Final answer:
The total time taken to fill the tank is 16 hours, considering the initial combined rate of both pipes and the reduced rate after the leak develops.
Step-by-step explanation:
When solving problems involving rates of work, such as two pipes filling a tank, a common approach is to determine the rate of each pipe and then combine these rates to find the total time taken to fill the tank. In this scenario, one pipe fills the tank in 20 hours and the other in 30 hours. We can consider the rate of the first pipe as 1/20 tank per hour and the second as 1/30 tank per hour.
When both pipes work together without the leak, their combined rate is (1/20 + 1/30) tank per hour, which simplifies to 1/12 tank per hour. Without the complication of the leak, the tank would be full in 12 hours. However, when the tank is 1/3 full, a leak develops that causes 1/3 of the water to leak out. This essentially negates the effect of one of the pipes, as only 2/3 of their combined effort goes into filling the tank.
To calculate the new effective rate, only 2/3 of the combined rate is used, which is (2/3) x (1/12) = 1/18 tank per hour. To fill the remaining 2/3 of the tank, at this new rate, it will take (2/3 tank) / (1/18 tank per hour) = 12 hours. As it took 4 hours to fill the first 1/3 (at a rate of 1/12 tank per hour), the total time taken to fill the tank is 16 hours.