Henry Ernest Dudeney/Puzzles and Curious Problems/70 - The Donkey Cart/Solution

by : $70$

 * The Donkey Cart
 * Atkins, Brown and Cranby had to go an journey of $40$ miles.
 * Atkins could walk $1$ mile an hour,
 * Brown could walk $2$ miles an hour,
 * and Cranby could go in his donkey-cart at $8$ miles an hour.
 * Cranby drove Atkins a certain distance, and, dropping him to walk the remainder,
 * drove back to meet Brown on the way and carried him to their destination,
 * where they all arrived at the same time.


 * How long did the journey take?


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

Solution
The whole journey takes $10 \tfrac 5 {41}$ hours.

Proof
Let Atkins, Brown and Cranby be denoted by $A$, $B$ and $C$ respectively.

We have that:
 * $C$ sets off with $A$
 * drops him off $d_1$ miles from the start
 * then returns to pick up $B$ at a point $d_2$ miles from the start.

Let $t_1$ be the time $C$ and $A$ reach $d_1$.

Let $t_2$ be the time $C$ reaches $d_2$ to pick up $B$.

Let $t$ be the time they all arrive at their destination.

We have:

Then we have: