4.3: Add, Subtract, Multiply and Divide Fractions
- Page ID
- 51005
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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:
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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.
- \(\dfrac{3}{4}+\dfrac{8}{7}\)
- \(\dfrac{8}{7}-\dfrac{3}{4}\)
- \(\dfrac{3}{4} \times \dfrac{8}{7}\)
- \(\dfrac{3}{4} \div \dfrac{8}{7}\)
- \(5 \dfrac{2}{5} \div 3 \dfrac{1}{6}\)
- \(5 \dfrac{2}{5}-3 \dfrac{1}{6}\)
- \(5 \dfrac{2}{5} \times 3 \dfrac{1}{6}\)
- \(5 \dfrac{2}{5} \div 3 \dfrac{1}{6}\)