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., 41⁄2 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 41⁄2 and 6 hours. Our answer of 51⁄7 sounds reasonable enough, from that point of view.
Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.
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