 # Problem of the Week Problem E and Solution Let’s Paint

## Problem

Painters R Us has been given a large painting job. Initially, Jim started painting by himself. In $$15$$ days, working $$9$$ hours each day, he was able to complete $$\frac{3}{8}$$ of the job. He decided to have Wanda join him for the remaining part of the job. Together they completed the job in another $$10$$ days, each working $$9$$ hours per day. If Wanda had originally done the job by herself, how many hours would it have taken her to finish the complete job? ## Solution

We must make some reasonable assumptions. We will assume that each painter worked at a constant rate each hour, every day. These rates may or may not have been the same for the two painters.

Since Jim completed $$\frac{3}{8}$$ of the job in $$15$$ days, he would complete $$\frac{1}{3}$$ of $$\frac{3}{8}$$, or $$\frac{1}{8}$$, of the job in $$5$$ days.

Since Jim had completed $$\frac{3}{8}$$ of the job when Wanda started to work, $$\frac{5}{8}$$ of the job was left to be completed. Together they completed $$\frac{5}{8}$$ of the job in $$10$$ days. Since Jim can complete $$\frac{1}{8}$$ of the job in $$5$$ days, he would have completed $$\frac{2}{8}$$ of the job in these $$10$$ days. Therefore, Wanda completed the remaining $$\frac{5}{8} - \frac{2}{8} = \frac{3}{8}$$ of the job in these $$10$$ days.

Since Wanda worked $$9$$ hours a day, this means she completed $$\frac{3}{8}$$ of the job in $$10\times 9 = 90$$ hours. Therefore, she completed $$\frac{1}{8}$$ of the job in $$30$$ hours. Therefore, she could have completed the entire job on her own in $$8\times 30=240$$ hours.

Jim completed $$\frac{1}{8}$$ of the job in $$5$$ days. The whole job could be completed by Jim in $$8\times 5=40$$ days or 360 hours.
As it was, Jim worked a total of 25 days at 9 hours per day and Wanda worked 10 days at 9 hours per day. They worked a total of $$25\times 9 + 10\times 9=315$$ hours.
We know that together Jim and Wanda completed $$\frac{5}{8}$$ of the job in 10 days. Then, in 2 days they would have completed $$\frac{1}{8}$$ of the job and in 16 days they would have completed the entire job. That is, working together from the start they would have completed the job in $$16\times 9=144$$ hours.