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\geometry{left=0.5in,right=0.5in,top=0.5in,bottom=0.3in} \pagestyle{fancy} \lhead{\color{blue} Lecture Notes} \chead{\color{black} \Large Fractions -- Part 2: Equivalent Fractions} \rhead{\small page \ \thepage} \cfoot{} \lfoot{\footnotesize \copyright $\;$ Hidegkuti, 2018} \rfoot{\footnotesize Last revised: August 25, 2018} \textwidth 7.6in \textheight 9.6in \setlength{\headheight}{28pt} \setlength{\parindent}{0in} \cfoot{} \renewcommand{\headrulewidth}{0.8pt} \newcommand{\defstart}{ \begin{boxedminipage}{7in} \setlength{\fboxrule}{15pt} \setlength{\fboxsep}{15pt} \setlength{\rightskip}{20pt} \setlength{\leftskip}{20pt} \color{white} . \color{black} } \newcommand{\defcont}{ \setlength{\hangindent}{57pt} } \newcommand{\definside}{ \setlength{\hangindent}{57pt} \setlength{\parindent}{57pt} } \newcommand{\defend}{ \end{boxedminipage} \setlength{\parindent}{0pt} \setlength{\hangindent}{0pt} \setlength{\parskip}{0.06in} \setstretch{1.08} } \newcommand{\exstart}{ \setlength{\parindent}{0pt} \setlength{\hangindent}{60pt} \setlength{\parskip}{0.04in} \setstretch{1.08} } \newcommand{\excont}{ \setlength{\parindent}{60pt} \setlength{\hangindent}{75pt} } \newcommand{\solstart}{ \setlength{\parindent}{0pt} \setlength{\hangindent}{63pt} } \newcommand{\solcont}{ \setlength{\parindent}{63pt} \setlength{\hangindent}{63pt} } \newcommand{\solrestart}{ \setlength{\parindent}{48pt} \setlength{\hangindent}{63pt} } \newcommand{\sol}{ \setlength{\parindent}{0pt} \setlength{\hangindent}{50pt} } \newcommand{\solinside}{ \setlength{\parindent}{50pt} \setlength{\hangindent}{50pt} \setstretch{1.08} } \newcommand{\vissza}{ \setlength{\parindent}{0pt} \setlength{\hangindent}{0pt} \setlength{\parskip}{0.1in} \setstretch{1.08} \normalsize } \newcommand{\matek}{ \setstretch{1.04} \normalsize \setlength{\parskip}{0.04in} \setlength{\parindent}{0pt} \setlength{\hangindent}{0pt} } \setstretch{1.04} \begin{document} %TCIMACRO{\TeXButton{\vissza}{\vissza}}% %BeginExpansion \vissza% %EndExpansion So far, we didn't look at a standalone fraction on its own. \ We only defined a fraction as expressing part of something. \ We will continue to do this. \ A warning to the reader: do not read this section while hungry. \ We will be talking a lot about cakes and pizzas. \ %TCIMACRO{\TeXButton{\begin{minipage}{4.5in}}{\begin{minipage}{4.5in}}}% %BeginExpansion \begin{minipage}{4.5in}% %EndExpansion Suppose that Ann, Bethany, Cecile and Desire are about to share a pizza. \ They cut the pizza into four equal slices. \ Ann happily looks at her slice (the shaded region). \ What\vspace{0.04in} fraction can express this slice? \ The answer is $\dfrac{1}{4}$.