Filling/Emptying a Pond The Miller's backyard pond holds $120$ gallons of water and is draining at a rate of $6$ gallons per minute. Little Suzy Miller, sad to see her fish pond emptying, is trying to refill the pond by pouring in a one-gallon bucket of water each minute. How long will it take for the pond to be half empty? * * * I'm not really good with these sort of filling/refilling algebraic problems, but I think there is a simple formula for this, more specifically, we have $$\frac {1}{x}-\frac {1}{y}=\frac {1}{z}$$ But I'm not too sure if this is the correct formula!

Here is how I would calculate this:

Half empty is at 60 gallons. Because Suzy's adding a gallon a minute, the whole pond is really only draining at 5 gallons a minute. Therefore, we can simply do 60/5 to get the number of minutes for the pond to be empty. It will take 12 minutes for half the pond to drain (and 24 for the whole pond to drain).

As Ethan Bolker says in the comments below, it is sometimes best not to rely on formulas, but to instead just think about the problem. That will reduce the amount of memorizing you have to do and the amount of worrying you will do over which formula and whether it was applied correctly. Besides, it can sometimes be a bit simpler.

Hope this helps!