Nice Question. Thanks for posting.
Figure out what will happen in 1 hour.
The input pipe will fill 1/4 of a tank if left to it's own devices. You can just reason this. It he fills 1 tank in 4 hours it stands to reason that 1 tank / 4 hours will be what is filled up in 1 hour.
The problem is, there is a pipe that takes the water out. It takes 6 hours to empty the tank.
So in one hour, the tank will empty out 1 tank / 6 hours or 1/6 of a tank.
The difference is how much the tank will actually get filled in 1 hour.
1/4 - 1/6 = 3/12 - 2/12 = 1/12. So every hour, 1/12 of the tank will get filled. Now you can set up a proportion.
1 hour : (1/12) tank = x: 1 tank Set up the proportion.
1/(1/12) = x / 1 Cross multiply
1*1 = (1/12) * x Combine the left
1 = (1/12)x Multiply both sides by 12
12 * 1 = 1 * x
x = 12 hours.
You can do this question with decimals. You have to take it out 4 places to get close to 12.
Net = In - out
Net = 1/4 - 1/6
Net = 0.25 - 0.16667
Net = 0.083333 This is how much you will get in during an hour.
1 hour / 0.08333 tank = x hours / 1 tank Cross multiply
1 tank hour = 0.083333 tanks x divide by 0.083333
1/0.083333 = x
This is good enough to give 12 = x
Answer: 12 hours to fill the tank under the given conditions.