Algebra

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Algebra

by Tagne » Wed Mar 16, 2011 7:34 pm
1. In a certain bathtub, both the cold water and the hot water leak. The cold water leak alone would fill an empty bucket in c hours and the hot water leak alone would fill same bucket in h hours, where c < h. if both fixtures began to leak the same time into the empty bucket at their respective constant rates and consequently it took t hours to fill the bucket, which of the following must be true?
a. 0 < t < h
b. C < t < h
c. c/2 < t < h /2
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by Anurag@Gurome » Wed Mar 16, 2011 8:19 pm
Solution:
In 1 hour, cold water leak will fill 1/c th part of the bucket.
In 1 hour, hot water leak will fill 1/h th part of the bucket.
So, in 1 hour (1/c + 1/h) th part of the bucket is filled, if both fixtures are opened together.
It is given that it takes t hours for both the fixtures to fill the bucket.
Or, 1/t = (1/c + 1/h).
Now 1/t > 1/h.
Or t < h.
Hence a. is always true.
To check b., take c = 5 and h = 10 hours each.
So, t = 10/3 = 3 1/3.
Here t < c.
So, b. is not true.
Now, 1/2t = {(1/c+1/h)}/2.
So 1/2t is the mean of 1/c and 1/h.
So it has to lie between them.
Since c < h, 1/c > 1/h.
Or 1/h < 1/2t < 1/c.
Or h > 2t > c.
Or c/2 < t < h/2.
So, c. is always true.

The correct answer is a. and c.
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