7.5: Fractions of One Percent
- understand the meaning of a fraction of one percent
- be able to make conversions involving fractions of one percent
Conversions Involving Fractions of One Percent
Percents such as \(\dfrac{1}{2}\)%, \(\dfrac{3}{5}\)%, \(\dfrac{5}{8}\)%, and \(\dfrac{7}{11}\)%. where 1% has not been attained, are fractions of 1%. This implies that
\(\begin{array} {l} {\dfrac{1}{2} \% = \dfrac{1}{2} \text{ of } 1\%} \\ {\dfrac{3}{5} \% = \dfrac{3}{5} \text{ of } 1\%} \\ {\dfrac{5}{8} \% = \dfrac{5}{8} \text{ of } 1\%} \\ {\dfrac{7}{11} \% = \dfrac{7}{11} \text{ of } 1\%} \end{array}\)
Since "percent" means "for each hundred," and "of" means "times," we have
\(\begin{array} {l} {\dfrac{1}{2} \% = \dfrac{1}{2} \text{ of } 1\% = \dfrac{1}{2} \cdot \dfrac{1}{100} = \dfrac{1}{200}} \\ {\dfrac{3}{5} \% = \dfrac{3}{5} \text{ of } 1\% = \dfrac{3}{5} \cdot \dfrac{1}{100} = \dfrac{3}{500}} \\ {\dfrac{5}{8} \% = \dfrac{5}{8} \text{ of } 1\% = \dfrac{5}{8} \cdot \dfrac{1}{100} = \dfrac{5}{800}} \\ {\dfrac{7}{11} \% = \dfrac{7}{11} \text{ of } 1\% = \dfrac{7}{11} \cdot \dfrac{1}{100} = \dfrac{7}{1100}} \end{array}\)
Convert \(\dfrac{2}{3}\)% to a fraction
Solution
\(\begin{array} {rcl} {\dfrac{2}{3} \% = \dfrac{2}{3} \text{ of } 1 \%} & = & {\dfrac{\begin{array} {c} {^1} \\ {\cancel{2}} \end{array}}{3} \cdot \dfrac{1}{\begin{array} {c} {\cancel{100}} \\ {^{50}} \end{array}}} \\ {} & = & {\dfrac{1 \cdot 1}{3 \cdot 50}} \\ {} & = & {\dfrac{1}{150}} \end{array}\)
Convert \(\dfrac{5}{8}\)% to a decimal
Solution
\(\begin{array} {rcl} {\dfrac{5}{8} \% = \dfrac{5}{8} \text{ of } 1 \%} & = & {\dfrac{5}{8} \cdot \dfrac{1}{100}} \\ {} & = & {0.625 \cdot 0.01} \\ {} & = & {0.00625} \end{array}\)
Practice Set A
Convert \(\dfrac{1}{4}\)% to a fraction
- Answer
-
\(\dfrac{1}{400}\)
Practice Set A
Convert \(\dfrac{3}{8}\)% to a fraction
- Answer
-
\(\dfrac{3}{800}\)
Practice Set A
Convert \(3 \dfrac{1}{3}\)% to a fraction
- Answer
-
\(\dfrac{1}{30}\)
Conversions Involving Nonterminating Fractions
We must be careful when changing a fraction of 1% to a decimal. The number \(\dfrac{2}{3}\), as we know, has a nonterminating decimal representation. Therefore, it cannot be expressed exactly as a decimal.
When converting nonterminating fractions of 1% to decimals, it is customary to express the fraction as a rounded decimal with at least three decimal places.
Converting a Nonterminating Fraction to a Decimal
To convert a nonterminating fraction of 1% to a decimal:
- Convert the fraction as a rounded decimal.
- Move the decimal point two digits to the left and remove the percent sign.
Convert \(\dfrac{2}{3}\)% to a three-place decimal.
Solution
-
Convert \(\dfrac{2}{3}\) to a decimal.
Since we wish the resulting decimal to have three decimal digits, and removing the percent sign will account for two of them, we need to round \(\dfrac{2}{3}\) to one place (2 + 1 = 3).
\(\dfrac{2}{3} \% = 0.7\%\) to one decimal place. \((\dfrac{2}{3} = 0.6666...)\) -
Move the decimal point two digits to the left and remove the % sign. We'll need to add zeros to locate the decimal point in the correct location.
\(\dfrac{2}{3} \% = 0.007\) to 3 decimal places
Convert \(5 \dfrac{4}{11}\)% to a four-place decimal.
Solution
-
Since we wish the resulting decimal to have four decimal places, and removing the percent sign will account for two, we to round \(\dfrac{4}{11}\) to two places.
\(5 \dfrac{4}{11} \% = 5.36\%\) to two decimal places. \((\dfrac{4}{11} = 0.3636...)\) -
Move the decimal point two places to the left and drop the percent sign.
\(5 \dfrac{4}{11} \% = 0.0536\) to four decimal places.
Convert \(28 \dfrac{5}{9}\)% to a decimal rounded to ten thousandths.
Solution
-
Since we wish the resulting decimal to be rounded to ten thousandths (four decimal places), and removing the percent sign will account for two, we need to round \(\dfrac{5}{9}\) to two places.
