Cindy leaves school at the same time every day. If she cycles at $20 \mathrm{~km} / \mathrm{h}$, she arrives home at $4: 30$ in the afternoon. If she cycles at $10 \mathrm{~km} / \mathrm{h}$, she arrives home at 5:15 in the afternoon. At what speed, in $\mathrm{km} / \mathrm{h}$, must she cycle to arrive home at 5:00 in the afternoon?
(A) $16 \frac{2}{3}$
(B) 15
(C) $13 \frac{1}{3}$
(D) 12
(E) $18 \frac{3}{4}$
Cindy leaves school at the same time every day. If she cycles at $20 \mathrm{~km} / \mathrm{h}$, she arrives home at 4:30 in the afternoon. If she cycles at $10 \mathrm{~km} / \mathrm{h}$, she arrives home at 5:15 in the afternoon. At what speed, in $\mathrm{km} / \mathrm{h}$, must she travel to arrive home at 5:00 in the afternoon?
(A) $16 \frac{2}{3}$
(B) 15
(C) $13 \frac{1}{3}$
(D) 12
(E) $18 \frac{3}{4}$

## Solution

Since the distance from Cindy's home to school is unknown, represent this distance by $d$, in kilometres. We will consider the problem in two separate cases, the first in which she travels at 20 $\mathrm{km} / \mathrm{h}$ and the second when she travels at $10 \mathrm{~km} / \mathrm{h}$.

Distance travelled at $20 \mathrm{~km} / \mathrm{h}=$ Distance travelled at $10 \mathrm{~km} / \mathrm{h}$

Let the time that Cindy takes travelling home at $20 \mathrm{~km} / \mathrm{h}$ be $t$ hours.

If Cindy arrives home $\frac{3}{4} \mathrm{~h}$ later when travelling at $10 \mathrm{~km} / \mathrm{h}$, then the length of time travelling is $\left(t+\frac{3}{4}\right)$ hours. The previous equation becomes

$$
\begin{aligned}
20 t & =10\left(t+\frac{3}{4}\right) \\
20 t & =10 t+\frac{30}{4} \\
10 t & =\frac{15}{2} \\
t & =\frac{15}{20} \text { or } \frac{3}{4}
\end{aligned}
$$

Therefore the distance from school to home is $d=20 \times \frac{3}{4}$, or $d=15 \mathrm{~km}$.

If Cindy arrives home at 5:00 in the afternoon, she would have travelled home in $\frac{3}{4}+\frac{1}{2}=\frac{5}{4}$ hours over a distance of 15 kilometres.

Therefore, $s=\frac{d}{t}=\frac{15}{\frac{5}{4}}=15 \times \frac{4}{5}=12 \mathrm{~km} / \mathrm{h}$.

Therefore, Cindy would have had to travel at $12 \mathrm{~km} / \mathrm{h}$ to arrive home at 5:00 p.m.

ANSWER: (D)