th of the tank in 6 minutes. A third set (Z) of 16 pipes can empty half of the tank in 20 minutes. If half of the pipes of set X are closed and only half of the pipes of set Y are open, and all pipes of the set (Z) are open, then how long will it take to fill 50% of the tank?Correct Option
The problem provides information about three sets of pipes (X, Y, Z) with given capacities to fill or empty a tank, and asks for the time to fill 50% of the tank under modified conditions of pipe usage.
- Set X: 20 pipes fill 70% of the tank in 14 minutes. Combined rate = 0.70 ÷ 14 = 0.05 tank/min; rate per pipe = 0.05 ÷ 20 = 0.0025 tank/min.
- Set Y: 10 pipes fill 3/8 of the tank in 6 minutes. Combined rate = 0.375 ÷ 6 = 0.0625 tank/min; rate per pipe = 0.0625 ÷ 10 = 0.00625 tank/min.
- Set Z: 16 pipes empty half the tank in 20 minutes. Combined emptying rate = 0.5 ÷ 20 = 0.025 tank/min; rate per pipe = 0.025 ÷ 16 = 0.0015625 tank/min.
- Under new conditions: half of X’s pipes are open (10 pipes) ⇒ filling rate = 10 × 0.0025 = 0.025 tank/min; half of Y’s pipes are open (5 pipes) ⇒ filling rate = 5 × 0.00625 = 0.03125 tank/min; all Z pipes are open (16 pipes) ⇒ emptying rate = 16 × 0.0015625 = 0.025 tank/min.
- Net filling rate = 0.025 + 0.03125 − 0.025 = 0.03125 tank/min.
- Time to fill 50% of the tank = 0.50 ÷ 0.03125 = 16 minutes.
Incorrect Options
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Option 1: 8 minutes
This underestimates the time required. It ignores the effect of the Z pipes emptying the tank, which reduces the net filling rate.
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Option 2: 10 minutes
This is incorrect as it does not account for the reduced number of X and Y pipes and the emptying contribution of Z. The actual net rate is lower, requiring more time.
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Option 3: 12 minutes
This overestimates the net filling rate. With the correct net rate, more time is needed to fill 50% of the tank, making 12 minutes insufficient.