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

by : $16$

 * Digging a Ditch
 * 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.

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

Suppose:
 * $A$ can dig all in $d_a$ hours and shovel all in $s_a$ hours.
 * $B$ can dig all in $d_b$ hours and shovel all in $s_b$ hours.

Then:
 * $r = \dfrac {d_a} {d_b} = \dfrac {s_a} {s_b}$

We also given that final assumption that:
 * $\dfrac {d_a} {s_a} = \dfrac {d_b} {s_b}$

We are given:
 * $d_a = s_b$
 * $d_b = 4 s_a$

Hence: