Multiple choice

Pipes A, B and C can fill a cistern in 6 hr, 12 hr and 18 hr, respectively. Another pipe D can empty it in 15 hr. In which of the following cases will the tank be filled in the least time?

  1. If B alone is opened.

  2. If A and D are opened together.

  3. If A, B and D are opened.

  4. If B, C and D are opened.

  5. If A, C and D are opened.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Rates (per hour): A=1/6, B=1/12, C=1/18, D=-1/15. A: 1/6 (rate 0.166). B: 1/12 (rate 0.083). A+D: 1/6 - 1/15 = 5/30 - 2/30 = 3/30 = 1/10 (rate 0.1). A+B+D: 1/6 + 1/12 - 1/15 = 10/60 + 5/60 - 4/60 = 11/60 (rate 0.183). B+C+D: 1/12 + 1/18 - 1/15 = 15/180 + 10/180 - 12/180 = 13/180 (rate 0.072). A+C+D: 1/6 + 1/18 - 1/15 = 15/90 + 5/90 - 6/90 = 14/90 (rate 0.155). The highest rate is A+B+D (11/60), which fills the tank in the least time.

AI explanation

Using the least common multiple method, let the capacity of the cistern be 180 units. The filling rates for pipes A, B, and C are 30, 15, and 10 units per hour respectively, while the emptying rate for pipe D is 12 units per hour. Checking the given cases, opening A and B together with D gives a net rate of 30 plus 15 minus 12, which is 33 units per hour. This results in a filling time of 180 divided by 33, or approximately 5.45 hours. Comparing this to the other options, which take 6, 10, or 12 hours or more, this case provides the least time. The answer is If A, B and D are opened.

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