\(28 \dfrac{5}{9} \% = 28.56 \%\) to two decimal places. \((\dfrac{5}{9} = 0.5555...)\) -
Move the decimal point to the left two places and drop the percent sign.
\(28 \dfrac{5}{9} \% = 0.2856\) correct to ten thousandths.
Practice Set B
Convert \(\dfrac{7}{9}\)% to a three-place decimal.
- Answer
-
0.008
Practice Set B
Convert \(51\dfrac{5}{11}\)% to a decimal rounded to ten thousandths.
- Answer
-
0.5145
Exercises
Make the conversions as indicated.
Exercise \(\PageIndex{1}\)
Convert \(\dfrac{3}{4}\)% to a fraction
- Answer
-
\(\dfrac{3}{400}\)
Exercise \(\PageIndex{2}\)
Convert \(\dfrac{5}{6}\)% to a fraction
Exercise \(\PageIndex{3}\)
Convert \(\dfrac{1}{9}\)% to a fraction
- Answer
-
\(\dfrac{1}{900}\)
Exercise \(\PageIndex{4}\)
Convert \(\dfrac{15}{19}\)% to a fraction
Exercise \(\PageIndex{5}\)
Convert \(\dfrac{5}{4}\)% to a fraction
- Answer
-
\(\dfrac{5}{400}\) or \(\dfrac{1}{80}\)
Exercise \(\PageIndex{6}\)
Convert \(\dfrac{7}{3}\)% to a fraction
Exercise \(\PageIndex{7}\)
Convert \(1 \dfrac{6}{7}\)% to a fraction
- Answer
-
\(\dfrac{13}{700}\)
Exercise \(\PageIndex{8}\)
Convert \(2 \dfrac{5}{16}\)% to a fraction
Exercise \(\PageIndex{9}\)
Convert \(25 \dfrac{1}{4}\)% to a fraction
- Answer
-
\(\dfrac{101}{400}\)
Exercise \(\PageIndex{10}\)
Convert \(50 \dfrac{5}{6}\)% to a fraction
Exercise \(\PageIndex{11}\)
Convert \(72\dfrac{3}{5}\)% to a fraction
- Answer
-
\(\dfrac{363}{500}\)
Exercise \(\PageIndex{12}\)
Convert \(99 \dfrac{1}{8}\)% to a fraction
Exercise \(\PageIndex{13}\)
Convert \(136 \dfrac{2}{3}\)% to a fraction
- Answer
-
\(\dfrac{41}{30}\)
Exercise \(\PageIndex{14}\)
Convert \(521 \dfrac{3}{4}\)% to a fraction
Exercise \(\PageIndex{15}\)
Convert \(10 \dfrac{1}{5}\)% to a decimal.
- Answer
-
\(\dfrac{51}{500} = 0.102\)
Exercise \(\PageIndex{16}\)
Convert \(12 \dfrac{3}{4}\)% to a decimal.
Exercise \(\PageIndex{17}\)
Convert \(3 \dfrac{7}{8}\)% to a decimal.
- Answer
-
\(\dfrac{31}{800} = 0.03875\)
Exercise \(\PageIndex{18}\)
Convert \(7 \dfrac{1}{16}\)% to a decimal.
Exercise \(\PageIndex{19}\)
Convert \(\dfrac{3}{7}\)% to a three-place decimal.
- Answer
-
0.004
Exercise \(\PageIndex{20}\)
Convert \(\dfrac{1}{9}\)% to a three-place decimal.
Exercise \(\PageIndex{21}\)
Convert \(6 \dfrac{3}{11}\)% to a four-place decimal.
- Answer
-
0.0627
Exercise \(\PageIndex{22}\)
Convert \(9 \dfrac{2}{7}\)% to a four-place decimal.
Exercise \(\PageIndex{23}\)
Convert \(24 \dfrac{5}{21}\)% to a three-place decimal.
- Answer
-
0.242
Exercise \(\PageIndex{24}\)
Convert \(45 \dfrac{8}{27}\)% to a three-place decimal.
Exercise \(\PageIndex{25}\)
Convert \(11 \dfrac{16}{17}\)% to a four-place decimal.
- Answer
-
0.1194
Exercise \(\PageIndex{26}\)
Convert \(5 \dfrac{1}{7}\)% to a three-place decimal.
Exercises for Review
Exercise \(\PageIndex{27}\)
Write \(8 \cdot 8 \cdot 8 \cdot 8 \cdot 8\) using exponents.
- Answer
-
\(8^5\)
Exercise \(\PageIndex{28}\)
Convert \(4 \dfrac{7}{8}\) to an improper fraction.
Exercise \(\PageIndex{29}\)
Find the sum. \(\dfrac{7}{10} + \dfrac{2}{21} + \dfrac{1}{7}\).
- Answer
-
\(\dfrac{197}{210}\)
Exercise \(\PageIndex{30}\)
Find the product. (4.21)(0.006).
Exercise \(\PageIndex{31}\)
Convert 8.062 to a percent.
- Answer
-
806.2%