1.6 Add and subtract fractions  (Page 2/4)

Solution

Do the fractions have a common denominator? No, so we need to find the LCD.

Find the LCD. $$
Notice, 15 is “missing” three factors of 2 and 24 is “missing” the 5 from the factors of the LCD. So we multiply 8 in the first fraction and 5 in the second fraction to get the LCD.
Rewrite as equivalent fractions with the LCD.
Simplify.
Subtract.	$\mathrm{-}\frac{39}{120}$
Check to see if the answer can be simplified.	$\mathrm{-}\frac{13\cdot 3}{40\cdot 3}$
Both 39 and 120 have a factor of 3.
Simplify.	$\mathrm{-}\frac{13}{40}$

Solution

The fractions have different denominators.

		$$
Find the LCD. $$
Rewrite as equivalent fractions with the LCD.		$$
Simplify.		$$
Add.		$$

Solution

First ask, “What is the operation?” Once we identify the operation that will determine whether we need a common denominator. Remember, we need a common denominator to add or subtract, but not to multiply or divide.

1. ⓐ What is the operation? The operation is subtraction.
   $\begin{array}{cccccc}\text{Do the fractions have a common denominator? No.}\hfill & & & & & \hfill \frac{5x}{6}-\frac{3}{10}\hfill \\ \\ \\ \text{Rewrite each fraction as an equivalent fraction with the LCD.}\hfill & & & & & \hfill \begin{array}{c}\frac{5x·5}{6·5}-\frac{3·3}{10·3}\hfill \\ \frac{25x}{30}-\frac{9}{30}\hfill \end{array}\hfill \\ \\ \\ \begin{array}{c}\text{Subtract the numerators and place the difference over the}\hfill \\ \text{common denominators.}\hfill \end{array}\hfill & & & & & \hfill \frac{25x-9}{30}\hfill \\ \\ \\ \begin{array}{c}\text{Simplify, if possible There are no common factors.}\hfill \\ \text{The fraction is simplified.}\hfill \end{array}\hfill & & & & & \end{array}$
   
2. ⓑ What is the operation? Multiplication.
   $\begin{array}{cccccc}& & & & & \hfill \frac{5x}{6}·\frac{3}{10}\hfill \\ \\ \\ \begin{array}{c}\text{To multiply fractions, multiply the numerators and multiply}\hfill \\ \text{the denominators.}\hfill \end{array}\hfill & & & & & \hfill \frac{5x·3}{6·10}\hfill \\ \\ \\ \begin{array}{c}\text{Rewrite, showing common factors.}\hfill \\ \text{Remove common factors.}\hfill \end{array}\hfill & & & & & \hfill \frac{\cancel{5}x·\cancel{3}}{2·\cancel{3}·2·\cancel{5}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & & & \hfill \frac{x}{4}\hfill \end{array}$

Notice we needed an LCD to add $\frac{5x}{6}-\frac{3}{10},$ but not to multiply $\frac{5x}{6}·\frac{3}{10}.$