Word problem: Working together

Question

Two workers, if they were working together, could finish a certain job in 12 days. If one of the workers does the first half of the job and then the other one does the second half, the job will take them 25 days. How long would it take each worker to do the entire job by himself?

I'm really bad at these type. Can someone tell me how to make an equation out of this so I can solve this? Thanks.

Answer 1

Suppose that when they work together, one worker does a proportion $x$ of the job and the other does $1-x$.

(For example, if the second person works twice as hard then the first person does $\frac13$ of the job and the second does $\frac23$, so $x=\frac13$.)

The number of days taken by each worker to do the whole job working at the same rate would be $$\frac{12}{x}\quad\hbox{and}\quad \frac{12}{1-x}\ .\tag{$*$}$$ To do half the job each, the total time is $$\frac6x+\frac6{1-x}$$ and you know that this is equal to $25$. From this you should be able to work out and solve a quadratic equation, then substitute back into $(*)$.

Answer:

$x=\frac25$ (or $\frac35$, doesn't matter which) so $12/x=30$ and $12/(1-x)=20$, or vice versa.