Question 1202892: Officials begin to release water from a full man-made lake at a rate that would empty the lake in 12
weeks, but a river that can fill the lake in 20
weeks is replenishing the lake at the same time. How many weeks does it take to empty the lake? Express your answer as a fraction reduced to lowest terms, if needed.
Found 2 solutions by math_tutor2020, greenestamps:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Answer: 30 weeks
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Explanation:
The capacity of the lake does not matter.
Let's say the capacity is 240 thousand gallons.
I picked this value because 12*20 = 240. Then I tacked on "thousand" to make the lake capacity seem more realistic.
The lake can be fully drained in 12 weeks.
The drain rate is 240/12 = 20 thousand gallons per week.
The fill rate is 240/20 = 12 thousand gallons per week.
This is a difference of 20-12 = 8 thousand gallons per week.
Therefore, the net effect is the lake loses 8 thousand gallons per week even when water flows into the lake from the river.
The faster drain rate wins overall.
x = number of weeks to drain the lake
8000x = number of gallons drained after x weeks
8000x = 240,000
x = (240,000)/8000
x = 30
It takes 30 weeks to drain the lake.
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
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.
If you wanted a formal algebraic solution, it might look something like this:
Let x be the number of gallons in the full lake. Then, according to the given information,
(1/12)x = gallons released in 1 week
(1/20)x = gallons filled by the river in 1 week
The number of gallons in the lake decreases by the difference between those two amounts.
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.
ANSWER: 30
And, if formal algebra is not required, here is an informal method for solving this kind of problem that avoids working with fractions.
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.
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.
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.
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.
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