Simple algebra, economics

Question

Using some algebraic manipulation the expression $\displaystyle\frac{P-W_1P_x}{W_2}$ is made into $\displaystyle \frac{1-W_1 \frac{P_x}{P}}{W_2}$. It says it is simply multiplied by $\displaystyle \frac{1}{P}$. I am not terribly fluent in algebra as I should be, but online resources are failing me in helping me see what rules were applied to make this happen. Help please?

Answer 1

Consider $\frac{P - W_{1}P_{x}}{W_{2}} = (P - W_{1}P_{x}) * \frac{1}{W_{2}}$. Now dividing by $P$ is equivalent to $\frac{(P - W_{1}P_{x})}{P} * \frac{1}{W_{2}} = (\frac{P}{P} - \frac{W_{1}P_{X}}{P}) * \frac{1}{W_{2}}$. The cancellations should be obvious from here.

Answer 2

It is the distributive property of multiplication and addition.

$$\frac{1}{P}\frac{P-W_1P_x}{W_2} = \frac{\frac{1}{P}\left(P-W_1P_x\right)}{W_2}= \frac{1-W_1\frac{P_x}{P}}{W_2}$$

Note that this is also the same as just writing

$$\frac{P-W_1P_x}{PW_2}$$