Question 1202892
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The response from the other tutor solves the problem by choosing a "nice" number for the number of gallons in the full lake.  That of course is a valid way of solving the problem.<br>
If you wanted a formal algebraic solution, it might look something like this:<br>
Let x be the number of gallons in the full lake.  Then, according to the given information,<br>
(1/12)x = gallons released in 1 week
(1/20)x = gallons filled by the river in 1 week<br>
The number of gallons in the lake decreases by the difference between those two amounts.<br>
{{{(1/12)x-(1/20)x = (5/60)x-(3/60)x = (2/60)x = (1/30)x}}}<br>
Since the capacity of the lake in gallons is x, and the number of gallons in the lake is decreasing by (1/30)x each week, the number of weeks needed to empty the lake is 30.<br>
ANSWER: 30<br>
And, if formal algebra is not required, here is an informal method for solving this kind of problem that avoids working with fractions.<br>
Consider the least common multiple of the two rates for filling and emptying the lake.  The least common multiple of 12 and 20 is 60.<br>
In 60 weeks, the lake could be emptied 60/12 = 5 times by releasing water; in those 60 weeks, the lake could be filled by the river 60/20 = 3 times.<br>
That means that if water is flowing into and out of the lake at the same time, in 60 weeks the lake could be emptied 5-3 = 2 times.<br>
Then, since the lake would be emptied twice in 60 weeks, the number of weeks needed to drain the lake (once) is 60/2 = 30.<br>