 1.4.1

Write the percent as a ratio.

According to a survey, $89%$ of college students have a smartphone.

Example 1.4.2

In 2007, according to a U.S. Department of Education report, 21 out of every 100 first-time freshmen college students at 4-year public institutions took at least one remedial course. Write this as a ratio and then as a percent.

Answer

The amount we want to convert is $21$ out of $100$ .	$21$ out of $100$
Write as a ratio.	$\frac{21}{100}$
Convert the 21 per 100 to percent.	$21%$

Your Turn 1.4.2

Write as a ratio and then as a percent: The American Association of Community Colleges reported that $62$ out of $100$ full-time community college students balance their studies with full-time or part-time employment.

Convert Percents to Fractions and Decimals

Since percents are ratios, they can easily be expressed as fractions. Remember that percent means "per $100",$ so the denominator of the fraction is $100 .$

How To

Convert a percent to a fraction.

Write the percent as a ratio with the denominator $100 .$
Simplify the fraction, if possible.

Example 1.4.3

Convert each percent to a fraction:

ⓐ $36%$
ⓑ $125%$

Answer

ⓐ
	$36%$
Write as a ratio with denominator 100.	$\frac{36}{100}$
Simplify by dividing out the common factor of 4.	$\frac{9}{25}$

ⓑ
	$125%$
Write as a ratio with denominator 100.	$\frac{125}{100}$
Simplify by dividing out the common factor of 25.	$\frac{5}{4}$

Your Turn 1.4.3

Convert each percent to a fraction:

ⓐ $48%$
ⓑ $110%$

The previous example shows that a percent can be greater than $1 .$ We saw that $125%$ means $\frac{125}{100},$ or $\frac{5}{4} .$ This is an improper fraction, since it's value is greater than one.

Example 1.4.4

Convert each percent to a fraction:

ⓐ $24.5%$
ⓑ $33 \frac{1}{3} %$

Answer

ⓐ
	$24.5%$
Write as a ratio with denominator 100.	$\frac{24.5}{100}$
Clear the decimal by multiplying numerator and denominator by 10.	$\frac{24.5 (10)}{100 (10)}$
Multiply.	$\frac{245}{1000}$
Rewrite showing common factors.	$\frac{5 \cdot 49}{5 \cdot 200}$
Simplify.	$\frac{49}{200}$

ⓑ
	$33 \frac{1}{3} %$
Write as a ratio with denominator 100.	$\frac{33 \frac{1}{3}}{100}$
Write the numerator as an improper fraction.	$\frac{\frac{100}{3}}{100}$
Rewrite as fraction division, replacing 100 with $\frac{100}{1}$ .	$\frac{100}{3} \div \frac{100}{1}$
Multiply by the reciprocal.	$\frac{100}{3} \cdot \frac{1}{100}$
Simplify.	$\frac{1}{3}$

Your Turn 1.4.4

Convert each percent to a fraction:

ⓐ $64.4%$
ⓑ $66 \frac{2}{3} %$

As a review, to convert a fraction to a decimal, you can divide in your calculator. For example, to convert $\frac{1}{5},$ you type 1 ÷ 5 into your calculator. The answer is 0.2. So $\frac{1}{5}$ = 0.2.

To convert a percent to a decimal, we first convert it to a fraction and then change the fraction to a decimal.

How To

Convert a percent to a decimal.

Write the percent as a ratio with the denominator $100 .$
Convert the fraction to a decimal by dividing the numerator by the denominator.

Example 1.4.5

Convert each percent to a decimal:

ⓐ $6%$
ⓑ $78%$

Answer

Because we want to change to a decimal, we will leave the fractions with denominator $100$ instead of removing common factors.

ⓐ
	$6%$
Write as a ratio with denominator 100.	$\frac{6}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.06$

ⓑ
	$78%$
Write as a ratio with denominator 100.	$\frac{78}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.78$

Your Turn 1.4.5

Convert each percent to a decimal:

ⓐ $9%$
ⓑ $87%$

Example 1.4.6

Convert each percent to a decimal:

ⓐ $135%$
ⓑ $12.5%$

Answer

ⓐ
	$135%$
Write as a ratio with denominator 100.	$\frac{135}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$1.35$

ⓑ
	$12.5%$
Write as a ratio with denominator 100.	$\frac{12.5}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.125$

Your Turn 1.4.6

Convert each percent to a decimal:

ⓐ $115%$
ⓑ $23.5%$

Let's summarize the results from the previous examples in Table 1.4.1, and look for a pattern we could use to quickly convert a percent number to a decimal number.

Percent	Decimal
$6%$	$0.06$
$78%$	$0.78$
$135%$	$1.35$
$12.5%$	$0.125$

Table 1.4.1

Do you see the pattern?

To convert a percent number to a decimal number, we move the decimal point two places to the left and remove the $%$ sign. (Sometimes the decimal point does not appear in the percent number, but just like we can think of the integer $6$ as $6.0,$ we can think of $6%$ as $6.0% .$ ) Notice that we may need to add zeros in front of the number when moving the decimal to the left.

Figure 1.4.2 uses the percents in Table 1.4.1 and shows visually how to convert them to decimals by moving the decimal point two places to the left.

The figures shows two columns and five rows . The first row is a header row and it labels each column “Percent” and “Decimal”. Under the “Percent” column are the values: 6%, 78%, 135%, 12.5%. Under the “Decimal” column are the values: 0.06, 0.78, 1.35, 0.125. There are two jumps for each percent to show how to convert it to a decimal. — Figure 1.4.2

Example 1.4.7

Among a group of business leaders, $77%$ believe that poor math and science education in the U.S. will lead to higher unemployment rates.

