5.3: Add, Subtract, Multiply and Divide Fractions
- Page ID
- 51875
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Partner Activity 1
Which expression would you rather add?
\(\dfrac{51}{684}+\dfrac{43}{684}+\dfrac{738}{684}\) OR \(\dfrac{1}{8}+\dfrac{4}{5}+\dfrac{1}{9}\)
Explain to a 3rd grader why:
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Now, we will explore why fractions behave the way they do for adding, subtracting, multiplying and dividing:
Example \(\PageIndex{1}\): Add Fractions with a Drawing, Number Line, then with Common Denominators
Add \(\dfrac{1}{2}+\dfrac{1}{3}\). Why 6 boxes?
Example \(\PageIndex{2}\): Subtract Fractions with a Drawing, Number Line, then with Common Denominators
Subtract \(\dfrac{1}{2} - \dfrac{1}{3}\). Why 6 boxes?
Example \(\PageIndex{3}\): Multiply Fractions with a Drawing, Number Line, then with Common Denominators
Add \(\dfrac{1}{2} \times \dfrac{1}{3}\). Why 6 boxes?
Example \(\PageIndex{4}\): Divide Fractions with a Drawing, Number Line, then with Common Denominators
Divide \(\dfrac{1}{2} \div \dfrac{1}{3}\). “Think portions, when it comes to division! Why 6 boxes?
Example \(\PageIndex{5}\)
Why does “multiply and flip the second fraction” work when dividing fractions?
Solution
We know that:
\[\dfrac{a}{b} \times \dfrac{b}{a}=\dfrac{a b}{a b}=1 \nonumber \]
\[\begin{aligned}
\dfrac{3}{4} \div \dfrac{2}{7}&= \left(\dfrac{3}{4} \times \dfrac{7}{2}\right) \div\left(\dfrac{2}{7} \times \dfrac{7}{2}\right) \\
&=\left(\dfrac{3}{4} \times \dfrac{7}{2}\right) \div 1 \\
&=\dfrac{3}{4} \times \dfrac{7}{2}\\
&=\dfrac{21}{8}
\end{aligned} \nonumber \]
Partner Activity 2
Which Operation is Correct?
A stretch of highway is \(3 \dfrac{1}{2}\) miles long. Each day, \(\dfrac{2}{3}\) of a mile is repaved. How many days are needed to repave the entire section? How would you explain to a 5th grader which operation is correct?
Do we add?
\(3 \dfrac{1}{2}+\dfrac{2}{3}=\dfrac{7}{2}+\dfrac{2}{3}=\dfrac{21}{6}+\dfrac{4}{6}=\dfrac{25}{6}=4 \dfrac{1}{6}\) days
Do we subtract?
\(3 \dfrac{1}{2}-\dfrac{2}{3}=\dfrac{7}{2}-\dfrac{2}{3}=\dfrac{21}{6}-\dfrac{4}{6}=\dfrac{17}{6}=2 \dfrac{5}{6}\) days
Do we multiply?
\(3 \dfrac{1}{2} \times \dfrac{2}{3}=\dfrac{7}{2} \times \dfrac{2}{3}=\dfrac{14}{6}=2 \dfrac{2}{6}=2 \dfrac{1}{3}\) days
Do we divide?
\(3 \dfrac{1}{2} \div \dfrac{2}{3}=\dfrac{7}{2} \times \dfrac{3}{2}=\dfrac{21}{4}=5 \dfrac{1}{4}\) days
Practice Problems
Add, subtract, multiply or divide the expressions. Use any method.
- \(\dfrac{3}{4}+\dfrac{8}{7}\)
- \(\dfrac{8}{7}-\dfrac{3}{4}\)
- \(\dfrac{3}{4} \times \dfrac{8}{7}\)
- \(\dfrac{3}{4} \div \dfrac{8}{7}\)
- \(5 \dfrac{2}{5} \div 3 \dfrac{1}{6}\)
- \(5 \dfrac{2}{5}-3 \dfrac{1}{6}\)
- \(5 \dfrac{2}{5} \times 3 \dfrac{1}{6}\)
- \(5 \dfrac{2}{5} \div 3 \dfrac{1}{6}\)