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10 Percentages

Topics Covered[1]

In case you missed something in class, or just want to review a specific topic covered in this Module, here is a list of topics covered:

  1. Definition of Percent
  2. Convert Percent to Faction or Decimals
  3. Convert Decimals and Fractions to Percents
  4. Key Concepts

Use the Definition of Percent[2]

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.

Percent

A percent is a ratio whose denominator is 100. We use the percent symbol %, to show percent.

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 image below shows. Out of the 100 squares on the grid, 57 are shaded, which we write as the ratio  \displaystyle \frac{57}{100}.

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

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

Try it! – Converting from Percent to Fraction

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

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

Try it! – Convert from Fraction to Percent

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

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

Convert Percents to Fractions and Decimals[3]

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.

  1. Write the percent as a ratio with the denominator 100.
  2. Simplify the fraction if possible.

Try it! – Convert a Percent to a Fraction

Convert each percent to a fraction:

a. 36%

b. 125%

Solution
a.
Steps Arithmetic
36%
Write as a ratio with denominator 100. \displaystyle \frac{36}{100}
Simplify. \displaystyle \frac{9}{25}

b.

Steps Arithmetic
125%
Write as a ratio with denominator 100. \displaystyle \frac{125}{100}
Simplify. \displaystyle \frac{5}{4}

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

Try it! – Convert a Percent to a Fraction

Convert each percent to a fraction:

a. 24.5%

b. 33\displaystyle \frac{1}{3}%

Solution

a.

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

b.

Steps Arithmetic
\displaystyle 33\frac{1}{3} \%
Write as a ratio with denominator 100. \displaystyle \frac{33\frac{1}{3}}{100}
Write the numerator as an improper fraction. \displaystyle \frac{\frac{100}{3}}{100}
Rewrite as fraction division, replacing 100 with 1001. \displaystyle \frac{100}{3} \cdot \frac{100}{1}
Multiply by the reciprocal. \displaystyle \frac{100}{3} \cdot \frac{1}{100}
Simplify. \displaystyle \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.

  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.

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 following table uses the percents to show 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.

Try it! – Convert a Percent to a Decimal

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. a fraction

b. a decimal

The figure shows someone holding a deck of cards.

(credit: Riles32807, Wikimedia Commons)

Solution

a.

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

b.

Steps Arithmetic
\displaystyle \frac{1}{4}
Change the fraction to a decimal by dividing the numerator by the denominator. 0.25

Convert Decimals and Fractions to Percents[4]

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.

  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.

Try it! – Convert a Decimal to a Percent

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

Solution
a.
Steps Arithmetic
0.05
Write as a fraction. The denominator is 100. \displaystyle \frac{5}{100}
Write this ratio as a percent. 5%
b.
Steps Arithmetic
0.83
The denominator is 100. \displaystyle \frac{83}{100}
Write this ratio as a percent. 83%

The following table uses the decimal percentages to show visually to convert them to decimals by moving the decimal point two places to the left.

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.

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.

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

 

Try it! – Convert a Fraction to a Percent

Convert each fraction or mixed number to a percent:
a. \displaystyle \frac{3}{4}                b. \displaystyle \frac{11}{8}                c. \displaystyle 2\frac{1}{5}

Solution

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

a.

Steps Arithmetic
Change to a decimal. \displaystyle \frac{3}{4}
Write as a percent by moving the decimal two places. .
75%

b.

Steps Arithmetic
Change to a decimal. \displaystyle \frac{11}{8}
Write as a percent by moving the decimal two places. .
137.5%

c.

Steps Arithmetic
Write as an improper fraction. \displaystyle 2\frac{1}{5}
Change to a decimal. \displaystyle \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.

 Try it! – Convert a Fraction to a Percent

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

Solution

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

Steps Arithmetic
\displaystyle \frac{5}{7}
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 \displaystyle \frac{4}{3} to a decimal, we wrote the answer as 1.3. We will use this same notation, as well as fraction notation, when we convert fractions to percents in the next example.

 Try it! – Convert a Fraction to a Percent

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

Solution
Steps Arithmetic
\displaystyle \frac{1}{3}
Change to a decimal. .
Write as a repeating decimal. 0.333…
Write as a percent. \displaystyle 33\frac{1}{3}

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

Key Concepts

  • Convert a percent to a fraction.
    1. Write the percent as a ratio with the denominator 100.
    2. Simplify the fraction if possible.
  • 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.
  • 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 a denominator of 100.
    3. Write this ratio as a percent.
  • Convert a fraction to a percent.
    1. Convert the fraction to a decimal.
    2. Convert the decimal to a percent.
      1. Access for free at https://openstax.org/books/prealgebra-2e/pages/1-introduction ↵
      2. Section material derived from Openstax Prealgebra: Percents-Understand Percent ↵
      3. Section material derived from Openstax Prealgebra: Percent-Understand Percent ↵
      4. Section material derived from Openstax Prealgebra: Percents-Understand Percent ↵
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