Henry Ernest Dudeney/Puzzles and Curious Problems/77 - Out and Home/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $77$

Out and Home
Mr Wilkinson walks from his country house into the neighbouring town at the rate of five miles per hour,
and, as he is a little tired, he makes the return journey at the rate of three miles per hour.
As the double journey takes him exactly seven hours, can you tell me the distance from his house to the town?


Solution

$13 \tfrac 1 8$ miles.


Proof

Let $D$ be the distance to town.

Let $t_1$ and $t_2$ be the times taken to walk to town and to walk back respectively.

We have:

\(\ds D\) \(=\) \(\ds 5 t_1\) Mr Wilkinson walks from his country house into the neighbouring town at the rate of five miles per hour,
\(\ds \) \(=\) \(\ds 3 t_2\) and, as he is a little tired, he makes the return journey at the rate of three miles per hour.
\(\ds t_1 + t_2\) \(=\) \(\ds 7\) As the double journey takes him exactly seven hours, ...
\(\ds \leadsto \ \ \) \(\ds t_2\) \(=\) \(\ds 7 - t_1\)
\(\ds \leadsto \ \ \) \(\ds D\) \(=\) \(\ds 5 t_1\)
\(\ds D\) \(=\) \(\ds 3 \paren {7 - t_1}\)
\(\ds \) \(=\) \(\ds 21 - 3 t_1\)
\(\ds \) \(=\) \(\ds 21 - 3 \dfrac D 5\)
\(\ds \leadsto \ \ \) \(\ds 5 D\) \(=\) \(\ds 105 - 3 D\)
\(\ds \leadsto \ \ \) \(\ds 8 D\) \(=\) \(\ds 105\)
\(\ds \leadsto \ \ \) \(\ds D\) \(=\) \(\ds \dfrac {105} 8\)
\(\ds \) \(=\) \(\ds 13 \tfrac 1 8\)

$\blacksquare$


Sources

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