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

by : $16$

 * Digging a Ditch

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: