Converting Ratios

Question

In a community of M men and W women, m men and w women smoke (m ≤ M,w ≤ W). If a person is selected at random and A and B are the events that the person is a man and smokes, respectively, under what conditions are A and B independent?

I have seen the question posted here before Conditional probability. under which conditions $E$ and $F$ are independent?

Nonetheless the answer in my textbook states something more:

$$\frac{m}{M+W}=\frac{M}{M+W}\frac{m+w}{M+W}\tag{1}$$

Which implies $$\frac{m}{M}=\frac{w}{W}\tag{2}$$I do not see how this follows from the 1st equation

Answer 1

Starting with the first equation $$ \frac{m}{M+W} = \frac{M}{M+W} \frac{m+w}{M+W} $$ we can multiply both sides by $\frac{M+W}{M}$ to get $$ \frac{m}{M} = \frac{m+w}{M+W} $$ or $$ (M+W)m = (m+w)M \qquad \Rightarrow \qquad Mm + Wm = Mm + Mw $$ When you cancel the term $Mm$, you get $$ Wm = Mw $$ which you can divide by $WM$ to get $$ \frac{m}{M} = \frac{w}{W} $$