5.9: Adding and Subtracting Fractions
- Page ID
- 48767
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Paul and Tony order a pizza which has been cut into eight equal slices. Thus, each slice is \(\frac{1}{8}\) of the whole pizza. Paul eats two slices (shaded in light gray in Figure \(\PageIndex{1}\)), or \(\frac{2}{8}\) of the whole pizza. Tony eats three slices (shaded in light red) in Figure \(\PageIndex{1}\)), or \(\frac{3}{8}\) of the whole pizza.
It should be clear that together Paul and Tony eat five slices, or \(\frac{5}{8}\) of the whole pizza. This reflects the fact that
\[ \frac{2}{8} + \frac{3}{8} = \frac{5}{8}.\nonumber \]
This demonstrates how to add two fractions with a common (same) denominator. Keep the common denominator and add the numerators. That is,
\[ \begin{align*} \frac{2}{8} + \frac{3}{8} &= \frac{2 + 3}{8} ~ && \textcolor{red}{ \text{ Keep denominator; add numerators.}} \\ &= \frac{5}{8} ~ && \textcolor{red}{ \text{ Simplify numerator.}} \end{align*} \]
Adding Fractions with Common Denominators
Let \(\dfrac{a}{c}\) and \(\dfrac{b}{c}\) be two fractions with a common (same) denominator. Their sum is defined as
\[ \frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}\nonumber \]
That is, to add two fractions having common denominators, keep the common denominator and add their numerators.
A similar rule holds for subtraction.
Subtracting Fractions with Common Denominators
Let \(\dfrac{a}{c}\) and \(\dfrac{b}{c}\) be two fractions with a common (same) denominator. Their difference is defined as
\[ \frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}.\nonumber \]
That is, to subtract two fractions having common denominators, keep the common denominator and subtract their numerators.
Example \(\PageIndex{1}\)
Find the sum of \(\dfrac{4}{9}\) and \(\frac{3}{9}\).
Solution
Keep the common denominator and add the numerators.
\[ \begin{aligned} \frac{4}{9} + \frac{3}{9} &= \frac{4+3}{9} ~ & \textcolor{red}{ \text{ Keep denominator; add numerators.}} \\ &= \frac{7}{9} ~ & \textcolor{red}{ \text{ Simplify numerator.}} \end{aligned}\nonumber \]
Try It \(\PageIndex{1}\)
Add: \( \dfrac{1}{8} + \dfrac{2}{8} \)
- Answer
-
\(\dfrac{3}{8}\)
Example \(\PageIndex{2}\)
Subtract \(\dfrac{5}{16}\) from \(\dfrac{13}{16}\).
Solution
Keep the common denominator and subtract the numerators.
\[ \begin{aligned} \frac{13}{16} - \frac{5}{16} &= \frac{13-5}{16} ~ & \textcolor{red}{ \text{ Keep denominator; subtract numerators.}} \\ &=\frac{8}{16} ~ & \textcolor{red}{ \text{ Simplify numerator.}} \end{aligned}\nonumber \]
Of course, we should always reduce our final answer to lowest terms. One way to accomplish that in this case is to divide numerator and denominator by 8, the greatest common divisor of 8 and 16.
\[ \begin{aligned} = \frac{8 \div 8}{16 \div 8} ~ & \textcolor{red}{ \text{ Divide numerator and denominator by 8.}} \\ = \frac{1}{2} ~ & \textcolor{red}{ \text{ Simplify numerator and denominator.}} \end{aligned}\nonumber \]
Try It \(\PageIndex{2}\)
Subtract: \( \dfrac{11}{12} - \dfrac{7}{12} \)
- Answer
-
\(\dfrac{1}{3}\)
Adding Fractions with Different Denominators
Consider the sum
\[ \dfrac{4}{9} + \dfrac{1}{6}.\nonumber \]
We cannot add these fractions because they do not have a common denominator. So, what to do?
Goals
In order to add two fractions with different denominators, we need to:
- Find a common denominator for the given fractions.
- Make fractions with the common denominator that are equivalent to the original fractions.
If we accomplish the two items in the “Goal,” we will be able to find the sum of the given fractions.
