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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 3: Improper Fractions and Mixed Numbers} \rhead{\small page \ \thepage} \cfoot{} \lfoot{\footnotesize \copyright $\;$ Hidegkuti, 2018} \rfoot{\footnotesize Last revised: September 3, 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 %TCIMACRO{\TeXButton{\setstretch{1.3}}{\setstretch{1.3}}}% %BeginExpansion \setstretch{1.3}% %EndExpansion Recall the definition of an improper fraction. %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 it\vspace{0.04in} a \textbf{proper fraction}. \ 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 it an \textbf{improper fraction}. \ 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 %TCIMACRO{\TeXButton{\setstretch{1.3}}{\setstretch{1.3}}}% %BeginExpansion \setstretch{1.3}% %EndExpansion Also recall that if we multiply both numerator and denominator of a fraction by the same non-zero number, the resulting fraction is equivalent to the original fraction. \ Equivalent fractions express the same amount. \ We can also write any integer as a fraction. \ For example, $1$ can be written as $% \dfrac{1}{1}$, $\dfrac{3}{3}$, $\dfrac{5}{5}$, $\dfrac{100}{100}$ or $100\%$% . \ The integer $5$ can be written as $\dfrac{5}{1}$, $\dfrac{10}{2}$, $% \dfrac{15}{3}$, or $500\%$. \ We can also divide both numerator and denominator by the same number. \ When a fraction's numerator and denominator share no divisor greater than $1$, the fraction is \textbf{in lowest terms}. %TCIMACRO{\TeXButton{\vissza}{\vissza}}% %BeginExpansion \vissza% %EndExpansion %TCIMACRO{\TeXButton{\setstretch{1.3}}{\setstretch{1.3}}}% %BeginExpansion \setstretch{1.3}% %EndExpansion With the introduction of improper fractions, our notation includes a new (and huge) duality. \ We can look at the expression $\dfrac{20}{4}$ as a division between two integers, i.e. two objects and an operation. \ We can also interpret the expression $\dfrac{20}{4}$ as an improper fraction, i.e. a single object. \ This ambiguity is only allowed because every question we can ever ask has the same answer, no matter which interpretations we used. \ As final result, fractions must be presented in their simplest form. \ For example, the fraction $\dfrac{15}{18}$ must be reduced to lowest terms and presented as $\dfrac{5}{6}$. \ The fraction $\dfrac{20}{4}$ must be presented as the integer $5$ because it is a much simpler presentation than $% \dfrac{5}{1}$. \ But how do we simplify (if we even can) the improper fraction $\dfrac{7}{3}$? The name improper fraction already suggests that there is something wrong with such a fraction. \ We strongly disagree with this notion. \ However, this might have not been the general opinion when the concept of mixed numbers was developed. A dime is a common nickname for the silver colored, small ten-cent coin. \ $% 10$ dimes are worth a dollar, so one dime can be represented as $\dfrac{1}{10% }$ of a dollar. \ \ Suppose we have $42$ dimes. \ How can we express this fact? \ As an improper fraction, we can of course write $\dfrac{42}{10}$. \ Suppose we go to the bank and change all dimes we can for dollar bills. \ In this case, we could exchange $40$ dimes for four dollar bills. \ Using this idea, we can write $\dfrac{42}{10}$ as a mixed number as $4\dfrac{2}{10}$. \ The integer part expresses the bills, the fraction part