\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}{Definition} \newtheorem{example}{Example} \newtheorem{exercise}[theorem]{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=0.4in,right=0.5in,top=0.5in,bottom=0.4in} \pagestyle{fancy} \lhead{\color{blue} \large Lecture Notes} \chead{\Large Fractions - Part 1} \rhead{} \cfoot{} \lfoot{\footnotesize \copyright \; Hidegkuti, 2013} \rfoot{\footnotesize Last revised: July 14, 2013} \textwidth 7.7in \textheight 9.7in \setlength{\headheight}{27pt} \rhead{\small page \thepage} \setlength{\parindent}{0pt} \begin{document} \begin{center} {\LARGE Definition of a Fraction}\vspace{0.1in} \end{center} What is a fraction? \ A fraction has three components as shown on the picture below. \ The important parts are the numerator above and the denominator below the little line. \FRAME{dtbpF}{2.1032in}{1.3301in}{0pt}{}{}{pic1.bmp}{\special{language "Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 2.1032in;height 1.3301in;depth 0pt;original-width 0.0839in;original-height 0.0571in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'pic1.bmp';file-properties "XNPEU";}} At first let us not even consider a fraction alone. \ We will just define a \textbf{fraction of something}.\vspace{0.1in} %TCIMACRO{% %\TeXButton{box start - 6.8in}{\begin{boxedminipage}{6.8in} %\setlength{\fboxrule}{12pt} %\setlength{\fboxsep}{15pt} %\color{blue} %\large}}% %BeginExpansion \begin{boxedminipage}{6.8in} \setlength{\fboxrule}{12pt} \setlength{\fboxsep}{15pt} \color{blue} \large% %EndExpansion \qquad \textbf{Definition:} \ $\dfrac{3}{5}$ of a quantity can be obtained as follows.\vspace{0.06in} \qquad \qquad \qquad Step 1. \ We first divide the quantity into $5$ equal shares.\vspace{0.06in} \qquad \qquad \qquad Step 2. \ Let us take $3$ such shares.\ That is $\dfrac{% 3}{5}$ of our quantity. %TCIMACRO{\TeXButton{end of box}{\end{boxedminipage}}}% %BeginExpansion \end{boxedminipage}% %EndExpansion \vspace{0.09in} So the numerator tells us how many shares we have. \ The denominator tells us how big each share is.\vspace{0.06in}\vspace{0.06in} \textbf{Example 1. }\ Find $\dfrac{3}{5}$ of $\$100$.\vspace{0.06in} Step 1. \ Divide $\$100$ into $5$ equal shares. \ \vspace{0.06in} \qquad We break the $\$100$ into five twenty dollar bills. \ In other words, $\dfrac{1}{5}$ of $\$100$ is $\$20$.\vspace{0.06in} Step 2. \ Let us take $3$ such shares.\vspace{0.06in} \qquad We take three twenty dollar bills, that is $\$60$. \ In other words, $% \dfrac{3}{5}$ of $\$100$ is $\$60$.\vspace{0.06in}\vspace{0.06in} \textbf{Example 2. \ }Compute\textbf{\ }$\dfrac{4}{7}$ of $42$.\vspace{0.06in% }\vspace{0.06in} Solution: \ \qquad Step 1. \ We divide $42$ into $7$ equal shares. \ $\dfrac{1}{7}$ of $% 42$ is $6$.\vspace{0.06in} \qquad Step 2. \ We take $4$ such shares. \ $\qquad \dfrac{4}{7}$ of $42$ is $4\cdot 6=24$. \ The answer is $24$.\vspace{0.06in} \rule[3pt]{510pt}{2pt} \textbf{Example 3.} \ Shade the region on the picture that corresponds to $% \dfrac{5}{8}$ of the circle.\FRAME{dtbpF}{0.998in}{0.998in}{0pt}{}{}{% pic2a.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 0.998in;height 0.998in;depth 0pt;original-width 0.0441in;original-height 0.0432in;cropleft "0";croptop "1.0321";cropright "1";cropbottom "0";filename 'pic2a.bmp';file-properties "XNPEU";}} \pagebreak Solution: \ Step 1. \ Divide the circle into $8$ equal shares.\qquad \qquad \qquad Step 2. \ Take $5$ such shares. \ \ \ \ \ \ \ \ \ \ \ \ \ \FRAME{itbpF}{1.2133in}{1.2133in}{0in}{}{}{% pic2b.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.2133in;height 1.2133in;depth 0in;original-width 0.0441in;original-height 0.0432in;cropleft "0";croptop "1.0326";cropright "1";cropbottom "0";filename 'pic2b.bmp';file-properties "XNPEU";}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \FRAME{itbpF}{1.1787in}{% 1.1787in}{0in}{}{}{pic2c.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.1787in;height 1.1787in;depth 0in;original-width 0.0441in;original-height 0.0432in;cropleft "0";croptop "1.0321";cropright "1";cropbottom "0";filename 'pic2c.bmp';file-properties "XNPEU";}}\bigskip \medskip \rule[3pt]{510pt}{2pt} \textbf{Example 4.} \ A cake was sliced into equal slices. \ Amy ate $2$ slices and Betsy ate $3$ slices. \ If $2$ slices were remaining, what fraction of the cake was eaten?\vspace{0.06in} Solution:\ \ We need to first figure out how many slices made up the cake. \ If $2$ were eaten by Amy and $3$ by Betsy and $2$ more were left, then there were all together $2+3+2=7$ slices. \ 5 slices were eaten which were $\dfrac{% 5}{7}$ of the cake. \ So the answer is $\dfrac{5}{7}$.\vspace{0.06in} \rule[3pt]{510pt}{2pt} \medskip \textbf{Example 5.} \ Compute $\dfrac{8}{100}$ of $\$2000$.\vspace{0.06in} Solution: \ We divide $\$2000$ into $100$ equal shares. \ Each share is $\$20 $. \ Then we take $8$ such shares, that is $8\cdot \$20=\$160$. \ Thus $% \dfrac{8}{100}$ of $\$2000$ is $\$160$.\vspace{0.1in}\vspace{0.1in} Note: \ We often use fractions with $100$ in the denominator. \ These fractions also called percents and denoted by $\%$. \ Thus, when we are asked to compute $8\%$ of a quantity, that is exactly the same as $\dfrac{8}{% 100}$ of it.\vspace{0.2in} \begin{center} \FRAME{itbpF}{0.6529in}{0.6772in}{0.2309in}{}{}{work.jpg}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 0.6529in;height 0.6772in;depth 0.2309in;original-width 4.6977in;original-height 4.875in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename '../../../../../../aa_book99/images/work.jpg';file-properties "XNPEU";}} \ \ {\large Practice Problems} \end{center} \begin{enumerate} \item %TCIMACRO{\TeXButton{2col begin}{\begin{multicols}{2}} }% %BeginExpansion \begin{multicols}{2} %EndExpansion Compute $\dfrac{4}{9}$ of $63$. \item Shade $\dfrac{5}{9}$ of the circle below. \ \ \ \ \FRAME{itbpF}{0.6996in}{0.6996in}{0in}{}{}{pic2a.bmp}{\special% {language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 0.6996in;height 0.6996in;depth 0in;original-width 0.0441in;original-height 0.0432in;cropleft "0";croptop "1.0321";cropright "1";cropbottom "0";filename 'pic2a.bmp';file-properties "XNPEU";}} \item Compute $\dfrac{5}{6}$ of $24$. \item The price of a TV is $\$400$. \ We want to raise \newline the price by $5\%$. \ What is the new price? \item The price of a couch is $\$700$. \ Next week, it will go \newline on a $15\%$ off sale. \ What is the new price? (A $15\%$ \newline sale means that the price of the item is lowered \newline by $15\%$.) \item Find $\dfrac{3}{4}$ of $56$. \item We placed $\$2000$ into a bank account with $6\%$ yearly interest rate. How much money do we have in the bank after one year? \item This problem is about a method of comparing fractions. a) \ Compute $\dfrac{3}{7}$ \ of \ $420$. b) \ Compute $\dfrac{4}{10}$ of $420$. c) \ Based on the results of parts a) and b), which fraction is larger, $% \dfrac{3}{7}$ \ or $\dfrac{4}{10}$? \item Bert has made $\$54\,000$ last year. \ If he has to pay $32\%$ of his income in taxes, how much taxes does he owe and how much of his income will he keep? \item Sally used to make $\$2400$ per month, but now she got a $3\%$ raise. \ How much is her monthly salary now? \item Mr. X won \thinspace $\$600\,000$ in the lottery two yers ago. \ By now he has spent some of the money. \ When he was asked what happened, he said the following. \ "\textit{I didn't spend it all. \ I spent }$\dfrac{1}{3% }$\textit{\ of the money by taking a luxury yacht trip around the world. \ Then I put the rest in the bank. \ Later I decided to buy a house I really liked. \ So I took }$\dfrac{3}{4}$\textit{\ of the money out of the bank and baught the house. \ For half of what was left, I purchased stocks that completely lost their value. \ Finally, I gave }$\dfrac{2}{5}$\textit{\ of what's left to my niece for her college education."} How much money is left from the winnings? %TCIMACRO{\TeXButton{multicol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \end{enumerate} \bigskip \bigskip \begin{center} \FRAME{itbpF}{0.7066in}{0.7066in}{0.2413in}{}{}{answers.jpg}{\special% {language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 0.7066in;height 0.7066in;depth 0.2413in;original-width 8.6455in;original-height 8.6455in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename '../../../../../../aa_book99/images/answers.jpg';file-properties "XNPEU";}} \ \ \ Answers \end{center} \begin{enumerate} \item %TCIMACRO{\TeXButton{2col begin}{\begin{multicols}{2}}}% %BeginExpansion \begin{multicols}{2}% %EndExpansion $16$ \ \ \ \ \ 2. \ See below \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \FRAME{itbpF}{1.3318in% }{1.3318in}{0in}{}{}{pic3a.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.3318in;height 1.3318in;depth 0in;original-width 0.045in;original-height 0.0432in;cropleft "0";croptop "1.0484";cropright "1";cropbottom "0";filename 'pic3a.bmp';file-properties "XNPEU";}} \item $20$ \ \ \ \ 4. \ $\$420$ \ \ \ \ 5. \ $\$595$ \ \ \ \ 6. \ $42$ \item[7.] $\$2120$ \ \ \ \ 8. \ a) \ $180$ \ \ b) \ $168$ \ \ c) \ $\dfrac{3% }{7}$ is $\ $larger \item[9.] $\$17\,280$ in taxes and will keep $\$36\,720$. \ \ \ \ \item[10.] $\$2472$ \ \ \ \ 11. \ $\$30\,000$% %TCIMACRO{\TeXButton{multicol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \end{enumerate} \end{document}