Understand Percent

6.1 Understand Percent

The topics covered in this section are:

  1. Use the definition of percent
  2. Convert percents to fractions and decimals
  3. Convert decimals and fractions to percents

6.1.1 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 $(2015)$, about $57 \%$ of community college students are female. This means $57$ out of every $100$ community college students are female, as the figure below shows. 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.
Amount 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

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.

Solution

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

Example 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.

Solution

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

6.1.2 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.

  1. Write the percent as a ratio with the denominator 100.100.
  2. Simplify the fraction if possible.

Example 3

Convert each percent to a fraction:

  1. $36 \%$
  2. $125 \%$
Solution

Part 1.
$36 \%$
Write a ratio with denominator $100$.$\frac{36}{100}$
Simplify.$\frac{9}{25}$
Part 2.
$125 \%$
Write a ratio with denominator $100$.$\frac{125}{100}$
Simplify.$\frac{5}{4}$

The previous example shows that a percent can be greater than $1$. We saw that $125 \%$ means $\frac{125}{100}$, or $\frac{5}{4}$. These are improper fractions, and their values are greater than one.

Example 4

Convert each percent to a fraction:

  1. $24.5 \%$
  2. $33 \frac{1}{3} \%$
Solution

Part 1.
$24.5 \%$
Write 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.$\frac{49}{200}$
Part 2.
$33 \frac{1}{3} \%$
Write 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.$\frac{1}{3}$

In Decimals, we learned how to convert fractions to decimals. 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.

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

Example 5

Convert each percent to a decimal:

  1. $6 \%$
  2. $78 \%$
Solution

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

Part 1.
$6 \%$
Write as a ratio with denominator $100$.$\frac{6}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.$0.06$
Part 2.
$78 \%$
Write as a ratio with denominator $100$.$\frac{78}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.$0.78$

Example 6

Convert each percent to a decimal:

  1. $135 \%$
  2. $12.5 \%$
Solution
Part 1.
$135 \%$
Write as a ratio with denominator $100$.$\frac{135}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.$1.35$
Part 2.
$12.5 \%$
Write as a ratio with denominator $100$.$\frac{12.5}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.$0.125$

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

PercentDecimal
$6 \%$$0.06$
$78 \%$$0.78$
$135 \%$$1.35$
$12.5 \%$$0.125$

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.

The figure below uses the percents in the table above 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.

Example 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:

  1. a fraction
  2. a decimal
Solution

Part 1.
$77 \%$
Write as a ratio with denominator $100$.$\frac{77}{100}$
Part 2.
$\frac{77}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.$0.77$

Example 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:

  1. a fraction
  2. a decimal
The figure shows someone holding a deck of cards.
(credit: Riles32807, Wikimedia Commons)
Solution

Part 1.
$25 \%$
Write as a ratio with denominator $100$.$\frac{25}{100}$
Simplify.$\frac{1}{4}$
Part 2.
$\frac{1}{4}$
Change the fraction to a decimal by dividing the numerator by the denominator.$0.25$

6.1.3 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.

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

Example 9

Convert each decimal to a percent:

  1. $0.05$
  2. $0.83$
Solution

Part 1.
$0.05$
Write as a fraction. The denominator is $100$.$\frac{5}{100}$
Write this ratio as a percent.$5 \%$
Part 2.
$0.83$
The denominator is $100$.$\frac{83}{100}$
Write this ratio as a percent.$83 \%$

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

Example 10

Convert each decimal to a percent:

  1. $1.05$
  2. $0.075$
Solution

Part 1.
$1.05$
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 $100 \%$.

Part 2.
$0.075$
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.$7.5 \%$

Let’s summarize the results from the pervious examples in Table 6.2 so we can look for a pattern.

DecimalPercent
$0.05$$5 \%$
$0.83$$83 \%$
$1.05$$105 \%$
$0.075$$7.5 \%$
Table 6.2

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.

Figure 6.5 uses the decimal numbers in Table 6.2 and shows visually to convert them to percents by moving the decimal point two places to the right and then writing the $\%$ sign.

The figure shows two columns and five rows. The  first row is a header row and it labels each column “Decimal” and “Percent”. Under the “Decimal” column are the values: 0.05, 0.83, 1.05, 0.075, 0.3. Under the “Percent” column are the values: 5%, 83%, 105%, 7.5%, 30%. There are two jumps for each decimal to show how to convert it to a percent.
Figure 6.5

In Decimals, we learned 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.

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

Example 11

Convert each fraction or mixed number to a percent:

  1. $\frac{3}{4}$
  2. $\frac{11}{8}$
  3. $2 \frac{1}{5}$
Solution

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

Part 1.
Change to a decimal.$\frac{3}{4}$
Write as a percent by moving the decimal two places..
$75 \%$
Part 2.
Change to a decimal.$\frac{11}{8}$
Write as a percent by moving the decimal two places..
$137.5 \%$
Part 3.
Write as an improper fraction.$2 \frac{1}{5}$
Change to a decimal.$\frac{11}{5}$
Write as a percent..
$220 \%$

Notice that we needed to add zeros at the end of the number when moving the decimal two places to the right.

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. For most other cases in this book we will round the number to the nearest thousandth, so the percent will be rounded to the nearest tenth.

Example 12

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

Solution

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

$\frac{5}{4}$
Change to a decimal—rounding to the nearest thousandth.$0.714$
Write as a percent.$71.4 \%$

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

Example 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.

Solution

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

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

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