\hfill \hfill } %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \geometry{left=0.5in,right=0.5in,top=0.5in,bottom=0.5in} \pagestyle{fancy} \lhead{\color{blue} \LARGE Lecture Notes} \chead{\color{black} \LARGE Review of Fractions and Percents} \rhead{\Large page \ \thepage} \cfoot{} \lfoot{\small \copyright $\;$ Hidegkuti, Powell, 2007} \rfoot{\small Last revised: December 20, 2013} \textwidth 7.4in \textheight 9.7in \setlength{\headheight}{28pt} \setlength{\parindent}{0in} \begin{document} \begin{center} {\LARGE Part 1: \ Decimal to Percent}\bigskip \end{center} Turning a decimal to a percent is easy if we know what percents mean. \ A percent is a standardized fraction, with denominator $100.$ \ For example, $% 14\%$ is exactly the same thing as $\dfrac{14}{100}$. \begin{example} Convert $0.6$ to a percent. \end{example} We first obtain a fraction by "growing a $1$". \ We can check mentally: any number divided by $1$ results in the original number.% \begin{equation*} 0.6=\dfrac{0.6}{1} \end{equation*}% Now we want to turn this fraction into one with denominator $100.$ \ The only thing we can do to a fraction without changing its value is to multiply upstairs and downstairs by the same number. \ Since our goal is to turn the denominator into $100$, we will use $100.$% \begin{equation*} 0.6=\dfrac{0.6}{1}=\dfrac{0.6\cdot 100}{1\cdot 100}=\dfrac{60}{100}=60\% \end{equation*}% Thus $0.6=60\%$. \begin{example} Convert \ $1.209$ to a percent. \end{example} We first obtain a fraction by "growing a $1$". \ Then we turn this fraction into one with denominator $100.$ \ The only thing we can do to a fraction without changing its value is to multiply upstairs and downstairs by the same number. \ Since our goal is to turn the denominator into $100$, we will use $100$.% \begin{equation*} 1.209=\dfrac{1.209}{1}=\dfrac{1.209\cdot 100}{1\cdot 100}=\dfrac{120.9}{100}% =120.9\% \end{equation*}% Thus $1.209=120.9\%$. \ The percent we obtained is larger than $100\%,$ indicating that we have started with a decimal that is larger than $1$% .\bigskip \begin{center} {\LARGE Part 2: \ Fraction to Percent}\bigskip \end{center} Again, the process is simple if we know what percents mean. \ A percent is a standardized fraction, with denominator $100.$ \ For example, $14\%$ is exactly the same thing as $\dfrac{14}{100}$.\bigskip {\Large Case 1}. \ If we are lucky, we have a fraction whose denominator is a factor of $100.$ \begin{example} Convert $\dfrac{7}{20}$ to a percent. \end{example} The only thing we can do to a fraction without changing its value is to multiply upstairs and downstairs by the same number. \ Since our goal is to turn the denominator into $100$, we will use $5.$ \ \begin{equation*} \dfrac{7}{20}=\dfrac{7\cdot 5}{20\cdot 5}=\dfrac{35}{100}=35\% \end{equation*}% Thus $\dfrac{7}{20}=35\%$.\pagebreak \begin{example} Convert \ $\dfrac{3}{4}$ to a percent. \end{example} The only thing we can do to a fraction without changing its value is to multiply upstairs and downstairs by the same number. \ Since our goal is to turn the denominator into $100$, we will use $25.$ \ \begin{equation*} \dfrac{3}{4}=\dfrac{3\cdot 25}{4\cdot 25}=\dfrac{75}{100}=75\% \end{equation*} Thus $\dfrac{3}{4}=75\%$. \begin{example} Convert \ $\dfrac{17}{10}$ to a percent. \end{example} The only thing we can do to a fraction without changing its value is to multiply upstairs and downstairs by the same number. \ Since our goal is to turn the denominator into $100$, we will use $10.$ \ \begin{equation*} \dfrac{17}{10}=\dfrac{17\cdot 10}{10\cdot 10}=\dfrac{170}{100}=170\% \end{equation*}% Thus $\dfrac{17}{10}=170\%$. \ The percent we obtained is larger than $100\%$% , indicating that we have started with a fraction that is larger than $1$% .\bigskip {\Large Case 2.} \ If we are given a fraction whose denominator is not a factor of $100$, we can not apply the method used above. \ The process is still simple: we convert the fraction to a decimal, and then convert the decimal to a percent.% \begin{equation*} \text{Fraction \ }\longmapsto \text{ \ Decimal \ }\longmapsto \text{ \ Percent} \end{equation*} \begin{example} Convert $\dfrac{29}{80}$ to a percent. \end{example} Step 1. We convert $\dfrac{29}{80}$ to a decimal by division.% \begin{equation*} 29\div 80=0.3625 \end{equation*}% The result is a terminating decimal. \ To preserve accuracy, we will carry all digits after the decimal point. Step 2. \ We convert the decimal to a percent as described in Part 1.% \begin{equation*} 0.3625=\dfrac{0.3625}{1}=\dfrac{0.3625\cdot 100}{1\cdot 100}=\dfrac{36.25}{% 100}=36.25\% \end{equation*} Thus $\dfrac{29}{80}=36.25\%$. \begin{example} Convert $\dfrac{4}{7}$ to a percent. \end{example} Step 1. We convert $\dfrac{4}{7}$ to a decimal by division.