So, how to start? We need to find a common denominator, but not just any common denominator. Let’s agree that we want to keep the numbers as small as possible and find a least common denominator.
Definition: Least Common Denominator
The least common denominator (LCD) for a set of fractions is the smallest number divisible by each of the denominators of the given fractions.
Consider again the sum we wish to find:
\[ \dfrac{4}{9} + \dfrac{1}{6} .\nonumber \]
The denominators are 9 and 6. We wish to find a least common denominator, the smallest number that is divisible by both 9 and 6. A number of candidates come to mind: 36, 54, and 72 are all divisible by 9 and 6, to name a few. But the smallest number that is divisible by both 9 and 6 is 18. This is the least common denominator for 9 and 6.
We now proceed to the second item in “Goal.” We need to make fractions having 18 as a denominator that are equivalent to \(\frac{4}{9}\) and \(\frac{1}{6}\). In the case of \(\frac{4}{9}\), if we multiply both numerator and denominator by 2, we get
\[ \begin{aligned} \frac{4}{9} &= \frac{4 \cdot 2}{9 \cdot 2} ~ & \textcolor{red}{ \text{ Multiply numerator and denominator by 2.}} \\ &= \frac{8}{18}. ~ & \textcolor{red}{ \text{ Simplify numerator and denominator.}} \end{aligned}\nonumber \]
In the case of \(\frac{1}{6}\), if we multiply both numerator and denominator by 3, we get
\[ \begin{aligned} \frac{1}{6} &= \frac{1 \cdot 3}{6 \cdot 3} ~ & \textcolor{red}{ \text{ Multiply numerator and denominator by 3.}} \\ &= \frac{3}{18}. ~ & \textcolor{red}{ \text{ Simplify numerator and denominator.}} \end{aligned}\nonumber \]
Typically, we’ll arrange our work as follows.
\[ \begin{aligned} \frac{4}{9} + \frac{1}{6} &= \frac{4 \cdot \textcolor{red}{2}}{9 \cdot \textcolor{red}{2}} + \frac{1 \cdot \textcolor{red}{3}}{6 \cdot \textcolor{red}{3}} ~ & \textcolor{red}{ \text{ Equivalent fractions with LCD = 18.}} \\ &= \frac{8}{18} + \frac{3}{18} ~ & \textcolor{red}{ \text{ Simplify numerators and denominators.}} \\ &= \frac{8+3}{18} ~ & \textcolor{red}{ \text{ Keep common denominator; add numerators.}} \\ &= \frac{11}{18} ~ & \textcolor{red}{ \text{ Simplify numerator.}} \end{aligned}\nonumber \]
Let’s summarize the procedure.
Adding or Subtracting Fractions with Different Denominators
- Find the LCD, the smallest number divisible by all the denominators of the given fractions.
- Create fractions using the LCD as the denominator that are equivalent to the original fractions.
- Add or subtract the resulting equivalent fractions. Simplify, including reducing the final answer to lowest terms.
Example \(\PageIndex{3}\)
Simplify: \( \displaystyle \frac{3}{5} - \frac{2}{3}\).
Solution
The smallest number divisible by both 5 and 3 is 15.
\[ \begin{aligned} \frac{3}{5} - \frac{2}{3} &= \frac{3 \cdot \textcolor{red}{3}}{5 \cdot \textcolor{red}{3}} - \frac{2 \cdot \textcolor{red}{5}}{3 \cdot \textcolor{red}{5}} ~ & \textcolor{red}{ \text{ Equivalent fractions with LCD = 15.}} \\ &= \frac{9}{15} - \frac{10}{15} ~ & \textcolor{red}{ \text{ Simplify numerators and denominators.}} \\ &= \frac{9-10}{15} ~ & \textcolor{red}{ \text{ Keep LCD; subtract numerators.}} \\ &= \frac{-1}{15} ~ & \textcolor{red}{ \text{ Simplify numerator.}} \end{aligned}\nonumber \]
Although this answer is perfectly acceptable, negative divided by positive gives us a negative answer, so we could also write
\[ = - \frac{1}{15}.\nonumber \]
Try It \(\PageIndex{3}\)
Subtract: \( \dfrac{3}{4} - \dfrac{7}{5} \)
- Answer
-
-\(\dfrac{13}{20}\)