Convert the percent to: ⓐ a fraction ⓑ a decimal

Answer

ⓐ
	$77%$
Write as a ratio with denominator 100.	$\frac{77}{100}$

ⓑ
	$\frac{77}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.77$

Your Turn 1.4.7

Convert the percent to: ⓐ a fraction and ⓑ a decimal

Twitter's share of web traffic jumped $24%$ when one celebrity tweeted live on air.

Example 1.4.8

There are four suits of cards in a deck of cards—hearts, diamonds, clubs, and spades. The probability of randomly choosing a heart from a shuffled deck of cards is $25% .$ Convert the percent to:

ⓐ a fraction
ⓑ a decimal

The figure shows someone holding a deck of cards. — Figure 1.4.3 (credit: Riles32807, Wikimedia Commons)

Answer

ⓐ
	$25%$
Write as a ratio with denominator 100.	$\frac{25}{100}$
Simplify.	$\frac{1}{4}$

ⓑ	$\frac{1}{4}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.25$

Your Turn 1.4.8

Convert the percent to: ⓐ a fraction, and ⓑ a decimal

The probability that it will rain Monday is $30% .$

Convert Decimals and Fractions to Percents

To convert a decimal to a percent, remember that percent means per hundred. If we change the decimal to a fraction whose denominator is $100,$ it is easy to change that fraction to a percent.

How To

Convert a decimal to a percent.

Write the decimal as a fraction.
If the denominator of the fraction is not $100,$ rewrite it as an equivalent fraction with denominator $100 .$
Write this ratio as a percent.

Example 1.4.9

Convert each decimal to a percent: ⓐ $0.05$ ⓑ $0.83$

Answer

ⓐ
	$0.05$
Write as a fraction. The denominator is 100.	$\frac{5}{100}$
Write this ratio as a percent.	$5%$

ⓑ
	$0.83$
The denominator is 100.	$\frac{83}{100}$
Write this ratio as a percent.	$83%$

Your Turn 1.4.9

Convert each decimal to a percent: ⓐ $0.01$ ⓑ $0.17 .$

To convert a mixed number to a percent, we first write it as an improper fraction.

Example 1.4.10

Convert each decimal to a percent: ⓐ $1.05$ ⓑ $0.075$

Answer

ⓐ
	$0.05$
Write as a fraction.	$1 \frac{5}{100}$
Write as an improper fraction. The denominator is 100.	$\frac{105}{100}$
Write this ratio as a percent.	$105%$

Notice that since $1.05 > 1,$ the result is more than $100%.$

ⓑ
	$0.075$
Write as a fraction. The denominator is 1,000.	$\frac{75}{1,000}$
Divide the numerator and denominator by 10, so that the denominator is 100.	$\frac{7.5}{100}$
Write this ratio as a percent.	$7.5%$

Your Turn 1.4.10

Convert each decimal to a percent: ⓐ $1.75$ ⓑ $0.0825$

Let's summarize the results from the previous examples in Table 1.4.2 so we can look for a pattern.

Decimal	Percent
$0.05$	$5%$
$0.83$	$83%$
$1.05$	$105%$
$0.075$	$7.5%$

Table 1.4.2

Do you see the pattern? To convert a decimal to a percent, we move the decimal point two places to the right and then add the percent sign.

We know how to convert fractions to decimals. Now, we also know how to change decimals to percents. So to convert a fraction to a percent, we first change it to a decimal and then convert that decimal to a percent.

How To

Convert a fraction to a percent.

Convert the fraction to a decimal.
Convert the decimal to a percent.

Example 1.4.11

Answer

To convert a fraction to a decimal, divide the numerator by the denominator.

ⓐ
Change to a decimal.	$\frac{3}{4}$ = 0.75
Write as a percent by moving the decimal two places.	$.$ = 75%

ⓑ
Change to a decimal.	$\frac{11}{8}$ = 1.375
Write as a percent by moving the decimal two places.	$.$ = 137.5%

ⓒ
Write as an improper fraction.	$2 \frac{1}{5}$ = $\frac{11}{5}$
Change to a decimal.	$\frac{11}{5}$ = 2.2
Write as a percent by moving the decimal two places.	$.$ = 220%

Notice that we needed to add a zero at the end of the number when moving the decimal two places to the right in example c.

Your Turn 1.4.11

Sometimes when changing a fraction to a decimal, the division continues for many decimal places and we will round off the quotient. The number of decimal places we round to will depend on the situation. If the decimal involves money, we round to the hundredths place.

Example 1.4.12

Convert $\frac{5}{7}$ to a percent.

Answer

To change a fraction to a decimal, we divide the numerator by the denominator.

	$\frac{5}{7}$
Change to a decimal—rounding to the nearest thousandth.	$0.714$
Write as a percent.	$71.4%$

Your Turn 1.4.12

Convert the fraction to a percent: $\frac{3}{7}$

When we first looked at fractions and decimals, we saw that some fractions converted to a repeating decimal. For example, when we converted the fraction $\frac{4}{3}$ to a decimal, we wrote the answer as $1 . \bar{3} .$ We will use this same notation, as well as fraction notation, when we convert fractions to percents in the next example.

Example 1.4.13

An article in a medical journal claimed that approximately $\frac{1}{3}$ of American adults are obese. Convert the fraction $\frac{1}{3}$ to a percent.

Answer

	$\frac{1}{3}$
Change to a decimal.	$.$
Write as a repeating decimal.	$0.333 \dots$
Write as a percent.	$33 \frac{1}{3} %$

We could also write the percent as $33 . \overline{3} %$ .

Your Turn 1.4.13

Convert the fraction to a percent:

According to the U.S. Census Bureau, about $\frac{1}{9}$ of United States housing units have just $1$ bedroom.