I just wanted to come back and play with this question.
I am just repeating what Rom has done, in the hope that I can do it myself next time. Thanks Rom.
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?
\(\mbox{Let worker 1 do }\quad r_1 \;\frac{jobs}{day} \quad \mbox{and take t days to do half a job.}\\ \mbox{Let worker 2 do }\quad r_2 \;\frac{jobs}{day} \quad \mbox{and take (25- t) days to do half a job.}\)
We have 3 unknowns and we have 4 equations
Since it takes 12 days for them to paint the house together
\(r_1+r_2=\frac{1}{12}\;\frac{job}{day} \qquad[1]\\ \frac{r_1t}{2}+\frac{r_2(25-t)}{2}=1\;job \quad[2]\\ r_1t=\frac{1}{2}\;\;job\qquad[3]\\ r_2(25-t)=\frac{1}{2}\;\;job\qquad[4]\\~\\~\\ r_1=\frac{1}{2t}\;\;job\qquad[3b]\\ r_2=\frac{1}{2(25-t)}\;\;job\qquad[4b]\\ \mbox{Sub [3b] and [4b] into [1] }\\ \frac{1}{2t}+\frac{1}{2(25-t)}=\frac{1}{12}\\ \frac{1}{t}+\frac{1}{(25-t)}=\frac{1}{6}\\ \frac{6(25-t)}{6t(25-t)}+\frac{6t}{6t(25-t)}=\frac{t(25-t)}{6t(25-t)}\\ 6(25-t)+6t=t(25-t)\\ 150=t(25-t)\\ t^2-25t+150=0\\ (t-15)(t-10)=0\\ t=15\qquad or \qquad t=10 \)
So one takes 15 days to do 1/2 a job and
the other takes 10 days to do half a job.
Hence, it would take them 30 days and 20 days respectively to do one whole job.
And I did not even need to use equation 2