# Jon leaves his house for a business trip driving at a rate of 45 miles per hour. Half an hour later his wife, Emily, realizes he forgot his cell phone and begins to follow him at a rate of 55 miles per hour. How long will it take for Emily to catch Jon?

$135$ minutes, or $2 \frac{1}{4}$ hours.

#### Explanation:

We're looking for the point where Jon and Emily have travelled the same distance.

Let's say that Jon travels for time $t$, so he travels $45 t$ before his wife catches up.

Emily travels faster, at 55 mph, but she does travel for so long. She travels for $t - 30$: $t$ for the time her husband travels and $- 30$ to account for her late start.

That gives us:

$45 t = 55 \left(t - 30\right)$

$45 t = 55 t - 1650$

$10 t = 1650 \implies t = 165$ minutes

(we know it's minutes because I used $t - 30$ with the 30 being 30 minutes. I could have said $t - \frac{1}{2}$ with $\frac{1}{2}$ being half an hour)

So Jon travels 165 minutes, or $2 \frac{3}{4}$ hours before Emily catches up.

Emily, for her part, travelled for $165 - 30 = 135$ minutes, or $2 \frac{1}{4}$ hours.

Pity Emily - she still has to drive home...