Henry Ernest Dudeney/Modern Puzzles/28 - Hill Climbing/Solution

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Modern Puzzles by Henry Ernest Dudeney: $28$

Hill Climbing
Weary Willie went up a certain hill at the rate of $1 \tfrac 1 2$ miles per hour
and came down at the rate of $4 \tfrac 1 2$ miles per hour,
so that it took him just $6$ hours to make the double journey.
Now, how far was it to the top of the hill?


Solution

$6 \tfrac 3 4$ miles.

He goes up in $4 \tfrac 1 2$ hours and back down again in $1 \tfrac 1 2$ hours.


Proof

Let $d$ miles be the distance to the top of the hill.

Let $t_1$ be the time taken to reach the top.

Let $t_2$ be the time taken to reach the bottom again.

The assumption has to be that no time is taken to rest at the top.


We have:

\(\text {(1)}: \quad\) \(\ds t_1 + t_2\) \(=\) \(\ds 6\)
\(\ds d\) \(=\) \(\ds t_1 \times 1 \tfrac 1 2\)
\(\ds \) \(=\) \(\ds t_2 \times 4 \tfrac 1 2\)
\(\text {(2)}: \quad\) \(\ds \leadsto \ \ \) \(\ds t_1\) \(=\) \(\ds 3 t_2\)
\(\ds \leadsto \ \ \) \(\ds 3 t_2 + t_2\) \(=\) \(\ds 6\) substituting for $t_1$ from $(2)$ into $(1)$
\(\ds \leadsto \ \ \) \(\ds t_2\) \(=\) \(\ds \dfrac 3 2\) simplifying
\(\ds \leadsto \ \ \) \(\ds d\) \(=\) \(\ds \dfrac 3 2 \times \dfrac 9 2\) simplifying
\(\ds \) \(=\) \(\ds \dfrac {27} 4\) simplifying
\(\ds \) \(=\) \(\ds 6 \tfrac 3 4\)

$\blacksquare$


Sources

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