Tank to Tank (Word Translations Question Bank)

[JaredT]

I don’t understand how the capacity of Tank 2 is determined. From the way it is described in the problem, Tank 2 seems to have a capacity equal to 2[z-(x-y)], rather than the 2(x-y) stated in the answer. Am I understanding the problem incorrectly?

Thanks in advance.

Jared


QUESTION:
Reserve tank 1 is capable of holding z gallons of water. Water is pumped into tank 1, which starts off empty, at a rate of x gallons per minute. Tank 1 simultaneously leaks water at a rate of y gallons per minute (where x > y). The water that leaks out of tank 1 drips into tank 2, which also starts out empty. If the total capacity of tank 2 is twice the number of gallons that remains in tank 1 after one minute, does tank 1 fill up before tank 2?

(1) zy < 2x2 - 4xy + 2y2

(2) The total capacity of tank 2 is less than one-half that of tank 1.

ANSWER:
If water is rushing into tank 1 at x gallons per minute while leaking out at y gallons per minute, the net rate of fill of tank 1 is x - y. To find the time it takes to fill tank 1, divide the capacity of tank 1 by the rate of fill: z / (x - y).

We know that the rate of fill of tank 2 is y and that the total capacity of tank 2 is twice the number of gallons remaining in tank 1 after one minute. After one minute, there are x - y gallons in tank 1, since the net fill rate is x - y gallons per minute. Thus, the total capacity of tank 2 must be 2(x - y).
The time it takes to fill tank two then is 2(x - y)/y.




The question asks us if tank 1 fills up before tank 2.

We can restate the question: Is z

x - y < 2(x - y)

y ?





(1) SUFFICIENT: We can manipulate zy < 2x2 - 4xy + 2y2:

zy < 2x2 - 4xy + 2y2
zy < 2(x2 - 2xy + y2)
zy < 2(x - y)(x - y) (dividing by x - y is okay since x - y > 0)
zy

x - y < 2(x - y)

(dividing by y is okay since y > 0)

z

x - y < 2(x - y)

y




This manipulation shows us that the time it takes to fill tank 1 is definitely longer than the time it takes to fill tank 2.

(2) INSUFFICIENT: We can express this statement algebraically as: 1/2(z) > 2(x - y). We cannot use this expression to provide us meaningful information about the question.

The correct answer is A.

[StaceyKoprince]

This is a tough question - the language is really tricky.

Tank 1's capacity is z.
Tank 2's capacity is "twice the number of gallons that remains in tank 1 after one minute" under the scenario described earlier in the problem.

In the scenario described earlier in the problem, tank 1 starts off empty. So tank 1 starts off with a capacity of 0, not z.
Water is added to tank 1 at x gal/min so, atter 1 min, x gallons have been added to tank 1. During that same minute, though, tank 1 is leaking y gal/min so, after 1 min, y gallons have been subtracted from tank 1. That's 0 + x - y (starting point + what's added - what's leaked). From what you asked, I'm guessing you thought the starting point was z gallons but the problem specifically says tank 1 starts off empty.

Let me know if that doesn't clarify the issue for you.

[mutagene]

The Condition (1) answer states that "This manipulation shows us that the time it takes to fill tank 1 is definitely longer than the time it takes to fill tank 2". In fact the time is definitely SHORTER.

[christiancryan]

Thanks, Mutagene -- we've corrected the language in the explanation to read "shorter."

JaredT, the language was indeed tricky: "twice the number of gallons that remains in tank 1 after one minute" means "twice the number of gallons ACTUALLY EXISTING AT THAT MOMENT IN tank 1 after 1 minute" -- NOT "twice the REMAINING (i.e., UNFILLED) capacity".

To clarify, I've changed the language to say "twice the number of gallons actually existing in tank 1 after 1 minute." Our original language, while technically correct, was unnecessarily confusing. Hope that's helpful!