Valen uses two pipes to fill a pool. The first pipe can fill the pool in 6 h. The second pipe can fill the pool in 8 h. Three hours after both pipes are turned on, Billy accidentally opens the pipe which can drain the pool completely in 12 h. With the 3rd pipe now remained open, what is the total time taken by Valen to fill the pool completely. Express your answer in h.
Let the pool be 1 P.
The rate of the first pipe per hour= ¹₆
The rate of the second pipe per hour= ¹₈
The rate of the third pipe per hour= ¹₁₂

Fraction of the pool filled by the 1st pipe and the 2nd pipe in two hours
= (¹₆ + ¹₈) x 3
= (⁴₂₄ + ³₂₄) x 3
= ⁷₂₄ x 3
= ²¹₂₄

Fraction of the remaining pool needed to be filled
= 1 - ²¹₂₄
= ³₂₄

Rate of filling up the pool per hour when 3rd pipe is open
= (1st pipe + 2nd pipe) - 3rd pipe
= ⁷₂₄ - ¹₁₂
= ⁷₂₄ - ²₂₄
= ⁵₂₄

Time taken to fill up the remaining pool
= ³₂₄ ÷ ⁵₂₄
= ³₂₄ x ²⁴₅
= ⁷²₁₂₀ h

Time taken to fill the pool
= ⁷²₁₂₀ + 2
= 2.6 h

Answer(s): 2.6 h