Two taps can fill a cistern in 2 hours and 37 hours, respectively. A third tap can empty it in 37 hours. How long (in hours) will…

Question

Two taps can fill a cistern in 2 hours and 37 hours, respectively. A third tap can empty it in 37 hours. How long (in hours) will it take to fill half of the empty cistern if all the taps are opened together?

1. A. 3
2. B. 1 (Correct answer)
3. C. 2
4. D. 4

Correct Answer

Option B — 1

Detailed Solution & Explanation

The correct answer is 1.

Shortcut Trick

• The second tap fills at $\dfrac{1}{37}$ per hour and the third empties at $\dfrac{1}{37}$ per hour — they cancel exactly.
• Only the first tap has any net effect: $\dfrac{1}{2}$ per hour, filling the cistern in 2 hours.
• Half the cistern therefore takes 1 hour.

Alternate Method

• Net rate $= \dfrac{1}{2} + \dfrac{1}{37} - \dfrac{1}{37} = \dfrac{1}{2}$ per hour.
• Time for a full cistern $= 2$ hours; time for half $= \dfrac{2}{2} = \textbf{1}$ hour.

Additional Information

• Spot the cancellation before computing. The equal fill and empty times of 37 hours are placed there deliberately; a candidate who starts finding a common denominator wastes a minute reaching the same result.
• Sign convention: inlets are positive, outlets negative, and the cistern fills only if the net rate is positive. Had the outlet been faster, the cistern would never fill.
• Note the question asks for half the cistern, not the whole — an easy final-step slip after the arithmetic is done.