\vspace{0.04in} \ We are talking about one slice, so big so that four equal slices make up the whole. \ %TCIMACRO{\TeXButton{\end{minipage}}{\end{minipage}}}% %BeginExpansion \end{minipage}% %EndExpansion %TCIMACRO{% %\TeXButton{\begin{minipage}{3in}}{\begin{minipage}{3in} %\setlength{\leftskip}{40pt}}}% %BeginExpansion \begin{minipage}{3in} \setlength{\leftskip}{40pt}% %EndExpansion %TCIMACRO{% %\FRAME{itbpF}{1.3292in}{1.3136in}{0in}{}{}{fractions2pic1.bmp}{% %\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.3292in;height 1.3136in;depth 0in;original-width 0.0441in;original-height 0.0432in;cropleft "0";croptop "1.0193";cropright "1";cropbottom "0";filename 'fractions2pic1.bmp';file-properties "XNPEU";}}}% %BeginExpansion \includegraphics[ natheight=0.0432in, natwidth=0.0441in, height=1.3136in, width=1.3292in] {//Mac/Dropbox/lnotes/1Prealgebra/fractions/fractions2/graphics/fractions2pic1__1.pdf}% %EndExpansion %TCIMACRO{\TeXButton{\end{minipage}}{\end{minipage}}}% %BeginExpansion \end{minipage}% %EndExpansion %TCIMACRO{\TeXButton{\begin{minipage}{4.5in}}{\begin{minipage}{4.5in}}}% %BeginExpansion \begin{minipage}{4.5in}% %EndExpansion Just before they start to eat, four other friends arrive. \ They decide to share, so everyone cuts their own slice in two equal pieces. \ Before she gives her friend half of her food, Ann looks at her plate. \ Her share now looks like this. \ What fraction could express this?\vspace{0.04in} \ Since we cut each slice in two, the four slices became eight slices. \ So, we are now looking at $\dfrac{2}{8}$. \ %TCIMACRO{\TeXButton{\end{minipage}}{\end{minipage}}}% %BeginExpansion \end{minipage}% %EndExpansion %TCIMACRO{% %\TeXButton{\begin{minipage}{3in}}{\begin{minipage}{3in} %\setlength{\leftskip}{40pt}}}% %BeginExpansion \begin{minipage}{3in} \setlength{\leftskip}{40pt}% %EndExpansion %TCIMACRO{% %\FRAME{itbpF}{1.3258in}{1.3145in}{0in}{}{}{fractions2pic2.bmp}{% %\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.3258in;height 1.3145in;depth 0in;original-width 0.0441in;original-height 0.0432in;cropleft "0";croptop "1.0238";cropright "1";cropbottom "0";filename 'fractions2pic2.bmp';file-properties "XNPEU";}}}% %BeginExpansion \includegraphics[ natheight=0.0432in, natwidth=0.0441in, height=1.3145in, width=1.3258in] {//Mac/Dropbox/lnotes/1Prealgebra/fractions/fractions2/graphics/fractions2pic2__2.pdf}% %EndExpansion %TCIMACRO{\TeXButton{\end{minipage}}{\end{minipage}}}% %BeginExpansion \end{minipage}% %EndExpansion The fractions $\dfrac{1}{4}$ and $\dfrac{2}{8}$ express the same part of a given quantity. \ Such fractions are called \textbf{equivalent} to each other. \ Given any fraction, we can create infitely many equaivalent fractions by multiplying both numerator and denominator by the same non-zero number. \ For example, if we start with $\dfrac{2}{3}$, the following fractions are all equivalent: %TCIMACRO{\TeXButton{\begin{minipage}{1.8in}}{\begin{minipage}{1.8in}} }% %BeginExpansion \begin{minipage}{1.8in} %EndExpansion $\ \ \ \ \ \ \ \ \ \ \ \ \dfrac{2}{3}=\dfrac{4}{6}$ \vspace{0.08in} multiply both numerator and denominator by $2$ %TCIMACRO{\TeXButton{\end{minipage}}{\end{minipage}} }% %BeginExpansion \end{minipage} %EndExpansion \ \ \ %TCIMACRO{\TeXButton{\begin{minipage}{1.8in}}{\begin{minipage}{1.8in}} }% %BeginExpansion \begin{minipage}{1.8in} %EndExpansion $\ \ \ \ \ \ \ \ \ \ \ \ \dfrac{2}{3}=\dfrac{6}{9}$ \vspace{0.08in} multiply both numerator and denominator by $3$ %TCIMACRO{\TeXButton{\end{minipage}}{\end{minipage}} }% %BeginExpansion \end{minipage} %EndExpansion \ \ \ %TCIMACRO{\TeXButton{\begin{minipage}{1.8in}}{\begin{minipage}{1.8in}} }% %BeginExpansion \begin{minipage}{1.8in} %EndExpansion $\ \ \ \ \ \ \ \ \ \ \ \ \dfrac{2}{3}=\dfrac{8}{12}$ \vspace{0.08in} multiply both numerator and denominator by $4$ %TCIMACRO{\TeXButton{\end{minipage}}{\end{minipage}} }% %BeginExpansion \end{minipage} %EndExpansion \ \ \ %TCIMACRO{\TeXButton{\begin{minipage}{1.8in}}{\begin{minipage}{1.8in}} }% %BeginExpansion \begin{minipage}{1.8in} %EndExpansion $\ \ \ \ \ \ \ \ \ \ \ \ \dfrac{2}{3}=\dfrac{10}{15}$ \vspace{0.08in} multiply both numerator and denominator by $5$ %TCIMACRO{\TeXButton{\end{minipage}}{\end{minipage}}}% %BeginExpansion \end{minipage}% %EndExpansion %TCIMACRO{\TeXButton{\defstart}{\defstart}}% %BeginExpansion \defstart% %EndExpansion %TCIMACRO{\TeXButton{\defcont}{\defcont}}% %BeginExpansion \defcont% %EndExpansion \textbf{Theorem: }(also called the Fundamental Property of Fractions) \ Given a fraction $\dfrac{a}{b}$ where $a$ and $b$ are integers, $b\not=0$, and any non-zero number $c$, $\dfrac{a}{b}$ and $\dfrac{ac}{bc}$ are equivalent fractions, i.e. \begin{equation*} \dfrac{a}{b}=\dfrac{ac}{bc} \end{equation*} %TCIMACRO{\TeXButton{\defend}{\defend}}% %BeginExpansion \defend% %EndExpansion \vspace{0.09in} Another way to express this is as follows: \ One thing we can always do to a fraction without changing its value is to multiply both numerator and denominator by the same number. \ Equivalent fractions are essentially different representations of the same number.\vspace{0.06in}\vspace{0.06in} %TCIMACRO{\TeXButton{\exstart}{\exstart}}% %BeginExpansion \exstart% %EndExpansion \textbf{Example 1. }\ Re-write $\dfrac{4}{5}$ with a denominator of $20$. \ \vspace{0.06in} %TCIMACRO{\TeXButton{\sol}{\sol}}% %BeginExpansion \sol% %EndExpansion \textbf{Solution:} \ The only thing we can do is to multiply both numerator and denominator by the same number. If we want a new denominator of $20$, that means that we have to multiply the old denominator, $5$, by $4$. \ Then we have no other option than multiplying the numerator by the same number. \ $4\cdot 4=16$ and $5\cdot 4=20$. \ Thus the equivalent fraction we are looking for is \fbox{$\dfrac{16}{20}$}.\vspace{0.06in}\vspace{0.06in} \pagebreak %TCIMACRO{\TeXButton{\exstart}{\exstart}}% %BeginExpansion \exstart% %EndExpansion \textbf{Example 2. \ }Re-write $\dfrac{4}{5}$ as a percent.\vspace{0.06in}% \vspace{0.06in} %TCIMACRO{\TeXButton{\sol}{\sol}}% %BeginExpansion \sol% %EndExpansion \textbf{Solution: }\ The only thing we can do is to multiply both numerator and denominator by the same number. Since we must write $\dfrac{4}{5}$ as a percent, that measn that we want a new denominator of $100.$ \ What number should we multiply by $5$ to get to $100$? \ This is a division problem, $% 100\div 5=20$. \ This means that we will multiply both numerator and denominator by $20$. \ \begin{equation*} \dfrac{4}{5}=\dfrac{4\cdot 20}{5\cdot 20}=\dfrac{80}{100} \end{equation*}% and $\dfrac{80}{100}$ is the same as \fbox{$80\%$}.