Question 1208808
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The fraction of the tub filled in 1 minute by the hot water tap alone is 1/18.<br>
The fraction of the tub filled in 1 minute by both taps together is 1/18 + 1/12 = 2/36 + 3/36 = 5/36.<br>
The fraction of the tub drained by the drain in 1 minute is 1/9.<br>
When both taps and the drain are all open, the fraction of the tub filled in 1 minute is 5/36 - 1/9 = 5/36 - 4/36 = 1/36.<br>
The hot water tap is run for 8 minutes.  The fraction of the tub filled is 8(1/18) = 8/18 = 4/9.<br>
Both taps are run for 2 minutes.  The fraction of the tub filled in those 2 minutes is 2(5/36) = 10/36 = 5/18.  The fraction of the tub now filled is 4/9 + 5/18 = 8/18 + 5/18 = 13/18; the fraction of the tub remaining to be filled is 5/18.<br>
The fraction of the tub remaining to be filled is 5/18; the rate at which the tub is being filled with both taps and the drain all open is 1/36.  The number of minutes needed to finish filling the tank after the drain is opened is (5/18)/(1/36) = (10/36)/(1/36) = 10.<br>
ANSWER: 10 minutes<br>
CHECK:
The hot water tap will have been running for 8+2+10 = 20 minutes; the cold water tap for 2+10 = 12 minutes; and the drain will have been open for 10 minutes.  The tub should be full at the end of that time -- i.e, the fraction of the tub that should be full is 1.<br>
20(1/18) + 12(1/12) - 10(1/9) = 20/18 + 1 - 10/9 = 1<br>