Henry Ernest Dudeney/Modern Puzzles/84 - The Three Brothers/Solution

by : $84$

 * The Three Brothers
 * The discussion arose before one of the tribunals as to which of a tradesman's three sons could best be spared for service in the Army.
 * "All I know as to their capabilities," said the father, "is this:
 * Arthur and Benjamin can do a certain quantity of work in eight days,
 * which Arthur and Charles will do in nine days,
 * and which Benjamin and Charles will take ten days over."


 * Of course, it was at once seen that as longer time was taken over the job whenever Charles was one of the pair,
 * he must be the slowest worker.
 * This was all they wanted to know, but it is an interesting puzzle to ascertain just how long each son would be required to do that job alone.


 * Can you discover?

Solution
Arthur, Benjamin and Charles would take respectively $14 \tfrac {34} {49}$ days, $17 \tfrac {23} {41}$ days, and $23 \tfrac 7 {31}$ days.

Proof
Let $a, b, c$ be the rate of working in jobs per day of (respectively) Arthur, Benjamin and Charles.

Let $t_a, t_b, t_c$ be the number of days it would take (respectively) Arthur, Benjamin and Charles to do the job alone.

We have:

and so:

Hence and so.

Also see

 * Henry Ernest Dudeney: Puzzles and Curious Problems: $138$ - The Three Workmen, exactly the same problem but with different names.