Working together, two pumps can drain a certain pool in 3 hours. If it takes the older pump 13 hours to drain the pool by itself,

Question

Working together, two pumps can drain a certain pool in 3 hours. If it takes the older pump 13 hours to drain the pool by itself, how long will it take the newer pump to drain the pool on its own?

 

Answer 1

Oh, yes, one of these! It always seems nasty until you know the trick, and then it's fairly easy!

Basically, there are three different factors at work, here: work, rate, and time, and they all like to hang out in the same formula:

Work = Rate * Time  In each of the cases they described, the "work" is draining pools, and in each case, you drain only one. You are also given the time, which means you didn't know the rates. Simple! Divide the work (1) by the respective times to get the rates:

Older pump: 1 / 13

Together: 1/3

New Pump: 1 / X

If they work together, they will still be draining one pool, so you aren't going to add work. Adding times would imply that the more pumps you have, the slower the job, and that isn't right either. What you can add (and use for an equation) is the rates:

Old + New = together

1/13 + 1/X = 1/3

My favorite way to solve this is by clearing fractions with the LCD, in this case 39X:

39X [ 1/13 + 1/X = 1/3]

3X + 39 = 13X

-3X            -3X

         39   = 10X

     39 / 10= 10X / 10

        3.9 = X

And there you have it! We're just adding the reciprocals of the times, clearing fractions and then solving normally.