expresses the coins. \ Of course we can also bring the fraction part to lowest terms and get $4% \dfrac{1}{5}$. \ For some, this is the only way to present the fraction $% \dfrac{42}{10}$ in its simplest form. %TCIMACRO{\TeXButton{\defstart}{\defstart}}% %BeginExpansion \defstart% %EndExpansion %TCIMACRO{\TeXButton{\defcont}{\defcont}}% %BeginExpansion \defcont% %EndExpansion \textbf{Definition: }A \textbf{mixed number} is an alternative representation for improper fractions that can not be simplified as integers. \ A mixed number has an integer part and a fraction part. For example, $2\dfrac{1}{3}$ is a mixed number where the integer part is $2$ and the fraction part is $\dfrac{1}{3}$. %TCIMACRO{\TeXButton{\defend}{\defend}}% %BeginExpansion \defend% %EndExpansion \vspace{0.09in} %TCIMACRO{\TeXButton{\vissza}{\vissza}}% %BeginExpansion \vissza% %EndExpansion %TCIMACRO{\TeXButton{\setstretch{1.3}}{\setstretch{1.3}}}% %BeginExpansion \setstretch{1.3}% %EndExpansion Not every country uses mixed number notation. \ In countries that don't use mixed numbers, $2\dfrac{1}{3}$ would appear as $2+\dfrac{1}{3}$. \ In the USA, mixed numbers are fairly common, but their usefullness is debated. \ Let us also note that this notation is an example where two objects are written next to each other with no operation between them, and it does \textit{not} represent multiplication. %TCIMACRO{\TeXButton{\exstart}{\exstart}}% %BeginExpansion \exstart% %EndExpansion \textbf{Example 1. }\ Re-write the improper fraction $\dfrac{87}{10}$ as a mixed number. \ \vspace{0.06in} %TCIMACRO{\TeXButton{\sol}{\sol}}% %BeginExpansion \sol% %EndExpansion \textbf{Solution:} \ We can think of this as a person with $87$ dimes in their pocket. \ How much money can be exchanged to dollar bills? \ Since $10$ dimes are worth a dollar, we can exchange $80$ dimes for eight dollars in paper money, and are left with seven dimes. \ This can be expressed as \fbox{% $8\dfrac{7}{10}$}. \ %TCIMACRO{\TeXButton{\exstart}{\exstart}}% %BeginExpansion \exstart% %EndExpansion \textbf{Example 2. }\ Re-write the improper fraction $\dfrac{513}{100}$ as a mixed number. \ \vspace{0.06in} %TCIMACRO{\TeXButton{\sol}{\sol}}% %BeginExpansion \sol% %EndExpansion \textbf{Solution:} \ We can think of this as a person with $403$ pennies in their pocket. \ How much money can be exchanged to dollar bills? \ Since $% 100 $ pennies are worth a dollar, we can exchange $500$ pennies for five dollars in paper money, and are left with thirteen pennies. \ This can be expressed as \fbox{$5\dfrac{13}{100}$}. %TCIMACRO{\TeXButton{\vissza}{\vissza}}% %BeginExpansion \vissza% %EndExpansion %TCIMACRO{\TeXButton{\setstretch{1.3}}{\setstretch{1.3}}}% %BeginExpansion \setstretch{1.3}% %EndExpansion Notice that in converting an improper fraction to a mixed number, we apply division with remainder. The division\ \newline $87\div 10=8$ R $7$ is behind the conversion $\dfrac{87}{10}=8\dfrac{7}{10}$ and the divsion $513\div 100=5$ R $13$ is behind the conversion $\dfrac{513}{% 100}=5\dfrac{13}{100}$. \ The idea behind mixed numbers is simplifying, which means that the integer part needs to be reduced to lowest terms. %TCIMACRO{\TeXButton{\exstart}{\exstart}}% %BeginExpansion \exstart% %EndExpansion \textbf{Example 3. }\ Re-write the improper fraction $\dfrac{28}{10}$ as a mixed number. \ \vspace{0.06in} %TCIMACRO{\TeXButton{\sol}{\sol}}% %BeginExpansion \sol% %EndExpansion \textbf{Solution:} \ We can think of this as a person with $28$ dimes in their pocket. \ We perform the division with remainder: \ \newline $28\div 10=2$ R $8$. \ This is the same as saying that $\dfrac{28}{10}=2% \dfrac{8}{10}$, just as in the previous examples. \ However, this mixed number needs to be simplified where we bring the fraction part to lowest terms. \ Clearly $\dfrac{8}{10}=\dfrac{4}{5}$, so $\dfrac{28}{10}=$ \fbox{$2% \dfrac{4}{5}$}. %TCIMACRO{\TeXButton{\vissza}{\vissza}}% %BeginExpansion \vissza% %EndExpansion %TCIMACRO{\TeXButton{\setstretch{1.3}}{\setstretch{1.3}}}% %BeginExpansion \setstretch{1.3}% %EndExpansion Let us see an example beyond coins. \ Suppose that we work in a pizza place where each pizza is cut into $6$ slices. \ So, one slice of pizza can be represented as $\dfrac{1}{6}$. \ A whole pizza can be represented as $1$ (imagine we are not cutting it into slices) or $\dfrac{6}{6}$ (if we do cut it into slices). \ How can we express $\dfrac{25}{6}$ as a mixed number? \ We perform the division with remainder: \ $25\div 6=4$ R $1$ and we convert $% \dfrac{25}{6}$ to the mixed number $4\dfrac{1}{6}$. \ We can interpret this as $4$ whole pizzas, and one additional slice. \ This is correct, $4$ whole pizzas can be represented as $4=\dfrac{4}{1}=\dfrac{24}{6}$ that is, $24$ slices. %TCIMACRO{\TeXButton{\exstart}{\exstart}}% %BeginExpansion \exstart% %EndExpansion \textbf{Example 4. }\ Re-write the improper fraction $\dfrac{17}{6}$ as a mixed number. %TCIMACRO{\TeXButton{\sol}{\sol}}% %BeginExpansion \sol% %EndExpansion \textbf{Solution:} \ We perform the division with remainder: \ $17\div 6=2$ R $5$. \ Thus $\dfrac{17}{6}=\fbox{$2\dfrac{5}{6}$}$. %TCIMACRO{\TeXButton{\exstart}{\exstart}}% %BeginExpansion \exstart% %EndExpansion \textbf{Example 5. }\ Draw a picture representing the improper fraction\ $% \dfrac{7}{3}$. \ Use your picture to convert $\dfrac{7}{3}$ to a mixed number.\vspace{0.06in} %TCIMACRO{\TeXButton{\sol}{\sol}}% %BeginExpansion \sol% %EndExpansion \textbf{Solution:} \ To represent a fraction with denominator three means that we will draw circles and slice them into three equal part. \ Because this is an improper fraction, we will need more than one circle. \ We keep starting new units until we have seven slices. \ Our picture shows that the mixed number corresponding to $\dfrac{7}{3}$ is $\fbox{$2\dfrac{1}{3}$}$. %TCIMACRO{\TeXButton{\solinside}{\solinside} }% %BeginExpansion \solinside %EndExpansion \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ %TCIMACRO{% %\FRAME{itbpF}{2.2226in}{0.6815in}{0.1609in}{}{}{fractions3pic1.bmp}{% %\special{language "Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 2.2226in;height 0.6815in;depth 0.1609in;original-width 0.1531in;original-height 0.0458in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'fractions3pic1.bmp';file-properties "XNPEU";}}}% %BeginExpansion \raisebox{-0.1609000000in}{ \includegraphics[ natheight=0.0458in, natwidth=0.1531in, height=0.6815in, width=2.2226in] {//Mac/Dropbox/lnotes/1Prealgebra/fractions/fractions3/graphics/fractions3pic1__1.pdf}% }% %EndExpansion %TCIMACRO{\TeXButton{\exstart}{\exstart}}% %BeginExpansion \exstart% %EndExpansion \textbf{Example 6. }\ Re-write the improper fraction\ $\dfrac{17}{4}$ as a mixed number. \ Use your result to plot $\dfrac{17}{4}$ on the number line.