Henry Ernest Dudeney/Modern Puzzles/44 - Find the Distance

by : $44$

 * Find the Distance
 * A man named Jones set out to walk from $A$ to $B$,
 * and on the road he met his friend Kenward, $10$ miles from $A$, who had left $B$ at exactly the same time.
 * Jones executed his commission at $B$ and, without delay, set out on his return journey,
 * while Kenward as promptly returned from $A$ to $B$.
 * They met $12$ miles from $B$.


 * Of course, each walked at a uniform rate throughout.


 * Now, how far is $A$ from $B$?


 * I will show the reader a simple rule by which the distance may be found by anyone in a few seconds without the use of a pencil.
 * In fact, it is quite absurdly easy -- when you know how to do it.

Solution

 * $18$ miles.

The "absurdly simple" rule being referred to by :


 * $3$ times the distance from $A$ that they first cross minus the distance from $B$ when they cross for the second time.

Proof
Let $d$ miles be the distance between $A$ and $B$.

Let $V_J$ and $V_K$ miles per hour be the speeds of Jones and Kenward respectively.

Then we have:

For the general case, let $d_1$ and $d_2$ be the distances from $A$ and $B$ that they crossed respectively.

We have:

So the general rule is:
 * $3 d_1 - d_2$