The bath with the plug out

Question

With the plug in, the taps fill the bath in 3 minutes. With the taps off, the drain empties a full bath in 5 minutes.

If the taps are on and the plug is out, how long does it take to fill the bath from empty? Then explain what would happen if the drain emptied it in 3 minutes rather than 5.

Accepted Answer

Seven and a half minutes. Rates. Taps: 1/3 of a bath per minute. Drain: 1/5 of a bath per minute. Net filling rate: 1/3 minus 1/5 = 5/15 minus 3/15 = 2/15 of a bath per minute. Time. One full bath at 2/15 per minute takes 15/2 = 7.5 minutes. If the drain matched the taps. With the drain emptying in 3 minutes, the net rate would be 1/3 minus 1/3 = 0: the water level would never rise, and the bath would never fill. With a drain faster than the taps, the bath cannot even start to fill from empty. The formula ab/(b minus a), with a the filling time and b the draining time, gives the filling time …

The bath with the plug out

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