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

by : $64$

 * 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 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 $C$?

The illustration in the book shows $C$ as being in completely the opposite direction from $B$ to $A$.

That is, $B$ is directly between $A$ and $C$.

The real puzzle is why he set off in the direction of $C$ when his destination was in fact $A$, which is going the wrong way.

Solution
$1 \tfrac 1 3$ miles 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: