Greek Anthology Book XIV: 7. - Problem/Variant

Problem

 * $A$ and $B$ together can do a piece of work in $6$ days.
 * $B$ and $C$ together can do it in $20$ days.
 * $C$ and $A$ together can do it in $7 \frac 1 2$ days.

How many days will each require to do the job separately?

Solution
It takes:
 * $8$ days for $A$ to do the job
 * $24$ days for $B$ to do the job
 * $120$ days for $C$ to do the job.

Proof
Let $a$, $b$ and $c$ denote the number of days it takes $A$, $B$ and $C$ to complete the job respectively.

Then each completes $\dfrac 1 a$, $\dfrac 1 b$ and $\dfrac 1 c$ of the job each day, respectively.

Thus:

Thus we have: