Math: an assessment style problem

Tutoring math, you work with people heading back to school who need to pass assessment tests.  The math tutor demonstrates a problem they often ask about.

The problem is as follows:

Joe and Bill are scene painters for a theatre set.  Joe can paint the scene background by himself in 9 hours; Bill can do it by himself in 12.  How long will they take if they work together?

First, you realize that if Joe can paint the scene in 9 hours, he must work at the rate of 1/9 of the scene each hour. Similarly, if Bill can paint the scene in 12 hours, he must complete 1/12 of the scene per hour.

Add their hourly rates together to get their combined rate:

1/9+1/12=4/36+3/36=7/36

Therefore, working together, they complete 7/36 of the job each hour. The time they will take to complete the one job is

1/(7/36)=1÷7/36=1*36/7=36/7 hours = 5 and 1/7 hours

To check if the answer is reasonable, imagine the following: First, if Bill was as fast as Joe – i.e., they each could finish the scene, working solo, in 9 hours – then it makes sense that working together, they’d take half the time – i.e., 412 hours. Then again, if they both worked at Bill’s pace, either of them needing 12 hours to finish the job solo, then working together, they’d take half the time, or 6 hours. So if one works at the nine hour rate, the other at the twelve hour rate, and they work together, they take somewhere between 412 and 6 hours. Our answer of 517 sounds reasonable enough, from that point of view.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

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