The Prodigal Son

Topics: Trigraph, The New Guy, Gh Pages: 15 (4530 words) Published: June 19, 2013
Work problems
Suppose one painter can paint the entire house in twelve hours, and the second painter takes eight hours. How long would it take the two painters together to paint the house? f the first painter can do the entire job in twelve hours and the second painter can do it in eight hours, then (this here is the trick!) the first guy can do 1/12 of the job per hour, and the second guy can do 1/8 per hour. How much then can they do per hour if they work together? To find out how much they can do together per hour, I add together what they can do individually per hour: 1/12 + 1/8 = 5/24. They can do 5/24 of the job per hour. Now I'll let "t" stand for how long they take to do the job together. Then they can do 1/t per hour, so 5/24 = 1/t. Flip the equation, and you get that t = 24/5 = 4.8 hours. That is: Hours to complete job:

  first painter: 12
  second painter: 8
  together: t
Completed per hour:
  first painter: 1/12
  second painter: 1/8
  together: 1/t
Adding their labor: 
1/12 + 1/8 = 1/t
5/24 = 1/t
24/5 = t
They can complete the job together in just less than five hours. As you can see in the above example, "work" problems commonly create rational equations. But the equations themselves are usually pretty simple. One pipe can fill a pool 1.25 times faster than a second pipe. When both pipes are opened, they fill the pool in five hours. How long would it take to fill the pool if only the slower pipe is used? Convert to rates:

Hours to complete job:
  fast pipe: f
  slow pipe: 1.25f
  together: 5
Completed per hour:
  fast pipe: 1/f
  slow pipe: 1/1.25f
  together: 1/5
Adding their labor:
1/f + 1/1.25f = 1/5
Multiplying through by 5f:
5 + 5/1.25 = f 
5 + 4 = f = 9
Then 1.25f = 11.25, so the slower pipe takes 11.25 hours. When the tub faucet is on full, it can fill the tub to overflowing in 20 minutes (we'll ignore the existence of the overflow drain). The drain can empty the tub in 15 minutes. Your four-year-old has managed to turn the faucet on full, and the drain was closed. Just as the tub starts to overflow, you run in and discover the mess. You grab the faucet handle, and it comes off in your hand, leaving the water running at full power. You yank the drain open, and run for towels to clean up the overflow. How long will it take for the tub to empty, with the faucet still on but the drain now open? Okay, yeah; in "real life" you'd go find the shut-off valve and turn off the water to the whole house, but this is math, not real life. Thinking about this problem, we see that the drain can empty 1/15of the tub per minute. The faucet can fill 1/20 of the tub per minute. Then, working together, they can empty 1/15 – 1/20 of the tub per minute. The subtraction indicates that the faucet is actually working against the drain.  Again, let "t" indicate how long it takes to drain the whole tub. Then 1/t is drained per minute. Then 1/15 – 1/20 = 1/t, so 1/60 = 1/t, and t = 60. That is: Minutes to complete job:

  faucet: 20 minutes to fill
  drain: 15 minutes to empty
  together: t minutes to empty
Completed per minute:
  faucet: 1/20 filled
  drain: 1/15 emptied
  together: 1/t emptied
Adding (or, in this case, subtracting) their labor:
1/15 – 1/20 = 1/t
1/60 = 1/t
60 = t
Recall that the time "t" is defined in minutes, so "60" is "60 minutes", or "1 hour". It will take an hour to drain the tub.
Another "typical" work problem is the "one guy did part of the job" type. It might run something like this: Two mechanics were working on your car. One can complete the given job in six hours, but the new guy takes eight hours. They worked together for the first two hours, but then the first guy left to help another mechanic on a different job. How long will it take the new guy to finish your car? The first guy can do 1/6 per hour. The new guy can do 1/8 per hour. Together, they can do 1/6 +1/8 = 7/24 per hour.  That is: Hours to complete job:

  first guy: 6
  new guy: 8
  together: t...
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