Jack and Jill

Jack and Jill leave home at noon and walk along a level road, then up a hill, then back down the same hill and home again, without stopping. They walk at $4$ miles an hour on the level, $3$ uphill and $6$ downhill, and they reach home at six o’clock. At what time did they reach the top of the hill? Give the answer to within half an hour.

Solution

Let the level stretch be $a$ miles each way and the hill $b$ miles each way. Going out they cover $a$ on the level at $4$ and $b$ uphill at $3$; coming back, $b$ downhill at $6$ and $a$ on the level at $4$. The whole journey takes $$\frac{a}{4} + \frac{b}{3} + \frac{b}{6} + \frac{a}{4} = \frac{a}{2} + \frac{b}{2} = \frac{a+b}{2}$$ hours. This equals six, so $a + b = 12$.

The time to reach the top is the outward part alone, $$t = \frac{a}{4} + \frac{b}{3} = \frac{12-b}{4} + \frac{b}{3} = 3 + \frac{b}{12}.$$ Since $b$ lies between $0$ and $12$, the time $t$ lies between $3$ and $4$ hours. So they reached the top between three and four o’clock, and the best single answer, good to within half an hour, is half past three.