#8 Word Translations Question Bank

[Guest]

I have pasted the original question below:

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 of water actually existing 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.

I'm having trouble with the second statement, which translates algebraically into 1/2(z) > 2(x - y). Hypothetically, if I use x =2 and y =1, then z > 4. This would lead me to conclude that it takes longer to fill tank 1 than tank 2. However, this is the opposite conclusion from statement 1. What am I doing wrong here?

Thanks!

[ilonabas]

I believe you got it a little bit wrong......

it should be 2(x-y) >1/2*z.... (where 2(x-y) is the capacity of the tank 2... which is less when 1/2 tank 1 (or z)....)

[sumit]

the qn can be rephrased to read is z/(x-y)> 2(x-y)/y or is zy> 2x^2 +2y^2-4xy

1) clearly says that ZY<2x^2+2y^2-4xy ; Sufficient- so we can rule out BCE

2) can be rephrased to read 2(x-y)<1/2 Z or Z>4(x-y)

=>Z/(x-y) will be greater than 4 . Also 2(x-y)/y will be less than 2 since x>y is given. Hence sufficient

So D

[StaceyKoprince]

This is a lot more complicated than anything I think the test would give you - just FYI.

First, to address the original question, sure - you could pick numbers that would tell you T1 could fill up first and you could pick numbers that would tell you that T2 could fill up first. That's why this particular statement is NOT sufficient - you can't actually tell either way from this info. :)

(Note that it does not contradict statement 1; it just offers more possibilities, including what statement 1 tells us - that T1 fills up first. A contradiction is something like x<0 in one statement and x>0 in another. There's no possible overlap there and you can't find one x that fulfills both statements!)

* * *
Now, on to some of the other comments: Correct answer here is A.

T1 holds z gallons. T1 starts empty and fills at x gal/min. T1 also leaks at y gal/min (where x > y, so more is coming in than is leaking out).

The leaking water goes into empty T2, so T2 is filling at y gal/min which is slower than the rate at which T1 is filling (because x>y).

Capacity of T2 is two times the water in T1 one minute after T1 starts filling.
* * *
So what can I figure out here before going further?
T1 is filling at the rate (x-y) gal/min. T2 is filling at the rate y gal/min.
To find out how long it takes T1 to fill, I'd take the total capacity (z) and divide it by the fill rate (x-y). So length of time to fill T1 is z/(x-y).

After one minute, there are (x-y) gallons in the T1 (because T1's fill rate is x-y gal/min). So T2's capacity is twice that, or 2(x-y). The time it takes to fill T2 is its total capacity, 2(x-y), divided by its fill rate, y.

We want to know if we can tell whether T1 fills up first, so the question is: is T1's time faster than T2's time (and "faster" of course means a smaller number):
Is z/(x-y) < [2(x-y)]/y?
If I can answer definitely yes or definitely no, the info is sufficient. If I can't, the info is not sufficient. We already discussed above why statement 2 is not sufficient, so let's look at statement 1 now. This is a ridiculous equation, but so is my rephrased question above, so maybe I can manipulate until they look more alike:

zy < 2x^2 - 4xy + 2y^2
zy < 2(x^2 - 2xy + y^2)
zy < 2(x - y)(x - y)
(zy)/(x-y) < 2(x-y)
z/(x-y) < [2(x-y)]/y

(Note that it's okay to divide by (x-y) and y here without worrying about the inequality because we know both of those are positive.)
So I've matched my original equation: turns out that T1's time to fill is faster than T2's time! Therefore, statement 1 is sufficient.

[prashantponde]

Great explanation Stacy!

[tim]

thanks