9.1 Percent: solving percent problems

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses fractions of one percent. By the end of the module students should understand the meaning of a fraction of one percent and be able to make conversions involving fractions of one percent.

Section overview

• Conversions Involving Fractions of One Percent
• Conversions Involving Nonterminating Fractions

Conversions involving fractions of one percent

Percents such as $\frac{1}{2}$ %, $\frac{3}{5}$ %, $\frac{5}{8}$ %, and $\frac{7}{\text{11}}$ %, where 1% has not been attained, are fractions of 1%. This implies that

Since "percent" means "for each hundred," and "of" means "times," we have

$\frac{1}{2}%\text{}=\frac{1}{2}\text{of 1%}=\frac{1}{2}\cdot \frac{1}{\text{100}}=\frac{1}{\text{200}}$

$\frac{3}{5}%\text{}=\frac{3}{5}\text{of 1%}=\frac{3}{5}\cdot \frac{1}{\text{100}}=\frac{3}{\text{500}}$

$\frac{5}{8}%\text{}=\frac{5}{8}\text{of 1%}=\frac{5}{8}\cdot \frac{1}{\text{100}}=\frac{5}{\text{800}}$

$\frac{7}{\text{11}}%\text{}=\frac{7}{\text{11}}\text{of 1%}=\frac{7}{\text{11}}\cdot \frac{1}{\text{100}}=\frac{7}{\text{1100}}$

Sample set a

Convert $\frac{2}{3}$ % to a fraction.

$\begin{array}{ccc}\hfill \frac{2}{3}%=\frac{2}{3}\phantom{\rule{2px}{0ex}}\text{of}\phantom{\rule{2px}{0ex}}1%& =& \frac{\stackrel{1}{\overline{)2}}}{3}\cdot \frac{1}{\underset{50}{\overline{)100}}}\hfill \\ & =& \frac{1\cdot 1}{3\cdot 50}\hfill \\ & =& \frac{1}{150}\hfill \end{array}$

Convert $\frac{5}{8}$ % to a decimal.

$\begin{array}{ccc}\hfill \frac{5}{8}%=\frac{5}{8}\phantom{\rule{2px}{0ex}}\text{of}\phantom{\rule{2px}{0ex}}1%& =& \frac{5}{8}\cdot \frac{1}{100}\hfill \\ & =& 0.625\cdot 0.01\hfill \\ & =& 0.00625\hfill \end{array}$

Practice set a

Convert $\frac{1}{4}$ % to a fraction.

$\frac{1}{\text{400}}$

Convert $\frac{3}{8}$ % to a fraction.

$\frac{3}{\text{800}}$

Convert $3\frac{1}{3}$ % to a fraction.

$\frac{1}{\text{30}}$

Conversions involving nonterminating fractions

We must be careful when changing a fraction of 1% to a decimal. The number $\frac{2}{3}$ , 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 $\frac{2}{3}$ % to a three-place decimal.

1. Convert $\frac{2}{3}$ 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 $\frac{2}{3}$ to one place $\left(\text{2}+\text{1}=\text{3}\right)$ .

$\frac{2}{3}%\text{}=0\text{.}7\text{}$ % to one decimal place. $\left(\frac{2}{3}=0\text{.}\text{6666}\dots \right)$

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.

$\frac{2}{3}\text{%}=\text{0}\text{.}\text{007}$ to 3 decimal places

Convert $5\frac{4}{\text{11}}$ % to a four-place decimal.

1. Since we wish the resulting decimal to have four decimal places, and removing the percent sign will account for two, we to round $\frac{4}{\text{11}}$ to two places.

$5\frac{4}{\text{11}}%\text{}=5\text{.}\text{36}\text{}%$ to two decimal places. $\left(\frac{4}{\text{11}}=0\text{.}\text{3636}\dots \right)$

2. Move the decimal point two places to the left and drop the percent sign.

$5\frac{4}{\text{11}}%\text{}=0\text{.}\text{0536}\text{}$ to four decimal places.

