Pipes and Swimming Pools

Everybody knows the famous water pipe question - two pipes fill a swimming pool in a certain time, so find how much time each pipe needs to fill the pool. Well, here is one question of this type.

Two pipes fill a swimming pool in 11 1/9 hours (eleven hours and one ninth of an hour) together. One pipe can fill the pool in 5 hours less than the other pipe. Find out how much times it takes each pipe to fill up the swimming pool separately.

First, as you know, we need variables. Let x be the time it takes the faster pipe to fill up the swimming pool. This means it takes the slower pipe x + 5 hours to fill up the swimming pool.

Since it takes the first pipe x hours to fill up the pool, each hour it fills 1/x of the pool. For the other pipe, the rate is 1/(x + 5) per hour. Since they fill up the pool in 100/9 hours together (I changed it into the more comfortable fraction form), let's multiply each rate by 100/9 and add them up to 1 (100% of the pool):
100/9x + 100/(9x + 45) = 1

Multiply by the lowest common denominator: 9x(x + 5):
100(x + 5) + 100x = 9x(x + 5)
100x + 500 + 100x = 9x² + 45x
9x² - 155x - 500 = 0
(x - 20)(9x + 25) = 0
x = 20, -25

Since x is a time, it cannot be negative, so it must be 20.

The fast pipe fills the pool in 20 hours. The slower one fills the pool in 25 hours.

Hope you liked it,
Nadav

nadavs