The problem is as follows:<\/p>\n
Joe and Bill are scene painters for a theatre set. \u00a0Joe can paint the scene background by himself in 9 hours; Bill can do it by himself in 12. \u00a0How long will they take if they work together?<\/em><\/p>\n 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.<\/p>\n Add their hourly rates together to get their combined rate:<\/p>\n 1\/9+1\/12=4\/36+3\/36=7\/36<\/p>\n Therefore, working together, they complete 7\/36 of the job each hour. The time they will take to complete the one job is<\/p>\n 1\/(7\/36)=1÷7\/36=1*36\/7=36\/7 hours = 5 and 1\/7 hours<\/p>\n 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<\/sup>\u20442<\/sub> 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<\/sup>\u20442<\/sub> and 6 hours. Our answer of 51<\/sup>\u20447<\/sub> sounds reasonable enough, from that point of view.<\/p>\n