Question

can anybody help me tackle these word problems?
http://oi57.tinypic.com/2przhc5.jpg

Answer 1

For the 1st one,
The 1st 7 hours, they work together and their work rate is
$$
\cfrac{1~roof}{12~hrs}+\cfrac{1~roof}{x~hrs}
$$
In 7 hours they can complete
$$
\left(\cfrac{1~roof}{12~hrs}+\cfrac{1~roof}{x~hrs}\right )\times7~hrs
$$
of a roof.

The last 8 hours is performed at the apprentice's rate of x hours per roof and he completes
$$
\cfrac{1~roof}{x~hrs}\times8~hrs
$$
of a roof during that time to finish the job.

So one roof is completed in 15 hours, with the first 7 hours completed at the combined rate above and the last hour completed at the apprentice's rate:
$$
1~roof=\left(\cfrac{1~roof}{12~hrs}+\cfrac{1~roof}{x~hrs}\right )\times7~hrs+\cfrac{1~roof}{x~hrs}\times8~hrs\\
\implies x=\cfrac{36~hrs}{roof}
$$
So the apprentice takes 36 hours to complete a roof.

Hope this helps.
For the second question, the combined rate time, T, to finish all jobs is
$$
\cfrac{1}{T}=\cfrac{1}{6}+\cfrac{1}{9}+\cfrac{1}{18}\\
\implies T=3~mins

$$