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"Work" Word Problems (page 4 of 4)

• Working together, Bill and Tom painted a fence in 8 hours. Last year, Tom painted the fence by himself. The year before, Bill painted it by himself, but took 12 hours less than Tom took. How long did Bill and Tom take, when each was painting alone?

Well, Bill can do it in "b" hours, so he does ¹/b per hour. Tom does it in "t" hours, so he does ¹/t per hour. Then, working together, they can do ¹/b + ¹/t = ¹/₈ per hour, since they took 8 hours. I also know that Tom takes 12 hours longer than Bill, so t = b + 12. Then I get ¹/b + ¹/(b + 12) =  ¹/₈. That is:

hours to complete job:
  Bill: b
  Tom: b + 12
  together: 8

completed per hour:
  Bill: ¹/b
  Tom: ¹/(b + 12)
  together: ¹/₈

adding their labor:

¹/b + ¹/(b + 12) = ¹/₈

To solve for "b", I'll multiply through by 8b(b + 12) to get rid of all of the denominators:

8(b + 12) + 8(b) = (b)(b + 12)
8b + 96 + 8b = b² + 12b
16b + 96 = b² + 12b
0 = b² – 4b – 96
(b – 12)(b + 8) = 0

(Review how to factor quadratics, if you're not sure how I just got to that last line.)

Then b = 12 and b = –8. Of course, for the purposes of our problem, b must be positive, so we'll ignore the "b = –8" solution; it's "extraneous" (pronounced "ek-STRAY-nee-uss", meaning "valid mathematically, but pointless as far as our situation is concerned").

Then, since b = 12, then Bill takes twelve hours. Tom takes twelve hours longer than Bill, so:

Bill takes 12 hours, and Tom takes 24 hours.

This next one is a bit different: Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

• Ben takes 2 hours to wash 500 dishes, and Frank takes 3 hours to wash 450 dishes. How long will they take, working together, to wash 1000 dishes?

For this exercise, we are given how many can be done in one time unit, rather than how much of a job can be completed. But the thinking process is otherwise the same.

Ben can do 250 dishes per hour, and Frank can do 150 dishes per hour. Working together, they can do 250 + 150 = 400 dishes an hour. That is:

Ben: 500 dishes / 2 hours = 250 dishes / hour
Frank: 450 dishes / 3 hours = 150 dishes / hour
together: (250 + 150) dishes / hour = 400 dishes / hour

Now that I have their hourly rate, I have to find the number of hours that it takes to wash 1000 dishes. Set things up so units cancel and you're left with "hours":

(1000 dishes) × (1 hour / 400 dishes)
   = (1000 / 400) hours
   = 2.5 hours

It will take two and a half hours for the two of them to wash 1000 dishes.

• If six men can do a job in fourteen days, how many would it take to do the job in twenty-one days?

Convert this to man-hours, or, in this case, man-days. If it takes six guys fourteen days, then:

(6 men) × (14 days) = 84 man-days

That is, the entire job requires 84 man-days. This exercise asks you to expand the time allowed from fourteen days to twenty-one days. Obviously, if they're giving you more time, then you'll need fewer guys. But how many guys, exactly?

(x guys) × (21 days) = 84 man-days

...or, in algebra:

21x = 84

x = 4

So you'll need four guys to do the job in twenty-one days.

You may have noticed that each of these problems used some form of the "how much can be done per time unit" construction, but other than that, each problem was done differently. That's how "work" problems are. You'll have to be alert and clever to do these. But as you saw above, if you label things neatly and do your work orderly, you should find your way to the solution.

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