Bunuel wrote:

At a community center, three separate pumps- A, B, and C- started to fill an empty swimming pool at 8:00 AM, and each pump worked at its own constant rate until the pool was full. Was the pool completely filled at 3:00 PM?

(1) Working alone, pump C could fill 3/10 of the pool in 126 minutes.

(2) Working together, pumps A and B could fill 1/8 of the pool in 55 minutes.

For word problems in general, and rate problems specifically, translate them into math terms as best you can before you continue.

In this problem, we have three pumps working together. So, we'll add their rates together to get an overall rate.

The problem is really an inequality problem! It doesn't ask you to find out exactly when the pool will be filled. It just asks whether it'll be filled by 3pm. 8am to 3pm is 7 hours, so the question is really asking:

Will the pool be filled within 7 hours?

It looks like I'll have to work with rates whether I'd like to or not! That's what you need to do if you have multiple people (or pumps) working together. So, I'll actually phrase the question in terms of rates:

Is the combined rate at least 1/7 pool per hour?

"Pool per hour" is a bit of a weird way to think about a rate, but it's correct here!

Now,

statement 1: This will give us pump C's rate. In some situations, I'd just assume this was insufficient.

However, this is an inequality problem. Knowing C's rate won't give me the exact rate, sure. But, what if C works REALLY fast - fast enough to fill the pool completely on its own? In that case, this statement actually could be sufficient. So, we have to check.

C's rate is (3/10 pool)/(126 minutes). 126 minutes is very close to 2 hours - can we estimate?

(3/10 pool)/2 hours = 3/20 pool per hour.

Hold on: 3/20 is actually greater than 1/7. (I recognize this because 3/21 = 1/7, and 3/20 has a smaller denominator, so it's bigger.) Is C's rate actually bigger than 1/7 by itself?

If I'm in a hurry, I might just assume that it is. But if I'm not, then I'm going to recall that I estimated. I'll go back and get an exact solution.

126 minutes = 2 hours + 6 minutes = 2 and 1/10 hours, or 21/10 hours

C's rate is (3/10 pool)/(21/10 hours) = 3/21 = 1/7 pool per hour.

Whoa - C would exactly fill the pool up on its own. So, this statement is (barely) sufficient! The pool will certainly be full, no matter what A and B are doing. Eliminate B, C, and E.

Statement 2: We'll approach it similarly. Pumps A and B have a rate of (1/8 pool)/(55 min), or just a bit more than 1/8 of the pool per hour. That's close enough to 1/7 that I'd like to get an exact answer, just to be safe.

55 minutes = 1 hour - 5 minutes = 1 hour - 1/12 hour = 11/12 hours

Rate = (1/8 pool)/(11/12 hours) = 12/88 pool per hour = 3/22 pool per hour.

This is

less than 1/7. So, we know that A and B can

almost fill the pool on their own. But if C was REALLY slow, A and B still couldn't quite do it. If C was fast, it would be fine. Since we don't know for sure, this statement must be insufficient.

The answer is A.

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