<< Chapter < Page | Chapter >> Page > |
A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Fractions .
When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.
If $a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c$ are numbers where $c\ne 0,$ then
To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
Find the sum: $\frac{x}{3}+\frac{2}{3}.$
$\begin{array}{cccccc}& & & & & \hfill \frac{x}{3}+\frac{2}{3}\hfill \\ \begin{array}{c}\text{Add the numerators and place the sum over}\hfill \\ \text{the common denominator.}\hfill \end{array}\hfill & & & & & \hfill \frac{x+2}{3}\hfill \end{array}$
Find the difference: $-\phantom{\rule{0.2em}{0ex}}\frac{23}{24}-\phantom{\rule{0.2em}{0ex}}\frac{13}{24}.$
$\begin{array}{cccccc}& & & & & \hfill -\phantom{\rule{0.2em}{0ex}}\frac{23}{24}-\phantom{\rule{0.2em}{0ex}}\frac{13}{24}\hfill \\ \begin{array}{c}\text{Subtract the numerators and place the}\hfill \\ \text{difference over the common denominator.}\hfill \end{array}\hfill & & & & & \hfill \frac{\mathrm{-23}-13}{24}\hfill \\ \text{Simplify.}\hfill & & & & & \hfill \frac{\mathrm{-36}}{24}\hfill \\ \text{Simplify. Remember,}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{a}{b}=\frac{\text{\u2212}a}{b}.\hfill & & & & & \hfill -\phantom{\rule{0.2em}{0ex}}\frac{3}{2}\hfill \end{array}$
Find the difference: $-\phantom{\rule{0.2em}{0ex}}\frac{19}{28}-\phantom{\rule{0.2em}{0ex}}\frac{7}{28}.$
$-\phantom{\rule{0.2em}{0ex}}\frac{26}{28}$
Find the difference: $-\phantom{\rule{0.2em}{0ex}}\frac{27}{32}-\phantom{\rule{0.2em}{0ex}}\frac{1}{32}.$
$-\phantom{\rule{0.2em}{0ex}}\frac{7}{8}$
Simplify: $-\phantom{\rule{0.2em}{0ex}}\frac{10}{x}-\phantom{\rule{0.2em}{0ex}}\frac{4}{x}.$
$\begin{array}{cccccc}& & & & & \hfill -\phantom{\rule{0.2em}{0ex}}\frac{10}{x}-\phantom{\rule{0.2em}{0ex}}\frac{4}{x}\hfill \\ \\ \\ \begin{array}{c}\text{Subtract the numerators and place the}\hfill \\ \text{difference over the common denominator.}\hfill \end{array}\hfill & & & & & \hfill \frac{\mathrm{-14}}{x}\hfill \\ \\ \\ \begin{array}{c}\text{Rewrite with the sign in front of the}\hfill \\ \text{fraction.}\hfill \end{array}\hfill & & & & & \hfill -\phantom{\rule{0.2em}{0ex}}\frac{14}{x}\hfill \end{array}$
Find the difference: $-\phantom{\rule{0.2em}{0ex}}\frac{9}{x}-\phantom{\rule{0.2em}{0ex}}\frac{7}{x}.$
$-\phantom{\rule{0.2em}{0ex}}\frac{16}{x}$
Find the difference: $-\phantom{\rule{0.2em}{0ex}}\frac{17}{a}-\phantom{\rule{0.2em}{0ex}}\frac{5}{a}.$
$-\phantom{\rule{0.2em}{0ex}}\frac{22}{a}$
Now we will do an example that has both addition and subtraction.
Simplify: $\frac{3}{8}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{5}{8}\right)-\phantom{\rule{0.2em}{0ex}}\frac{1}{8}.$
$\begin{array}{cccccc}\begin{array}{c}\text{Add and subtract fractions\u2014do they have a}\hfill \\ \text{common denominator? Yes.}\hfill \end{array}\hfill & & & & & \hfill \frac{3}{8}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{5}{8}\right)-\phantom{\rule{0.2em}{0ex}}\frac{1}{8}\hfill \\ \\ \\ \begin{array}{c}\text{Add and subtract the numerators and place}\hfill \\ \text{the result over the common denominator.}\hfill \end{array}\hfill & & & & & \hfill \frac{3+\left(\mathrm{-5}\right)-1}{8}\hfill \\ \\ \\ \text{Simplify left to right.}\hfill & & & & & \hfill \frac{\mathrm{-2}-1}{8}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & & & \hfill -\phantom{\rule{0.2em}{0ex}}\frac{3}{8}\hfill \end{array}$
Simplify: $\frac{2}{5}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{4}{9}\right)-\phantom{\rule{0.2em}{0ex}}\frac{7}{9}.$
$\mathrm{-1}$
Simplify: $\frac{5}{9}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{4}{9}\right)-\phantom{\rule{0.2em}{0ex}}\frac{7}{9}.$
$-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}$
As we have seen, to add or subtract fractions, their denominators must be the same. The least common denominator (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.
The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.
After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!
Notification Switch
Would you like to follow the 'Elementary algebra' conversation and receive update notifications?