Clara takes 2 hours to ride her bicycle from Appsley to Bancroft. The reverse trip takes her 2 hours and 15 minutes. If she travels downhill at $24 \mathrm{~km} / \mathrm{h}$, on level road at $16 \mathrm{~km} / \mathrm{h}$ and uphill at $12 \mathrm{~km} / \mathrm{h}$, what is the distance, in kilometres, between the two towns?
Suppose that, on the way from Appsley to Bancroft, Clara rides $x \mathrm{~km}$ downhill, $y \mathrm{~km}$ on level road, and $z \mathrm{~km}$ uphill.

Then on the reverse trip, she travels $x \mathrm{~km}$ uphill, $y \mathrm{~km}$ on level road, and $z \mathrm{~km}$ downhill.

Clara rides downhill at $24 \mathrm{~km} / \mathrm{h}$, on level road at $16 \mathrm{~km} / \mathrm{h}$, and uphill at $12 \mathrm{~km} / \mathrm{h}$.

Since the original trip takes 2 hours, then $\frac{x}{24}+\frac{y}{16}+\frac{z}{12}=2$.

Since the reverse trip takes 2 hours and 15 minutes (or 2.25 hours), then $\frac{x}{12}+\frac{y}{16}+\frac{z}{24}=2.25$.

Adding these two equations, we obtain

$$
\left(\frac{1}{24}+\frac{1}{12}\right) x+\left(\frac{1}{16}+\frac{1}{16}\right) y+\left(\frac{1}{12}+\frac{1}{24}\right) z=4.25
$$

Since $\frac{1}{24}+\frac{1}{12}=\frac{1}{24}+\frac{2}{24}=\frac{3}{24}=\frac{1}{8}$, then the last equation is equivalent to

$$
\frac{x}{8}+\frac{y}{8}+\frac{z}{8}=4.25
$$

Multiplying both sides by 8 , we obtain $x+y+z=34$.

The distance from Appsley to Bancroft is $(x+y+z) \mathrm{km}$, or $34 \mathrm{~km}$.

ANSWER: 34