Question 1208808
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A tub can be filled with hot water in 18 minutes, or with cold water in 12 minutes. The tub can also be emptied when the plug is pulled out, in 9 minutes. Farley ran the hot water in his tub for 8 minutes, then both the cold and hot water together for 2 minutes, and then with the two taps still running, he accidentally pulled the plug. How long did it take, in minutes, for the tub to fill after the plug was pulled?

Time tub can be filled with hot water: 18 minutes, which means that {{{1/18}}} of tub can be filled with hot water in 1 minute

Time tub can be filled with cold water: 12 minutes, which means that {{{1/12}}} of tub can be filled with cold water in 1 minute

With hot water being run for 8 minutes, and both hot and cold water for 2 minutes, fraction of tub thatw's filled with hot
and cold water in those 10 minutes = {{{matrix(3,1, 8(1/18) + 2(1/18 + 1/12), 8/18 + 2/18 + 2/12, 10/18 + 2/12)}}}
                                           {{{matrix(1,5, 20/36 + 6/36, "=", 26/36, "=", 13/18)}}}
{{{13/18}}} of tub was filled in the 8 minutes that the hot water ran for, alone, and the 2 minutes thereafter that both hot and cold ran, together

Therefore, {{{matrix(1,3, 1 - 13/18, "=", 5/18)}}} of tub was left to be filled

Time tub can be emptied after plug is pulled: 9 minutes, which means that {{{1/9}}} of tub can be EMPTIED in 1 minute

Now, with the plug being pulled, and fill-time being T, we get: {{{matrix(4,3, (1/18 + 1/12 - 1/9)/(5/18), "=", 1/T, (2/36 + 3/36 - 4/36)/(5/18), "=", 1/T,
(1/36)(18/5), "=", 1/T, 1/10, "=", 1/T)}}}

Cross-multiplying, we get the <font color = red><font size = 4><b>TIME taken to fill the tub, after the plug was pulled,</font></font></b> or T = <font color = red><font size = 4><b>10 minutes</font></font></b>.</pre>