7.4: Percent
- understand the relationship between ratios and percents
- be able to make conversions between fractions, decimals, and percents
Ratios and Percents
Ratio, Percent
We defined a
ratio
as a comparison, by division, of two pure numbers or two like denominate numbers. A most convenient number to compare numbers to is 100. Ratios in which one number is compared to 100 are called
percents
. The word
percent
comes from the Latin word "per centum." The word "per" means "for each" or "for every," and the word "centum" means "hundred." Thus, we have the following definition.
Percent means “for each hundred," or "for every hundred."
The symbol % is used to represent the word percent.
The ratio 26 to 100 can be written as 26%. We read 26% as "twenty-six percent."
The ratio \(\dfrac{165}{100}\) can be written as 165%.
We read 165% as "one hundred sixty-five percent."
The percent 38% can be written as the fraction \(\dfrac{38}{100}\).
The percent 210% can be written as the fraction \(\dfrac{210}{100}\) or the mixed number \(2\dfrac{1)}{100}\) or 2.1.
Since one dollar is 100 cents, 25 cents is \(\dfrac{25}{100}\) of a dollar. This implies that 25 cents is 25% of one dollar.
Practice Set A
Write the ratio 16 to 100 as a percent.
- Answer
-
16%
Practice Set A
Write the ratio 195 to 100 as a percent.
- Answer
-
195%
Practice Set A
Write the percent 83% as a ratio in fractional form.
- Answer
-
\(\dfrac{83}{100}\)
Practice Set A
Write the percent 362% as a ratio in fractional form.
- Answer
-
\(\dfrac{362}{100}\) or \(\dfrac{181}{50}\)
The Relationship Between Fractions, Decimals, and Percents – Making Conversions
Since a percent is a ratio, and a ratio can be written as a fraction, and a fraction can be written as a decimal, any of these forms can be converted to any other.
Before we proceed to the problems in Sample Set B and Practice Set B, let's summarize the conversion techniques.
| To Convert a Fraction | To Convert a Decimal | To Convert a Percent |
| To a decimal: Divide the numerator by the denominator | To a fraction: Read the decimal and reduce the resulting fraction | To a decimal: Move the decimal point 2 places to the left and drop the % symbol |
| To a percent: Convert the fraction first to a decimal, then move the decimal point 2 places to the right and affix the % symbol. | To a percent: Move the decimal point 2 places to the right and affix the % symbol | To a fraction: Drop the % sign and write the number “over” 100. Reduce, if possible. |
Convert 12% to a decimal.
Solution
\(12\% = \dfrac{12}{100} = 0.12\)
Note that
The % symbol is dropped, and the decimal point moves 2 places to the left.
Convert 0.75 to a percent.
Solution
\(0.75 = \dfrac{75}{100} = 75\%\)
Note that
The % symbol is affixed, and the decimal point moves 2 units to the right.
Convert \(\dfrac{3}{5}\) to a percent.
Solution
We see in Example above that we can convert a decimal to a percent. We also know that we can convert a fraction to a decimal. Thus, we can see that if we first convert the fraction to a decimal, we can then convert the decimal to a percent.
\(\dfrac{3}{5} \to \begin{array} {r} {.6} \\ {5\overline{)3.0}} \\ {\underline{3\ 0}} \\ {0} \end{array} \text{ or } \dfrac{3}{5} = 0.6 = \dfrac{6}{10} = \dfrac{60}{100} = 60\%\)
Convert 42% to a fraction.
Solution
\(42\% = \dfrac{42}{100} = \dfrac{21}{50}\)
or
\(42\% = 0.42 = \dfrac{42}{100} = \dfrac{21}{50}\)
Practice Set B
Convert 21% to a decimal.
- Answer
-
0.21
Practice Set B
Convert 461% to a decimal.
- Answer
-
4.61
Practice Set B
Convert 0.55 to a percent.
- Answer
-
55%
Practice Set B
Convert 5.64 to a percent.
- Answer
-
564%
Practice Set B
Convert \(\dfrac{3}{20}\) to a percent.
- Answer
-
15%
Practice Set B
Convert \(\dfrac{11}{8}\) to a percent.
- Answer
-
137.5%
Practice Set B
Convert \(\dfrac{3}{11}\) to a percent.
- Answer
-
\(27.\overline{27}\)%
Exercises
For the following 12 problems, convert each decimal to a percent.
Exercise \(\PageIndex{1}\)
0.25
- Answer
-
25%
Exercise \(\PageIndex{2}\)
0.36
Exercise \(\PageIndex{3}\)
0.48
- Answer
-
48%
Exercise \(\PageIndex{4}\)
0.343
Exercise \(\PageIndex{5}\)
0.771
- Answer
-
77.1%
Exercise \(\PageIndex{6}\)
1.42
Exercise \(\PageIndex{7}\)
2.58
- Answer
-
258%
Exercise \(\PageIndex{8}\)
4.976
Exercise \(\PageIndex{9}\)
16.1814
- Answer
-
1,618.14%
Exercise \(\PageIndex{10}\)
533.01
Exercise \(\PageIndex{11}\)
2
- Answer
-
200%
Exercise \(\PageIndex{12}\)
14
For the following 10 problems, convert each percent to a decimal.