% \vspace{0.06in} %TCIMACRO{\TeXButton{\sol}{\sol}}% %BeginExpansion \sol% %EndExpansion \textbf{Solution:} \ We perform the divsion with remainder: $17\div 4=4$ R $% 1 $. \ Therefore, $\dfrac{17}{4}=\fbox{$4\dfrac{1}{4}$}$. \ When we plot $4% \dfrac{1}{4}$, we first find $4$ and $5$. \ We split the line segment between $4$ and $5$ into four equal part, and count one from $4$. %TCIMACRO{\TeXButton{\solinside}{\solinside} }% %BeginExpansion \solinside %EndExpansion \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ %TCIMACRO{% %\FRAME{dtbpF}{3.339in}{0.5249in}{0pt}{}{}{fractions3pic2.bmp}{% %\special{language "Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 3.339in;height 0.5249in;depth 0pt;original-width 4.9796in;original-height 0.7913in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'fractions3pic2.bmp';file-properties "XNPEU";}}}% %BeginExpansion \begin{center} \includegraphics[ natheight=0.7913in, natwidth=4.9796in, height=0.5249in, width=3.339in] {//Mac/Dropbox/lnotes/1Prealgebra/fractions/fractions3/graphics/fractions3pic2__2.pdf}% \end{center} %EndExpansion %TCIMACRO{\TeXButton{\exstart}{\exstart}}% %BeginExpansion \exstart% %EndExpansion \textbf{Example 7. }\ Re-write the mixed number $3\dfrac{2}{5}$ as an improper fraction. \vspace{0.06in} %TCIMACRO{\TeXButton{\sol}{\sol}}% %BeginExpansion \sol% %EndExpansion \textbf{Solution:} \ We can imagine a pizza place where each pizza is cut into five slices. \ Then the question is: if someone orders three full pizzas and two more slices, how many slices were ordered? \ Three full pizzas will account for $15$ slices, so all together we have $17$ slices. \ Algebraically, $3=\dfrac{3}{1}=\dfrac{15}{3}$ and so $3\dfrac{2}{5}=\fbox{$% \dfrac{17}{3}$}$. %TCIMACRO{\TeXButton{\exstart}{\exstart}}% %BeginExpansion \exstart% %EndExpansion \textbf{Example 8. }\ Re-write the mixed number $2\dfrac{5}{8}$ as an improper fraction. \vspace{0.06in} %TCIMACRO{\TeXButton{\sol}{\sol}}% %BeginExpansion \sol% %EndExpansion \textbf{Solution:} \ We can imagine a pizza place where each pizza is cut into eight slices. \ Then the question is: if someone orders two full pizzas and five more slices, how many slices were ordered? \ Two full pizzas will account for $16$ slices, so all together we have $16+5=21$ slices. \ Algebraically, $2=\dfrac{2}{1}=\dfrac{16}{8}$ and so $2\dfrac{5}{8}=\fbox{$% \dfrac{21}{8}$}$. %TCIMACRO{\TeXButton{\vissza}{\vissza}}% %BeginExpansion \vissza% %EndExpansion %TCIMACRO{\TeXButton{\setstretch{1.3}}{\setstretch{1.3}}}% %BeginExpansion \setstretch{1.3}% %EndExpansion In this course, we will always have to present fractions in their simplest possible form as final answers. \ \textbf{However, we will not view mixed numbers as more simplified than improper fractions.} \ This means that improper fractions can always be presented in their improper form as a final answer, as long as it is in lowest terms. \ For example, $\dfrac{42}{10}$ is not acceptable as final answer, but $\dfrac{21}{5}$ is. \ The mixed number $4% \dfrac{2}{10}$ is not simplified, $4\dfrac{1}{5}$ is. \ However, $4\dfrac{1}{% 5}$ is not considered more simplified than $\dfrac{21}{5}$. \ They are equally acceptable, and, from an algebraic point of view, $\dfrac{21}{5}$ is actually preferred. \ We will see that with just a very few exceptions, improper fractions have nicer properties than mixed numbers. \ \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/fractions3/graphics/work__3.jpg}% }% %EndExpansion \ \ {\large Practice Problems} \begin{enumerate} \item Re-write each of the following improper fractions as mixed numbers. \ Bring the fraction part to lowest terms. a) \ $\dfrac{10}{3}$ \ \ \ \ b) \ $150\%$ \ \ \ \ c) \ $\dfrac{35}{10}$ \ \ \ \ \ d) \ $\dfrac{120}{7}$ \ \ \ \ \ e) \ $\dfrac{42}{8}$ \ \ \ \ \ \ f) \ $% \dfrac{715}{100}$ \ \ \ g) $\ 320\%$ \item Re-write each of the mixed numbers as improper fractions in lowest terms. a) \ $3\dfrac{4}{5}$ \ \ \ \ \ \ b) \ $1\dfrac{3}{8}$ \ \ \ \ c) \ $5\dfrac{6% }{7}$ \ \ \ \ \ d) \ $3\dfrac{4}{10}$ \ \ \ \ \ e) \ $10\dfrac{5}{7}$ \ \ \ \ \ f) \ $5\dfrac{1}{4}$ \item Draw a picture to represent each of the given fractions. a) \ $\dfrac{13}{4}$ \ \ \ \ \ b) \ $2\dfrac{3}{5}$ \ \ \ \ c) \ $\dfrac{7}{2% }$ \ \ \ \ \ d) \ $\dfrac{10}{7}$ \item Plot each of the given fractions on the number line. a) \ $\dfrac{8}{3}$ \ \ \ \ \ b) \ $3\dfrac{2}{5}$ \ \ \ \ c) \ $\dfrac{9}{2} $ \ \ \ \ \ d) \ $\dfrac{7}{4}$ \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/fractions3/graphics/answers__4.jpg}% }% %EndExpansion \ \ \ {\large Answers} \begin{enumerate} \item a) \ $3\dfrac{1}{3}$ \ \ \ \ b) \ $1\dfrac{1}{2}$ \ \ \ \ \ \ \ c) \ $3% \dfrac{1}{2}$ \ \ \ \ \ \ \ d) \ $17\dfrac{1}{7}$ \ \ \ \ \ \ e) \ $5\dfrac{1% }{4}$ \ \ \ \ f) \ $7\dfrac{3}{20}$ \ \ \ \ g) \ $3\dfrac{1}{5}$ \item a) \ $\dfrac{19}{5}$ \ \ \ \ \ \ b) \ $\dfrac{11}{8}$ \ \ \ \ \ c) \ $% \dfrac{41}{7}$ \ \ \ \ \ d) \ $\dfrac{17}{5}$ \ \ \ \ \ \ e) \ $\dfrac{75}{7} $ \ \ \ \ \ \ f) \ $\dfrac{21}{4}$ \item a) \ $\dfrac{13}{4}=3\dfrac{1}{4}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ b) \ $2\dfrac{3}{5}=\dfrac{13}{5}$ %TCIMACRO{% %\FRAME{itbpF}{2.7in}{0.6988in}{-0.3208in}{}{}{fractions3pic3.bmp}{% %\special{language "Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 2.7in;height 0.6988in;depth -0.3208in;original-width 0.1228in;original-height 0.0311in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'fractions3pic3.bmp';file-properties "XNPEU";}}}% %BeginExpansion \raisebox{0.3208000000in}{ \includegraphics[ natheight=0.0311in, natwidth=0.1228in, height=0.6988in, width=2.7in] {//Mac/Dropbox/lnotes/1Prealgebra/fractions/fractions3/graphics/fractions3pic3__5.pdf}% }% %EndExpansion \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ %TCIMACRO{% %\FRAME{itbpF}{1.497in}{1.19in}{0in}{}{}{fractions3pic4.bmp}{% %\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.497in;height 1.19in;depth 0in;original-width 0.0726in;original-height 0.0519in;cropleft "0";croptop "1.1069";cropright "1";cropbottom "0";filename 'fractions3pic4.bmp';file-properties "XNPEU";}}}% %BeginExpansion \includegraphics[ natheight=0.0519in, natwidth=0.0726in, height=1.19in, width=1.497in] {//Mac/Dropbox/lnotes/1Prealgebra/fractions/fractions3/graphics/fractions3pic4__6.pdf}% %EndExpansion c) \ $\dfrac{7}{2}=3\dfrac{1}{2}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ d) \ $\dfrac{10}{7}=1\dfrac{3}{7}$ %TCIMACRO{% %\FRAME{itbpF}{2.7008in}{0.7126in}{-0.2413in}{}{}{fractions3pic5.bmp}{% %\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.7008in;height 0.7126in;depth -0.2413in;original-width 0.1228in;original-height 0.0311in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'fractions3pic5.bmp';file-properties "XNPEU";}}}% %BeginExpansion \raisebox{0.2413000000in}{ \includegraphics[ natheight=0.0311in, natwidth=0.1228in, height=0.7126in, width=2.7008in] {//Mac/Dropbox/lnotes/1Prealgebra/fractions/fractions3/graphics/fractions3pic5__7.pdf}% }% %EndExpansion \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ %TCIMACRO{% %\FRAME{itbpF}{2.0609in}{1.0741in}{-0.2006in}{}{}{fractions3pic6.bmp}{% %\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.0609in;height 1.0741in;depth -0.2006in;original-width 0.0951in;original-height 0.0363in;cropleft "0";croptop "1.3559";cropright "1";cropbottom "0";filename 'fractions3pic6.bmp';file-properties "XNPEU";}}}% %BeginExpansion \raisebox{0.2006000000in}{ \includegraphics[ natheight=0.0363in, natwidth=0.0951in, height=1.0741in, width=2.0609in] {//Mac/Dropbox/lnotes/1Prealgebra/fractions/fractions3/graphics/fractions3pic6__8.pdf}% }% %EndExpansion \item a) \ $\dfrac{8}{3}=2\dfrac{2}{3}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ b) \ $3\dfrac{2}{5}$ %TCIMACRO{% %\FRAME{itbpF}{3.0096in}{0.32in}{0in}{}{}{fractions3pic7.bmp}{% %\special{language "Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 3.0096in;height 0.32in;depth 0in;original-width 7.4166in;original-height 0.5725in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'fractions3pic7.bmp';file-properties "XNPEU";}}}% %BeginExpansion \includegraphics[ natheight=0.5725in, natwidth=7.4166in, height=0.32in, width=3.0096in] {//Mac/Dropbox/lnotes/1Prealgebra/fractions/fractions3/graphics/fractions3pic7__9.pdf}% %EndExpansion \ \ \ \ \ \ \ \ \ \ \ \ \ \ %TCIMACRO{% %\FRAME{itbpF}{3.2292in}{0.4402in}{0in}{}{}{fractions3pic8.bmp}{% %\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 3.2292in;height 0.4402in;depth 0in;original-width 6.3538in;original-height 0.8441in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'fractions3pic8.bmp';file-properties "XNPEU";}}}% %BeginExpansion \includegraphics[ natheight=0.8441in, natwidth=6.3538in, height=0.4402in, width=3.2292in] {//Mac/Dropbox/lnotes/1Prealgebra/fractions/fractions3/graphics/fractions3pic8__10.pdf}% %EndExpansion c) \ $\dfrac{9}{2}=4\dfrac{1}{2}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ d) \ $\dfrac{7}{4}=1\dfrac{3% }{4}$ %TCIMACRO{% %\FRAME{itbpF}{3.0338in}{0.4315in}{0in}{}{}{fractions3pic9.bmp}{% %\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 3.0338in;height 0.4315in;depth 0in;original-width 6.3123in;original-height 0.8752in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'fractions3pic9.bmp';file-properties "XNPEU";}}}% %BeginExpansion \includegraphics[ natheight=0.8752in, natwidth=6.3123in, height=0.4315in, width=3.0338in] {//Mac/Dropbox/lnotes/1Prealgebra/fractions/fractions3/graphics/fractions3pic9__11.pdf}% %EndExpansion \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ %TCIMACRO{% %\FRAME{itbpF}{3.48in}{0.3788in}{0in}{}{}{fractions3pic10.bmp}{% %\special{language "Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 3.48in;height 0.3788in;depth 0in;original-width 7.2809in;original-height 0.6564in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'fractions3pic10.bmp';file-properties "XNPEU";}}}% %BeginExpansion \includegraphics[ natheight=0.6564in, natwidth=7.2809in, height=0.3788in, width=3.48in] {//Mac/Dropbox/lnotes/1Prealgebra/fractions/fractions3/graphics/fractions3pic10__12.pdf}% %EndExpansion \vspace{1.5in}\vspace{0.7in} \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}