Word problem by modelling a system of equations

Question

A crew of workers did one job for a certain number of days. If they were three more, they would have finished $2$ days earlier; if they were $12$ more, the same work would have done $5$ days earlier. I should determine the number of workers in the crew and the days for which the work was completed.

I think it is best if $x$ is the number of workers and $y$ are the days for which the work is done. Now I am trying to find a relationship between the number of workers and the days that are required. Can you give me a hint?

Answer 1

We can assume that the amount of work done (in worker-days) is the product of the number of workers and the number of days, so the work done by $x$ workers in $y$ days is $xy$.

If we increase the number of workers by $3$ and decrease the number of days by $2$ then the work done is $(x+3)(y-2)$. But this should be the same as the work done by $x$ workers for $y$ days. So we have

$xy = (x+3)(y-2)$

In the same way, we also have

$xy = (x+12)(y-5)$

By re-arranging these two simultaneous equations you can find the values of $x$ and $y$.

Answer 2

Hint: define $f$ to be the fraction of the job that one crew member is able to accomplish in one day.

Translate every sentence into an equation involving $f$ (as well as your $x$ and $y$), and solve for $x$ and $y$.