Roscoe saves $60 in 4 months. At this unit rate, how long will it take him to save $112.50?

We have the proportion $\frac{\text{dollars}}{\text{time}} = \frac{\text{dollars}}{\text{time}}.$ We can set up the proportion $\frac{60}{4} = \frac{112.50}{x}.$ Solving for $x$, we find $x = \boxed{7.5}$ months.