Extracting common factor

Reading from right to left the equality given by the distributive property, we have the expression$$\displaystyle \frac{a}{b}\cdot \frac{c}{d}+\frac{a}{b}\cdot \frac{n}{m}$$which can be written:$$\displaystyle \frac{a}{b}\cdot \Big(\frac{c}{d}+\frac{n}{m}\Big)$$

We call this process extracting common factor since we have found a factor, that is a number that is multiplying, common to both addends of the expression.

Namely$$\displaystyle \frac{a}{b}\cdot \frac{c}{d}+\frac{a}{b}\cdot \frac{n}{m}=\frac{a}{b}\cdot \Big(\frac{c}{d}+\frac{n}{m}\Big)=\frac{a}{b}\cdot \Big(\frac{c}{d}+\frac{n}{m}\Big)$$which means that the sum of two products has been converted into the product of a number by a sum.

Let's see how to extract the common factor in the expression: $$\displaystyle \frac{1}{5}\cdot\frac{2}{3}+\frac{1}{5}\cdot 4 -\frac{1}{5}\cdot \frac{1}{2}$$

The common factor in all three addends is the fraction $\displaystyle \frac{1}{5}$, so:$$\displaystyle \frac{1}{5}\cdot\frac{2}{3}+\frac{1}{5}\cdot4-\frac{1}{5}\cdot \frac{1}{2}= \frac{1}{5}\cdot \frac{1}{5}\cdot 4+\frac{1}{5}\cdot \Big(-\frac{1}{2}\Big)= \\ =\frac{1}{5}\cdot \Big[\frac{2}{3}+4+\Big(-\frac{1}{2}\Big)\Big]=\frac{1}{5}\cdot\Big[\frac{2}{3}+4-\frac{1}{2}\Big]$$