2.13: Percents
- Page ID
- 176513
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Definitions and Theorems
A percent expresses a quantity as a number of parts out of one hundred. Because every percent is measured against the same total of 100, percents provide a common scale on which proportions drawn from different totals can be compared directly.
A percent is a number or ratio expressed as a fraction of 100. The percent sign, %, denotes this relationship, so that for any number \(p\), the expression \(p\%\) means \(\frac{p}{100}\).
For example, \(45\% = \frac{45}{100} = 0.45\), and a class in which \(50\%\) of the students are female has 50 female students for every 100 students. While most percents encountered in practice lie between \(0\) and \(100\), there is no such restriction; values such as \(150\%\) or \(0\%\) are permitted.
For any number \(x\),\[x = (100x)\%, \nonumber\]and, equivalently, for any number \(p\),\[p\% = \dfrac{p}{100}. \nonumber\]Thus a number is written as a percent by multiplying it by 100, and a percent is written as an ordinary number by dividing by 100.
This single relationship governs every conversion among percents, decimals, and fractions. To write a decimal or a fraction as a percent, multiply by 100; to write a percent as a decimal or a fraction, divide by 100.
Writing the percent sign and dividing by 100 at the same time changes the value. For instance, \(25\%\) already means \(\frac{25}{100} = 0.25\); the expression \(\frac{25\%}{100}\) instead equals \(0.0025\). Convert a percent to an ordinary number once, not twice.
If a part is \(p\%\) of a whole, then\[\dfrac{\text{part}}{\text{whole}} = \dfrac{p}{100}, \nonumber\]equivalently,\[\text{part} = \dfrac{p}{100} \cdot \text{whole}. \nonumber\]
When a percent problem is read aloud, the number following the word "is" corresponds to the part, and the number following the word "of" corresponds to the whole. The percent proportion then takes the convenient form\[\dfrac{\text{is}}{\text{of}} = \dfrac{p}{100}. \nonumber\]Any one of the three quantities—the part, the whole, or the percent—may be found when the other two are known.
Examples
Write \(0.45\) as a percent.
- Solution
- By the conversion theorem, multiply by 100:\[0.45 = (100 \cdot 0.45)\% = 45\%. \nonumber\]Multiplying by 100 moves the decimal point two places to the right.
Write \(\dfrac{5}{8}\) as a percent.
- Solution
- Multiply by 100:\[\dfrac{5}{8} = \left( \dfrac{5}{8} \cdot 100 \right)\% = \dfrac{500}{8}\% = 62.5\%, \nonumber\]since \(500 \div 8 = 62.5\).
Write \(8\%\) as a decimal and as a fraction in lowest terms.
- Solution
- Divide by 100. As a decimal,\[8\% = \dfrac{8}{100} = 0.08. \nonumber\]As a fraction, reduce \(\dfrac{8}{100}\) by dividing numerator and denominator by their common factor 4:\[\dfrac{8}{100} = \dfrac{2}{25}. \nonumber\]
What is \(20\%\) of \(80\)?
- Solution
- The percent is \(20\) and the whole (the number following "of") is \(80\); the part is unknown. Let \(x\) denote the part. By the percent proportion,\[\dfrac{x}{80} = \dfrac{20}{100}. \nonumber\]Solving, \(x = \frac{20}{100} \cdot 80 = 16\). Therefore \(20\%\) of \(80\) is \(16\).
\(12\) is what percent of \(48\)?
- Solution
- The part (following "is") is \(12\) and the whole (following "of") is \(48\); the percent is unknown. Let \(p\) denote the percent. Then\[\dfrac{12}{48} = \dfrac{p}{100}. \nonumber\]Solving, \(p = \frac{12}{48} \cdot 100 = 25\). Therefore \(12\) is \(25\%\) of \(48\).
\(15\) is \(30\%\) of what number?
- Solution
- The part (following "is") is \(15\) and the percent is \(30\); the whole (following "of") is unknown. Let \(x\) denote the whole. Then\[\dfrac{15}{x} = \dfrac{30}{100}. \nonumber\]Cross multiplying gives \(30x = 15 \cdot 100\), so \(x = \frac{1500}{30} = 50\). Therefore \(15\) is \(30\%\) of \(50\).
Sources
Several parts of this text use modifications from the following source:
- Wikipedia article: "Percentage"
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