\vspace{0.06in} %TCIMACRO{\TeXButton{\vissza}{\vissza}}% %BeginExpansion \vissza% %EndExpansion Now that we know that each fraction can be represented in infinitely many forms, we should find an "official" one, hopefully the simplest. \ \vspace{% 0.06in} We now know that a fraction's value does not change if we multiply both numerator and denominator by the same non-zero number.% \begin{equation*} \dfrac{a}{b}=\dfrac{ac}{bc}\text{ \ \ \ \ \ \ \ \ \ \ }b,c\not=0 \end{equation*}% Reading the same equality from right to left as \ $\dfrac{ac}{bc}=\dfrac{a}{b% }$\ \ tells\vspace{0.04in} us that we are also allowed to \textit{divide} both numerator and denominator by the same non-zero number, and we did not change the value of the fraction. \ This gives us the concept of the simplest form of a fraction. \ %TCIMACRO{\TeXButton{\defstart}{\defstart}}% %BeginExpansion \defstart% %EndExpansion %TCIMACRO{\TeXButton{\defcont}{\defcont}}% %BeginExpansion \defcont% %EndExpansion \textbf{Definition: }Given a fraction $\dfrac{a}{b}$ where\vspace{0.04in} $a$ and $b$ are integers, $b\not=0$, the fraction is \textbf{reduced} or \textbf{% in lowest terms} if $a$ and $b$ have no common divisor greater than $1$. \ If this is the case, we also say that $a$ and $b$ are \textbf{relatively prime}. \ %TCIMACRO{\TeXButton{\defend}{\defend}}% %BeginExpansion \defend% %EndExpansion \vspace{0.09in} %TCIMACRO{\TeXButton{\vissza}{\vissza}}% %BeginExpansion \vissza% %EndExpansion It feels true that every fraction has a unique reduced form.\vspace{0.04in} \ Since every fraction can be reduced, and the reduced form is clearly its simplest form, \textbf{we always present fractions in lowest terms as final answers}. \ %TCIMACRO{\TeXButton{\exstart}{\exstart}}% %BeginExpansion \exstart% %EndExpansion \textbf{Example 3.} \ Reduce $\dfrac{12}{28}$ to lowest terms. \ %TCIMACRO{\TeXButton{\sol}{\sol}}% %BeginExpansion \sol% %EndExpansion \textbf{Solution: \ }We need to look for a common factor between $12$ and $% 28 $.\vspace{0.04in} \ The greatest common divisor is $4.$ \ Indeed, when we divide both numerator and denominator by $4$, we get that $\dfrac{12}{28}=% \dfrac{3}{7}$. \ \vspace{0.04in} %TCIMACRO{\TeXButton{\solinside}{\solinside}}% %BeginExpansion \solinside% %EndExpansion Note\vspace{0.04in} that we do not have to get it right for the first time. \ Suppose we only notice the common factor $2$. \ Then we have $\dfrac{12}{28% }=\dfrac{6}{14}$.\vspace{0.04in} \ This is not reduced yet, because both $6$ and $12$ are even. \ So we go again: we divide both both numerator and denominator by $2$ again: $\dfrac{12}{28}=\dfrac{6}{14}=\dfrac{3}{7}$.% \vspace{0.04in} \ Now the fraction is in lowest terms, so the answer is \fbox{$\dfrac{3}{7}$}.\vspace{0.04in} %TCIMACRO{\TeXButton{\exstart}{\exstart}}% %BeginExpansion \exstart% %EndExpansion \textbf{Example 4.} \ Re-write $35\%$ as a fraction in lowest terms. \ %TCIMACRO{\TeXButton{\sol}{\sol}}% %BeginExpansion \sol% %EndExpansion \textbf{Solution: \ }First, $35\%=\dfrac{35}{100}$. \ Clearly both numerator and denominator are divisible by $5,$ so let's get rid of that $5$ by dividing both numerator and denominator by $5$. \ $\dfrac{35}{100}=\dfrac{7}{% 20}$. \ This is now in lowest form since $7$ and $20$ do not share any divisor besides the obvious one, $1$. So $35\%$ as a reduced fraction is \fbox{$\dfrac{7}{20}$}.