Two taps can fill a cistern in 2 hours and 52 hours, respectively. A third tap…

Concept

For taps/pipes acting on a tank together, each tap's own rate = capacity ÷ its own time. A filling tap adds its rate as positive; an emptying tap adds its rate (computed the same way, from its own time) as negative. The combined net rate is the sum of these individual signed rates, and the time to complete the job together is capacity ÷ (net rate).

Application

1. Take the cistern capacity as the LCM of the given times, 2 and 52, i.e. 52 units — this makes every individual rate a whole number.

2. Tap A fills the cistern in 2 hours, so its rate = 52 ÷ 2 = 26 units/hour.

3. Tap B fills the cistern in 52 hours, so its rate = 52 ÷ 52 = 1 unit/hour.

4. Tap C empties the cistern in 52 hours, so its rate (draining) = -(52 ÷ 52) = -1 unit/hour.

5. Combined net rate when all three are opened together = 26 + 1 - 1 = 26 units/hour.

6. Time to fill the full 52-unit cistern at this net rate = 52 ÷ 26 = 2 hours.

Cross-check

Redo it with plain fractions of the cistern per hour: Tap A = 1/2, Tap B = 1/52, Tap C (emptying) = -1/52. Net rate = 1/2 + 1/52 - 1/52 = 1/2 per hour, so time = 1 ÷ (1/2) = 2 hours — the same result. This confirms the general pattern here: Tap B and Tap C both act over exactly 52 hours in opposite directions, so their rates cancel exactly, leaving only Tap A's own fill time to determine the answer.

Result

The empty cistern gets filled in 2 hours when all three taps are opened together.