Pipes filling and draining a tank

CAT 2021 Slot 2 · QA · Medium · Pipes & Cisterns

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Two pipes A and B are attached to an empty water tank. Pipe A fills the tank while pipe B drains it. If pipe A is opened at 2 pm and pipe B is opened at 3 pm, then the tank becomes full at 10 pm. Instead, if pipe A is opened at 2 pm and pipe B is opened at 4 pm, then the tank becomes full at 6 pm. If pipe B is not opened at all, then the time, in minutes, taken to fill the tank is

  1. A.

    140

  2. B.

    120

  3. C.

    144

  4. D.

    264

Answer

C

Explanation

Let filling rate of A be aa tank/hour and draining rate of B be bb tank/hour.

Case 1: A works from 2 pm to 10 pm (8 hours), B works from 3 pm to 10 pm (7 hours). 8a−7b=18a - 7b = 1

Case 2: A works from 2 pm to 6 pm (4 hours), B works from 4 pm to 6 pm (2 hours). 4a−2b=14a - 2b = 1

Multiply Case 2 by 2: 8a−4b=28a - 4b = 2

Subtract Case 1 from this: (8a−4b)−(8a−7b)=2−1  ⟹  3b=1  ⟹  b=13(8a - 4b) - (8a - 7b) = 2 - 1 \implies 3b = 1 \implies b = \frac{1}{3}

Substituting b=13b = \frac{1}{3} into 4a−2b=14a - 2b = 1: 4a−23=1  ⟹  4a=53  ⟹  a=5124a - \frac{2}{3} = 1 \implies 4a = \frac{5}{3} \implies a = \frac{5}{12}

Time taken by pipe A alone = 1a=125\frac{1}{a} = \frac{12}{5} hours =125×60=144= \frac{12}{5} \times 60 = 144 minutes.

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