\vspace{0.06in} \pagebreak %TCIMACRO{\TeXButton{\vissza}{\vissza}}% %BeginExpansion \vissza% %EndExpansion Until now, we only looked at fractions that were a part of a whole. \ If we cut a pizza\vspace{0.04in} into six equal slices and take all six slices, how can we express this as a fraction? \ Based on what we learned so far, $% \dfrac{6}{6}$ \medskip is the fraction expressing this. \ Dividing both numerator and denominator by $6$, we get $\dfrac{6}{6}=\dfrac{1}{1}$ or simply $1$. \ $1$ whole\medskip\ pizza. So, the reduced form of fractions such as $\dfrac{3}{3}$ or $\dfrac{7}{7}$ or $\dfrac{12}{12}$ is simply $1$. \ \vspace{0.04in}\vspace{0.04in} What if we buy two pizzas, we slice them both into $6$ equal slices and take $10$ such slices?\vspace{0.04in} \ That could be expressed as $\dfrac{10}{6}$% .\vspace{0.04in} \ And if we take all $12$ slices? \ That would be $\dfrac{12% }{6}=\dfrac{2}{1}=2$, or two \ $2$ whole pizzas. \ This means that every positive integer can be expressed as a fraction, with infinitely many equaivalent forms. \ For example, $5=\dfrac{5}{1}=\dfrac{30}{% 6}$.\vspace{0.04in} \ Indeed, if we buy $5$ whole pizzas and cut each into $% 6 $ equal slices, we will have 30 slices, each so big so that $6$ of them makes an entire pizza. \ %TCIMACRO{\TeXButton{\defstart}{\defstart}}% %BeginExpansion \defstart% %EndExpansion %TCIMACRO{\TeXButton{\defcont}{\defcont}}% %BeginExpansion \defcont% %EndExpansion \textbf{Definition: }If the numerator of a fraction is less than its denominator, we call such\vspace{0.04in} a fraction \textbf{proper}. \ A proper fraction always expresses less than a unit of a given quantity. \ Examples of proper fractions are $\dfrac{2}{3}$, $\dfrac{3}{10}$, $\dfrac{6}{% 15}$, or $55\%$.\vspace{0.04in}\vspace{0.04in} %TCIMACRO{\TeXButton{\definside}{\definside}}% %BeginExpansion \definside% %EndExpansion If the numerator of a fraction is greater than or equal to its denominator, we call the fraction \textbf{improper}. \ An improper fraction always expresses a whole unit\vspace{0.04in} or more of a quantity. \ \ Examples of improper fractions are $\dfrac{5}{5}$, $\dfrac{12}{10}$, $\dfrac{40}{2}$, or $120\%$.\vspace{0.04in}\vspace{0.04in} %TCIMACRO{\TeXButton{\defend}{\defend}}% %BeginExpansion \defend% %EndExpansion \vspace{0.09in} %TCIMACRO{\TeXButton{\vissza}{\vissza}}% %BeginExpansion \vissza% %EndExpansion In spite of the suggested value judgement, we don't see anything improper about an improper fraction. \ Improper fractions can be presented as final values, as long as they are in lowest terms. \ \vspace{0.04in}\vspace{0.04in} %TCIMACRO{\TeXButton{\exstart}{\exstart}}% %BeginExpansion \exstart% %EndExpansion \textbf{Example 5.} \ Reduce $\dfrac{30}{12}$ to lowest terms. %TCIMACRO{\TeXButton{\sol}{\sol}}% %BeginExpansion \sol% %EndExpansion \textbf{Solution:} \ Both $30$ and $12$ are divisible by $6.$ \ So we divide both numerator and denominator by $6$ and we obtain that $\dfrac{30}{12}=% \dfrac{5}{2}$.