Water tank problem

Question

There is an empty tank that has a hole in it. Water can enter the tank at the rate of 1 gallon per second. Water leaves the tank through the hole at the rate of 1 gallon per second for each 100 gallons in the tank. How long, in seconds, will it take to fill the 50 gallons of water.

Answer 1

It is unclear if the water leaves the tank UNTIL there is 100 gallons.

If so, then it will take exactly 50 seconds to fill the tank with 50 gallons

at a rate of 1 gallon per second, and there is no change of any water to escape.

However, if the 1/100 ratio holds regardless of the amount of water in

the tank, then the formula becomes much more complicated.

At ANY point in time there is t - (1/100)t = (99/100)t gallons in the tank.

(99/100)t = 50

t = 50*100/99 = 5000/99 = 50 and 50/99 = 50.5050505050505050.... seconds

or just of 50 and 1/2 seconds

the extra half second is needed to replace the little bit of water that escapes