Henry Ernest Dudeney/Modern Puzzles/16 - Digging a Ditch/Solution

by : $16$

 * Digging a Ditch

Solution
Abraham should receive $\tfrac 1 3$ of $\pounds 2$, and Benjamin should receive $\tfrac 2 3$.

Proof
Let $A$ and $B$ denote Abraham and Benjamin respectively

Let $r$ denote the ratio of the ability of Abraham to work relative to Benjamin.

Suppose:
 * $B$ can dig all in $r$ hours and shovel all in $s$ hours.
 * $A$ can dig all in $s$ hours and shovel all in $4 r$ hours.

Then as their relative abilities for work is the same in digging and shovelling:
 * $\dfrac r s = \dfrac s {4 r}$

then: $\dfrac r s = \dfrac 1 2$

So $B$ accomplishes twice as much work as $A$.

Thus $B$ takes twice the pay that $A$ does.

The result follows.