\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=0.6in,right=0.6in,top=0.6in,bottom=0.6in} \pagestyle{fancy} \lhead{\color{cyan} \Large Lecture Notes} \chead{\LARGE Fractions - Part 1} \rhead{} \cfoot{} \lfoot{\small \copyright \; Hidegkuti, 2013} \rfoot{\small Last revised: July 14, 2013} \textwidth 7.5in \textheight 9.5in \setlength{\headheight}{27pt} \rhead{page \thepage} \setlength{\parindent}{0pt} \begin{document} \begin{center} {\huge Definition of a Fraction}\textbf{\bigskip }\bigskip \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}{3.0753in}{1.945in}{0pt}{}{}{pic1.bmp}{\special{language "Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 3.0753in;height 1.945in;depth 0pt;original-width 0.0839in;original-height 0.0571in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename '../../../../../../../Desktop/pic1.bmp';file-properties "XNPEU";}} At first let us not even consider a fraction alone. \ We will just define a \textbf{fraction of something}.\bigskip %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.\bigskip \qquad \qquad \qquad Step 1. \ We first divide the quantity into $5$ equal shares.\bigskip \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 \bigskip So the numerator tells us how many shares we have. \ The denominator tells us how big each share is. \medskip \bigskip \bigskip \bigskip \bigskip \textbf{Example 1. }\ Find $\dfrac{3}{5}$ of $\$100$.\bigskip Step 1. \ Divide $\$100$ into $5$ equal shares. \ \bigskip \qquad We break the $\$100$ into five twenty dollar bills. \ In other words, $\dfrac{1}{5}$ of $\$100$ is $\$20$.\bigskip Step 2. \ Let us take $3$ such shares. \qquad We take three twenty dollar bills, that is $\$60$. \ In other words, $% \dfrac{3}{5}$ of $\$100$ is $\$60$.\pagebreak \textbf{Example 2. \ }Compute\textbf{\ }$\dfrac{4}{7}$ of $42$.\bigskip \qquad Step 1. \ We divide $42$ into $7$ equal shares. \ $\dfrac{1}{7}$ of $% 42$ is $6$.\bigskip \qquad Step 2. \ We take $4$ such shares. \ $\qquad \dfrac{4}{7}$ of $42$ is $4\cdot 6=24$. \ The answer is $24$. \medskip \rule[3pt]{510pt}{2pt} \medskip \textbf{Example 3.} \ Shade the region on the picture below that corresponds to $\dfrac{5}{8}$ of the circle.\FRAME{dtbpF}{1.5056in}{1.5056in}{0pt}{}{}{% pic2a.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.5056in;height 1.5056in;depth 0pt;original-width 0.0441in;original-height 0.0432in;cropleft "0";croptop "1.0321";cropright "1";cropbottom "0";filename '../../../../../../../Desktop/pic2a.bmp';file-properties "XNPEU";}} Step 1. \ Divide the circle into $8$ equal shares.\qquad \qquad \qquad Step 2. \ Take $5$ such shares. \ \ \ \ \ \ \ \ \ \ \ \ \ \FRAME{itbpF}{1.5056in}{1.5056in}{0in}{}{}{% pic2b.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.5056in;height 1.5056in;depth 0in;original-width 0.0441in;original-height 0.0432in;cropleft "0";croptop "1.0326";cropright "1";cropbottom "0";filename '../../../../../../../Desktop/pic2b.bmp';file-properties "XNPEU";}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \FRAME{itbpF}{1.5056in}{1.5056in}{0in}{}{}{pic2c.bmp}{\special% {language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.5056in;height 1.5056in;depth 0in;original-width 0.0441in;original-height 0.0432in;cropleft "0";croptop "1.0321";cropright "1";cropbottom "0";filename '../../../../../../../Desktop/pic2c.bmp';file-properties "XNPEU";}}\bigskip \medskip \rule[3pt]{510pt}{2pt} \medskip \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?\bigskip 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 $\dfrac{5}{7}$.\medskip \rule[3pt]{510pt}{2pt} \medskip \textbf{Example 5.} \ Compute $\dfrac{8}{100}$ of $\$2000$.\bigskip 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 $\$160$.\bigskip \bigskip 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. \end{document}