#
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.

**For Your
Information:**

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.