Convert $\text{28}\frac{5}{9}$ % to a decimal rounded to ten thousandths.

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 $\frac{5}{9}$ to two places.

$\text{28}\frac{5}{9}%\text{}=\text{28}\text{.}\text{56%}\text{}$ to two decimal places. $\left(\frac{5}{9}=0\text{.}\text{5555}\dots \right)$

2. Move the decimal point to the left two places and drop the percent sign.

$\text{28}\frac{5}{9}%\text{}=0\text{.}\text{2856}$ correct to ten thousandths.

Practice set b

Convert $\frac{7}{9}$ % to a three-place decimal.

0.008

Convert $\text{51}\frac{5}{11}%$ to a decimal rounded to ten thousandths.

0.5145

Exercises

Make the conversions as indi­cated.

Convert $\frac{3}{4}$ % to a fraction.

$\frac{3}{\text{400}}$

Convert $\frac{5}{6}$ % to a fraction.

Convert $\frac{1}{9}$ % to a fraction.

$\frac{1}{\text{900}}$

Convert $\frac{\text{15}}{\text{19}}$ % to a fraction.

Convert $\frac{5}{4}$ % to a fraction.

$\frac{5}{\text{400}}\text{or}\frac{1}{\text{80}}$

Convert $\frac{7}{3}$ % to a fraction.

Convert $1\frac{6}{7}$ % to a fraction.

$\frac{\text{13}}{\text{700}}$

Convert $2\frac{5}{\text{16}}$ % to a fraction.

Convert $\text{25}\frac{1}{4}$ % to a fraction.

$\frac{\text{101}}{\text{400}}$

Convert $\text{50}\frac{1}{2}$ % to a fraction.

Convert $\text{72}\frac{3}{5}$ % to a fraction.

$\frac{\text{363}}{\text{500}}$

Convert $\text{99}\frac{1}{8}$ % to a fraction.

Convert $\text{136}\frac{2}{3}$ % to a fraction.

$\frac{\text{41}}{\text{30}}$

Convert $\text{521}\frac{3}{4}$ % to a fraction.

Convert $\text{10}\frac{1}{5}$ % to a decimal.

$\frac{\text{51}}{\text{500}}=0\text{.}\text{102}$

Convert $\text{12}\frac{3}{4}$ % to a decimal.

Convert $3\frac{7}{8}$ % to a decimal.

$\frac{\text{31}}{\text{800}}=0\text{.}\text{03875}$

Convert $7\frac{1}{\text{16}}$ % to a decimal.

Convert $\frac{3}{7}$ % to a three-place decimal.

$0\text{.}\text{004}$

Convert $\frac{1}{9}$ % to a three-place decimal.

Convert $6\frac{3}{\text{11}}$ % to a four-place decimal.

0.0627

Convert $9\frac{2}{7}$ % to a four-place decimal.

Convert $\text{24}\frac{5}{\text{21}}$ % to a three-place decimal.

0.242

Convert $\text{45}\frac{8}{\text{27}}$ % to a three-place decimal.

Convert $\text{11}\frac{\text{16}}{\text{17}}$ % to a four-place decimal.

0.1194

Convert $5\frac{1}{7}$ % to a three-place decimal.

Exercises for review

( [link] ) Write $\text{8}\cdot \text{8}\cdot \text{8}\cdot \text{8}\cdot \text{8}$ using exponents.

${8}^{5}$

( [link] ) Convert $4\frac{7}{8}$ to an improper fraction.

( [link] ) Find the sum. $\frac{7}{\text{10}}+\frac{2}{\text{21}}+\frac{1}{7}$ .

$\frac{\text{197}}{\text{210}}$

( [link] ) Find the product. $\left(4\text{.}\text{21}\right)\left(0\text{.}\text{006}\right)$ .

( [link] ) Convert 8.062 to a percent.

$\text{806}\text{.}2\text{}$ %

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