# Filling Pool with Fluids

A common word problem in a science fiction guise:

Creatures from a nearby dimension penetrate our reality and want to fill a large swimming pool with their lime-scented nutritional fluid. From Hose A pours a green slime that would, by itself, take 30 minutes to fill the pool. From Hose B surges a crimson slime that, by itself, would take 20 minutes to fill the pool. How long would it take to fill the pool if both hoses poured at the same time? The police will arrive in 15 minutes. If the creatures can fill the pool in under 15 minutes, they will deposit their spores in the liquid, multiply at fantastic rates, and take over Earth. Will the creatures succeed in their plan for world conquest?

Solution

### References

1. C. A. Pickover, A Passion for Mathematics, John Wiley & Sons, Inc. 2005

### Solution

Creatures from a nearby dimension penetrate our reality and want to fill a large swimming pool with their lime-scented nutritional fluid. From Hose A pours a green slime that would, by itself, take 30 minutes to fill the pool. From Hose B surges a crimson slime that, by itself, would take 20 minutes to fill the pool. How long would it take to fill the pool if both hoses poured at the same time? The police will arrive in 15 minutes. If the creatures can fill the pool in under 15 minutes, they will deposit their spores in the liquid, multiply at fantastic rates, and take over Earth. Will the creatures succeed in their plan for world conquest?

### Solution 1

If only Hose A is open, the portion of the pool filled by Hose A in time T is T/30,T/30,30/T,30T,T+30. The portion of the pool being filled by Hose B in time T is T/20,20/T,20T,T/20,T+20. The two hoses together in time T will fill T/30 + T/20 of the pool. When will this become the whole of the pool, i.e., when shall we have T/30 + T/20,T/(30 + 20),2T/(30+20),T/30 + T/20,? This is an equation for time T we are about to solve.

There are several ways to proceed. One is to multiply the equation by 60,20,30,60: 2T + 3T = 60, or 5T = 60, T = 12.

### Solution 2

In, say, 1 hour, Hose A can fill 2 pools, while Hose B can fill 3 pools. Working alongside each other the two hoses will fill 5,5,2,3,6 pools in 60 minutes (which is 1 hour, right?). It follows that the two hoses will fill a single pool in 1/5th,1/5th,half,1/3rd,1/6th of the time, i.e., 60/5 = 12 minutes.