\vspace{0.05in} Since $5$ and $2$ are obviously relatively% \vspace{0.05in} primes (i.e. share no divisor greater than $1$), this is the final answer. \ \ So the reduced form of $\dfrac{30}{12}$ is \fbox{$\dfrac{5% }{2}$}.\vspace{0.1in}\vspace{0.1in} %TCIMACRO{\TeXButton{\exstart}{\exstart}}% %BeginExpansion \exstart% %EndExpansion \textbf{Example 6.} \ Re-write $160\%$ as a fraction in lowest terms. \vspace{0.05in}\vspace{0.05in} %TCIMACRO{\TeXButton{\sol}{\sol}}% %BeginExpansion \sol% %EndExpansion \textbf{Solution:} \ $160\%=\dfrac{160}{100}$. \vspace{0.05in}\ We can easily divide both numerator and denominator by $10,$ that is just chopping off the last zero. \ $160\%=\dfrac{160}{100}=\dfrac{16}{10}$.\vspace{0.05in} \ This is still not reduced because both $16$ and $10$ are even. \ So we divide both numerator and denominator by $2$ and get $160\%=\dfrac{160}{100}=% \dfrac{16}{10}=\dfrac{8}{5}$.\vspace{0.05in} Since $5$ and $8$ are obviously relatively primes (i.e. share no divisor greater than $1$), this is the final answer. \ The reduced fraction form of $160\%$ is \fbox{$\dfrac{8}{5}$}% .\vspace{0.1in} \pagebreak Consider now the expression \ $\dfrac{15}{5}$.\vspace{0.04in} \ Are we looking at two integers with an operation, namely division between them, or are we looking at a single fraction, expressing $15$ slices, where the slices are so big that five of them makes up a whole? \ There is a tremendeous duality here: we can read the same expression in two fundamentally different way. \ This is only allowed in mathematics if the duality never causes conflict or confusion. \ That is, every question has the same answer, no matter which of the two interpretations are we using. \ Instead of confusion, this duality is really a source of power. \ We can always look at a fraction as a division between integers, and we can always look at a division between integers as if it was a fraction.\vspace{0.1in} Before the time of calculators, people used a lot of mental techniques to help in computation. \ The following is just an example of such a technique using the duality described above.\vspace{0.1in} %TCIMACRO{\TeXButton{\exstart}{\exstart}}% %BeginExpansion \exstart% %EndExpansion \textbf{Example 7.} \ Perform the division $\dfrac{140}{5}$ without the help of a calculator. %TCIMACRO{\TeXButton{\sol}{\sol}}% %BeginExpansion \sol% %EndExpansion \textbf{Solution: }\ Even though this is a division between two integers, we will look at it for a while as if it was a fraction. \ If so, we are allowed to multiply numerator and denominator by the same number, and our chosen number is $2$. \ This is because division by $10$ is very easy: just chop off a zero at the end. \ \begin{equation*} \dfrac{140}{5}=\dfrac{140\cdot 2}{5\cdot 2}=\dfrac{280}{10}=28 \end{equation*} \vspace{0.2in} %TCIMACRO{% %\TeXButton{vissza, matek}{\vissza %\matek}}% %BeginExpansion \vissza \matek% %EndExpansion %TCIMACRO{% %\FRAME{itbpF}{0.7169in}{0.7437in}{0.1807in}{}{}{work.jpg}{% %\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 0.7169in;height 0.7437in;depth 0.1807in;original-width 4.6977in;original-height 4.875in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'work.jpg';file-properties "XNPEU";}}}% %BeginExpansion \raisebox{-0.1807000000in}{ \includegraphics[ natheight=4.875in, natwidth=4.6977in, height=0.7437in, width=0.7169in] {//Mac/Dropbox/lnotes/1Prealgebra/fractions/fractions2/graphics/work__3.jpg}% }% %EndExpansion \ \ {\large Practice Problems} \begin{enumerate} \item a) \ Re-write $\dfrac{2}{9}$ with a denominator of $45$, i.e. find $x$ so that $\dfrac{2}{9}=\dfrac{x}{45}$. \ b) \ Verify your result by taking $\dfrac{2}{9}$ of $90$ and $\dfrac{x}{45}$ of $90$. \ If you get the same result, your answer is probably correct. \item a) \ Re-write $\dfrac{5}{6}$ with a numerator of $30$. \ b) \ Re-write $\dfrac{5}{6}$ with a denominator of $30$. \ \item Re-write each of the given percents as a reduced fraction. a) \ $60\%$ \ \ \ \ \ b) \ $25\%$ \ \ \ \ \ \ c) \ $100\%$ \ \ \ \ \ \ d) \ $% 50\%$ \ \ \ \ \ e) \ $150\%$ \ \ \ f) \ $220\%$ \item Re-write each of the given fractions as a percent. a) \ $\dfrac{3}{10}$ \ \ \ \ \ \ b) \ $\dfrac{17}{20}$ \ \ \ \ \ \ \ c) \ $% \dfrac{60}{25}$ \ \ \ \ \ \ \ d) \ $\dfrac{3}{50}$ \ \ \ \ \ \ \ e) \ $5$ \item Use the duality between division of integers and standalone fractions to perform the given divisions without the aid of a calculator. a) \ $\dfrac{120}{5}$ \ \ \ \ \ b) \ $\dfrac{2050}{50}$ \ \ \ \ c) \ $\dfrac{% 1200}{25}$ \item a) \ Bring the fractions $\dfrac{7}{10}$ \ and \ $\dfrac{2}{3}$\vspace{% 0.05in} to a common denominator. \ Compare the numerators. \ \newline Which fraction is greater, $\dfrac{7}{10}$ or $\dfrac{2}{3}$?\vspace{0.05in} b) \ Compute both $\dfrac{7}{10}$ \ and $\dfrac{2}{3}$ \ of $60$. \ Is your answer in conflict with part a)? \pagebreak \end{enumerate} %TCIMACRO{% %\FRAME{itbpF}{0.7766in}{0.7766in}{0.2006in}{}{}{answers.jpg}{% %\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 0.7766in;height 0.7766in;depth 0.2006in;original-width 8.6455in;original-height 8.6455in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'answers.jpg';file-properties "XNPEU";}}}% %BeginExpansion \raisebox{-0.2006000000in}{ \includegraphics[ natheight=8.6455in, natwidth=8.6455in, height=0.7766in, width=0.7766in] {//Mac/Dropbox/lnotes/1Prealgebra/fractions/fractions2/graphics/answers__4.jpg}% }% %EndExpansion \ \ \ {\large Answers} \begin{enumerate} \item a) \ $\dfrac{2}{9}=\dfrac{10}{45}$ \ \ \ \ b) \ $\dfrac{2}{9}$ of $90$ is $20$, and so is $\dfrac{10}{45}$ of $90$. \item a) \ $\dfrac{30}{36}$ \ \ \ \ b) \ $\dfrac{25}{30}$ \item a) \ $\dfrac{3}{5}$ \ \ \ \ b) \ $\dfrac{1}{4}$ \ \ \ c) \ $1$ \ \ \ \ d) \ $\dfrac{1}{2}$ \ \ \ \ e) \ $\dfrac{3}{2}$ \ \ \ f) \ $\dfrac{11}{5}$ \item a) \ $30\%$ \ \ \ \ b) \ $85\%$ \ \ \ \ \ \ c) \ $240\%$ \ \ \ \ \ d) \ $6\%$ \ \ \ \ e) \ $500\%$ \item a) \ $24$ \ \ \ \ \ b) \ $41$ \ \ \ \ c) \ $48$ \item a) \ $\dfrac{7}{10}=\dfrac{21}{30}$ \ \ and \ $\dfrac{2}{3}=\dfrac{20}{% 30}.$ \ Clearly, \ $\dfrac{21}{30}>\dfrac{20}{30}$ and \ so $\dfrac{7}{10}>% \dfrac{2}{3}$. b) \ $\dfrac{7}{10}$ of $60$ is $42$, and $\dfrac{2}{3}$ of $60$ is $40.$ \ Since \ $42>40$, we get again that $\dfrac{7}{10}$ is greater.\vspace{2.7in}% \vspace{3in} \end{enumerate} {\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}