% \begin{equation*} 4\div 7=0.571428571428...... \end{equation*}% The result is a non-terminating repeating decimal. \ To preserve accuracy, we will carry five (or more) digits after the decimal point. Step 2. \ We convert the decimal to a percent as described in Part 1.% \begin{equation*} 0.57142857=\dfrac{0.57142857}{1}=\dfrac{0.57142857\cdot 100}{1\cdot 100}=% \dfrac{57.142857}{100}=57.142857\% \end{equation*} We round the result according to the accuracy requested in the particular problem. \ For now, we will round to three decimals after the decimal point. \ The result is $\dfrac{4}{7}\approx 57.143\%$. \ The sign $\ \approx $ \ means "approximately equal". \begin{example} Convert $\dfrac{43}{15}$ to a percent. \end{example} Step 1. We convert $\dfrac{43}{15}$ to a decimal by division.% \begin{equation*} 43\div 15=2.866666...... \end{equation*}% The result is a non-terminating repeating decimal. \ To preserve accuracy, we will carry five (or more) digits after the decimal point. Step 2. \ We convert the decimal to a percent as described in Part 1.% \begin{equation*} 2.866666=\dfrac{2.866666}{1}=\dfrac{2.866666\cdot 100}{1\cdot 100}=\dfrac{% 286.6666}{100}=286.6666\% \end{equation*} We round the result according to the accuracy requested in the particular problem. \ For now, we will round to three decimals after the decimal point. \ The result is $\dfrac{43}{15}\approx 286.667\%$. \ The sign $\ \approx $ \ means "approximately equal". \ The result is a percent larger than $100\%$, indicating that we have started with a fraction larger than $1$.\bigskip \bigskip \bigskip \begin{center} {\LARGE Part 3: \ Percent to Decimal}\bigskip \end{center} \begin{example} Convert $64\%$ \ to a decimal. \end{example} First we re-write the percent as a fraction. \ The we perform the division indicated by the fraction \begin{equation*} 64\%=\dfrac{64}{100}=0.64 \end{equation*}% Thus $64\%=0.64$. \begin{example} Convert $150\%$ to a decimal. \end{example} First we re-write the percent as a fraction. \ The we perform the division indicated by the fraction \begin{equation*} 150\%=\dfrac{150}{100}=1.5 \end{equation*}% Thus $150\%=1.5$. \begin{example} Convert $8.5\%$ to a decimal. \end{example} First we re-write the percent as a fraction. \ The we perform the division indicated by the fraction \begin{equation*} 8.5\%=\dfrac{8.5}{100}=0.085 \end{equation*}% Thus $8.5\%=0.085$\pagebreak \begin{center} {\LARGE Part 4: \ Percent to Fraction}\bigskip \end{center} We have left the easiest one for last. \ Percents already are fractions, with denominator $100$. \begin{example} Convert $55\%$ to a reduced fraction. \end{example} We rewrite the percent as a fraction and simplify.% \begin{equation*} 55\%=\dfrac{55}{100}=\dfrac{\NEG{5}\cdot 11}{\NEG{5}\cdot 20}=\dfrac{11}{20} \end{equation*} \begin{example} Convert $240\%$ to a reduced fraction. \end{example} We rewrite the percent as a fraction and simplify.% \begin{equation*} 240\%=\dfrac{240}{100}=\dfrac{20\cdot 12}{20\cdot 5}=\dfrac{12}{5}=2\dfrac{2% }{5} \end{equation*}% Thus $240\%=2\dfrac{2}{5}$. \ The result is a mixed number, indicating that we started with a percent larger than $100\%$. \begin{example} Convert $37.5\%$ to a reduced fraction. \end{example} When we rewrite the percent as a fraction, the numerator is not an integer. \ We fix this by multiplying both numerator and denominator by $10$. \ The rest is as before.% \begin{equation*} 37.5\%=\dfrac{37.5}{100}=\dfrac{37.5\cdot 10}{100\cdot 10}=\dfrac{375}{1000}=% \dfrac{125\cdot 3}{125\cdot 8}=\dfrac{3}{8} \end{equation*} Thus $37.5\%=\dfrac{3}{8}$.\bigskip \begin{center} {\LARGE Exercises}\bigskip \end{center} Compute the missing values in the table shown below. \begin{center} \FRAME{dtbpF}{2.7838in}{4.4512in}{0pt}{}{}{table1.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.7838in;height 4.4512in;depth 0pt;original-width 3.7135in;original-height 5.9733in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'table1.bmp';file-properties "XNPEU";}}\pagebreak {\LARGE Answers for Exercises}\bigskip \FRAME{dtbpF}{2.7172in}{4.3068in}{0pt}{}{}{table2.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.7172in;height 4.3068in;depth 0pt;original-width 3.7464in;original-height 5.9733in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'table2.bmp';file-properties "XNPEU";}} \end{center} \vspace{4.3in} {\Large %TCIMACRO{\TeXButton{\small}{\small}}% %BeginExpansion \small% %EndExpansion } \href{https://teaching.martahidegkuti.com/shared/lnotes/lecturenotes.html}{% For more documents like this, visit our page at\ https://teaching.martahidegkuti.com and click on Lecture Notes. \ E-mail questions or comments to mhidegkuti@ccc.edu.} \end{document}