Two taps and a drain

Question

An empty pool can be filled by tap A alone in 2 hours, or by tap B alone in 3 hours. The drain, if opened, empties a full pool in 6 hours.

If both taps and the drain are all opened at once, how long does it take to fill the pool from empty?

Accepted Answer

One and a half hours. Rates. Tap A fills 1/2 of a pool per hour. Tap B fills 1/3 per hour. The drain removes 1/6 per hour. Together the net rate is 1/2 + 1/3 minus 1/6 = 3/6 + 2/6 minus 1/6 = 4/6 = 2/3 of a pool per hour. Time. A full pool at 2/3 per hour takes 3/2 hours, 90 minutes. Check. In 1.5 hours tap A delivers 3/4 of a pool, tap B delivers 1/2, and the drain removes 1/4: 3/4 + 1/2 minus 1/4 = 1. Full. The rule. Whenever several things work at once, add their rates (amount per unit time) and then invert to get the time; never add or average the times themselves. If you had averaged 2, …

Two taps and a drain

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