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

Modern Puzzles by Henry Ernest Dudeney: $16$

Here is a curious question that is more perplexing than it looks at first sight.

Abraham, an infirm old man, undertook to dig a ditch for $2$ pounds.

He engaged Benjamin, an able-bodied fellow, to assist him and share the money fairly according to their capacities.

Abraham could dig as fast as Benjamin could shovel out the dirt,

and Benjamin could dig four times as fast as Abraham could do the shovelling.

How should they divide the money?

Of course, we must assume their relative abilities for work to be the same in digging or shovelling.

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

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.

$\blacksquare$