Hello, pagethru!
Here's my approach . . . it's different. \(\displaystyle \;\) (Is anyone surprised?)
Mark can overhaul an engine in 20 hours and Phil can do the same job by himself in 30 hours.
If they both work together for a time and then Mark finishes the job himself in 3 hours,
how long did they work together?
Let \(\displaystyle x\) = number of hours they worked together.
Mark can do the job in 20 hours.
\(\displaystyle \;\;\)In one hour, he can do \(\displaystyle \frac{1}{20}\) of the job.
\(\displaystyle \;\;\)in \(\displaystyle x\) hours, he can do \(\displaystyle \frac{x}{20}\) of the job.
Phil can do the job in 30 hours.
\(\displaystyle \;\;\)In one hour, he can do \(\displaystyle \frac{1}{30}\) of the job.
\(\displaystyle \;\;\)In \(\displaystyle x\) hours, he can do \(\displaystyle \frac{x}{30}\) of the job.
Together, in \(\displaystyle x\) hours, they did \(\displaystyle \,\frac{x}{20}\,+\,\frac{x}{30}\) of the job.
Then Mark worked alone for 3 hours; he did \(\displaystyle \frac{3}{20}\) of the job alone.
The sum of this work is one (1) job: \(\displaystyle \:\frac{x}{20}\,+\,\frac{x}{30}\,+\,\frac{3}{20}\;=\;1\;\)
. . . there!