Exercise \(\PageIndex{13}\)
15%
- Answer
-
0.15
Exercise \(\PageIndex{14}\)
43%
Exercise \(\PageIndex{15}\)
16.2%
- Answer
-
0.162
Exercise \(\PageIndex{16}\)
53.8%
Exercise \(\PageIndex{17}\)
5.05%
- Answer
-
0.0505
Exercise \(\PageIndex{18}\)
6.11%
Exercise \(\PageIndex{19}\)
0.78%
- Answer
-
0.0078
Exercise \(\PageIndex{20}\)
0.88%
Exercise \(\PageIndex{21}\)
0.09%
- Answer
-
0.0009
Exercise \(\PageIndex{22}\)
0.001%
For the following 14 problems, convert each fraction to a percent.
Exercise \(\PageIndex{23}\)
\(\dfrac{1}{5}\)
- Answer
-
20%
Exercise \(\PageIndex{24}\)
\(\dfrac{3}{5}\)
Exercise \(\PageIndex{25}\)
\(\dfrac{5}{8}\)
- Answer
-
62.5%
Exercise \(\PageIndex{26}\)
\(\dfrac{1}{16}\)
Exercise \(\PageIndex{27}\)
\(\dfrac{7}{25}\)
- Answer
-
28%
Exercise \(\PageIndex{28}\)
\(\dfrac{16}{45}\)
Exercise \(\PageIndex{29}\)
\(\dfrac{27}{55}\)
- Answer
-
\(49.\overline{09}\)%
Exercise \(\PageIndex{30}\)
\(\dfrac{15}{8}\)
Exercise \(\PageIndex{31}\)
\(\dfrac{41}{25}\)
- Answer
-
164%
Exercise \(\PageIndex{32}\)
\(6 \dfrac{4}{5}\)
Exercise \(\PageIndex{33}\)
\(9 \dfrac{9}{20}\)
- Answer
-
945%
Exercise \(\PageIndex{34}\)
\(\dfrac{1}{200}\)
Exercise \(\PageIndex{35}\)
\(\dfrac{6}{11}\)
- Answer
-
\(54.\overline{54}\)%
Exercise \(\PageIndex{36}\)
\(\dfrac{35}{27}\)
For the following 14 problems, convert each percent to a fraction.
Exercise \(\PageIndex{37}\)
80%
- Answer
-
\(\dfrac{4}{5}\)
Exercise \(\PageIndex{38}\)
60%
Exercise \(\PageIndex{37}\)
25%
- Answer
-
\(\dfrac{1}{4}\)
Exercise \(\PageIndex{38}\)
75%
Exercise \(\PageIndex{37}\)
65%
- Answer
-
\(\dfrac{13}{20}\)
Exercise \(\PageIndex{38}\)
18%
Exercise \(\PageIndex{37}\)
12.5%
- Answer
-
\(\dfrac{1}{8}\)
Exercise \(\PageIndex{38}\)
37.5%
Exercise \(\PageIndex{37}\)
512.5%
- Answer
-
\(\dfrac{41}{8}\) or \(5 \dfrac{1}{8}\)
Exercise \(\PageIndex{38}\)
937.5%
Exercise \(\PageIndex{37}\)
\(9.\overline{9}\)%
- Answer
-
\(\dfrac{1}{10}\)
Exercise \(\PageIndex{38}\)
\(55.\overline{5}\)%
Exercise \(\PageIndex{37}\)
\(22.\overline{2}\)%
- Answer
-
\(\dfrac{2}{9}\)
Exercise \(\PageIndex{38}\)
\(63.\overline{6}\)%
Exercises for Review
Exercise \(\PageIndex{39}\)
Find the quotient. \(\dfrac{40}{54} \div 8 \dfrac{7}{21}\).
- Answer
-
\(\dfrac{2}{9}\)
Exercise \(\PageIndex{40}\)
\(\dfrac{3}{8}\) of what number is \(2\dfrac{2}{3}\)?
Exercise \(\PageIndex{41}\)
Find the value of \(\dfrac{28}{15} + \dfrac{7}{10} - \dfrac{5}{12}\).
- Answer
-
\(\dfrac{129}{60}\) or \(2 \dfrac{9}{60} = 2 \dfrac{3}{20}\)
Exercise \(\PageIndex{42}\)
Round 6.99997 to the nearest ten thousandths.
Exercise \(\PageIndex{43}\)
On a map, 3 inches represent 40 miles. How many inches represent 480 miles?
- Answer
-
36 inches