7.5: Fractions of One Percent

Learning Objectives

• 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

\(\begin{array} {l} {\dfrac{1}{2} \% = \dfrac{1}{2} \text{ of } 1\% = \dfrac{1}{2} \cdot \dfrac{1}{100} = \dfrac{1}{200}} \\ {\dfrac{3}{5} \% = \dfrac{3}{5} \text{ of } 1\% = \dfrac{3}{5} \cdot \dfrac{1}{100} = \dfrac{3}{500}} \\ {\dfrac{5}{8} \% = \dfrac{5}{8} \text{ of } 1\% = \dfrac{5}{8} \cdot \dfrac{1}{100} = \dfrac{5}{800}} \\ {\dfrac{7}{11} \% = \dfrac{7}{11} \text{ of } 1\% = \dfrac{7}{11} \cdot \dfrac{1}{100} = \dfrac{7}{1100}} \end{array}\)

Sample Set A

Convert \(\dfrac{2}{3}\)% to a fraction

Solution

\(\begin{array} {rcl} {\dfrac{2}{3} \% = \dfrac{2}{3} \text{ of } 1 \%} & = & {\dfrac{\begin{array} {c} {^1} \\ {\cancel{2}} \end{array}}{3} \cdot \dfrac{1}{\begin{array} {c} {\cancel{100}} \\ {^{50}} \end{array}}} \\ {} & = & {\dfrac{1 \cdot 1}{3 \cdot 50}} \\ {} & = & {\dfrac{1}{150}} \end{array}\)

Sample Set A

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:

1. Convert the fraction as a rounded decimal.
2. Move the decimal point two digits to the left and remove the percent sign.

Sample Set B

Convert \(\dfrac{2}{3}\)% to a three-place decimal.

Solution

1. 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. \((\dfrac{2}{3} = 0.6666...)\)
2. 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

Sample Set B

Convert \(5 \dfrac{4}{11}\)% to a four-place decimal.

Solution

1. 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. \((\dfrac{4}{11} = 0.3636...)\)
2. Move the decimal point two places to the left and drop the percent sign.
   \(5 \dfrac{4}{11} \% = 0.0536\) to four decimal places.

Sample Set B

Convert \(28 \dfrac{5}{9}\)% to a decimal rounded to ten thousandths.

Solution

1. 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. \((\dfrac{5}{9} = 0.5555...)\)
2. 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%