TITLE: Meaning of $\frac{x-y}{y}$ versus $\frac{x}{y}-1$
QUESTION [14 upvotes]: I'm trying to understand what is probably a fairly simple math concept, but this is escaping me for some reason.  Why are the results of these two expressions equal? Thanks for any responses.
$$\frac{x-y}{y}$$
$$\frac{x}{y}-1$$

REPLY [2 votes]: Let's assume our fraction $\dfrac{x-y}{y}$ equals $a$.
$$\frac{x-y}{y}=a$$
Multiplying by $y$ on both sides...
$$x-y=ay$$
Isolating $x$...
$$x=ay+y$$
Factoring the right hand side:...
$$x=y(a+1)$$
Dividing by $y$ on both sides...
$$\frac{x}{y}=a+1$$
Moving $1$ back to the left hand side...
$$\frac{x}{y}-1=a$$
Because $\dfrac{x-y}{y}=a$ and $\dfrac{x}{y}-1=a$:
$$\displaystyle \therefore \frac{x-y}{y}=\frac{x}{y}-1$$
Thank you for reading.