Henry Ernest Dudeney/Puzzles and Curious Problems/64 - The Bath Chair/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $64$

The Bath Chair
A correspondent informs us that a friend's house at $A$, where he was invited to lunch at $1$ p.m., is a mile from his own house at $B$.
He is an invalid, and at $12$ noon started in his Bath chair from $B$ towards $C$.
His friend, who had arranged to join him and help push back, left $A$ at $12.15$ p.m., walking at $5$ miles per hour towards $C$.
He joined him, and with his help they went back at $4$ miles per hour, and arrived at $A$ at exactly $1$ p.m.
How far did our correspondent go towards $C$?


Solution

$\dfrac 2 3$ of a mile from $B$.

They met at $12:35$.


Proof

Let $d$ miles be the distance towards $C$ at which our correspondent encountered the friend.

Let $t$ hours past $12$ be the time at which they met at that point.

We have:

\(\text {(1)}: \quad\) \(\ds d + 1\) \(=\) \(\ds 5 \paren {t - \dfrac 1 4}\) His friend ... left $A$ at $12.15$ p.m., walking at $5$ miles per hour towards $C$.
\(\text {(2)}: \quad\) \(\ds d + 1\) \(=\) \(\ds 4 \paren {1 - t}\) ... they went back at $4$ miles per hour, and arrived at $A$ at exactly $1$ p.m.''
\(\ds \leadsto \ \ \) \(\ds 5 t - \dfrac 5 4\) \(=\) \(\ds 4 - 4 t\)
\(\ds \leadsto \ \ \) \(\ds 9 t\) \(=\) \(\ds \dfrac {21} 4\)
\(\ds \leadsto \ \ \) \(\ds t\) \(=\) \(\ds \dfrac {21} {36} = \dfrac 7 {12}\)
\(\ds \leadsto \ \ \) \(\ds d + 1\) \(=\) \(\ds 4 \paren {1 - \dfrac 7 {12} }\)
\(\ds \leadsto \ \ \) \(\ds d\) \(=\) \(\ds \dfrac 5 3 - 1 = \dfrac 2 3\)

$\blacksquare$


Sources

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