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.
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- StaceyKoprince
- ManhattanGMAT Staff
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- Joined: Wed Oct 19, 2005 9:05 am
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MGMAT Word Translations Question Bank #8
This is a very difficult and time consuming question - you are unlikely to see much on the actual test that is this computation-intensive. FYI - for most people, studying this one is not a good use of your time.
First, let's deal with the info in the question stem. Tank 1 has a capacity of z. It fills at a rate of x but leaks at a rate of y for a combined fill rate of (x-y). Tank 2 fills at a rate of y.
Finally, we have the confusing piece of info about the capacity of tank 2. First, we have to figure out how many gallons are in tank 1 after one minute. We're told tank 1 starts out empty and we know the combined fill rate for tank 1 is (x-y) per minute - so after one minute, there are (x-y) gallons in tank 1. Therefore, tank 2's capacity is 2(x-y).
Okay, so we know the fill rates for both tanks 1 and 2, and we also know the capacity of both tanks. (Of course, we only know variables for all of this, not real numbers.) We're asked whether tank 1 fills up before tank 2 (a yes/no question), so basically we have to figure out either how long it takes to fill each tank or else we have to have enough info to know definitively that it will take one longer than the other, even if we don't know exactly how long each will take.
Now, to calculate how long it will take to fill a particular tank, we need to take the capacity (gallons) and divide it by the fill rate (gallons / minute); this will give us minutes.
For tank 1: z / (x-y)
For tank 2: [2(x-y)] / y
And the question is whether z / (x-y) > [2(x-y)] / y
Statement 1 gives us an inequality, so let's see if we can manipulate it to equal the above. (Note: when you want to raise something to an exponent, please use the carat symbol - ^ - so people know what you're trying to write.)
zy < 2x^2 - 4xy + 2y^2 factor out a 2
zy < 2(x^2 - 2xy + y^2) notice that the stuff in the parentheses is one of our 3 common quadratics
zy < 2(x-y)(x-y) try to match it with what we're looking for above - divide both sides by y *
z < [2(x-y)(x-y)] / y again try to match it - divide both sides by (x-y) *
z / (x-y) > [2(x-y)] / y we've just matched what we're looking for - sufficient
** Note: remember when dividing an inequality by a variable - if you don't know whether that variable is positive or negative, you have to split the problem into two, because you have two possibilities (since you have to switch the sign if you divide by a negative). In this case, we're okay, because both y and (x-y) are positive.
Statement 2 gives us an inequality we can write in math terms, so do that first:
2(x-y) < z/2
We have another equation so see if you can manipulate this one to come up with what we're looking for. We can't, basically - it has some of the same parts as the above inequality that we want to find, but it doesn't have enough to match that inequality.
First, let's deal with the info in the question stem. Tank 1 has a capacity of z. It fills at a rate of x but leaks at a rate of y for a combined fill rate of (x-y). Tank 2 fills at a rate of y.
Finally, we have the confusing piece of info about the capacity of tank 2. First, we have to figure out how many gallons are in tank 1 after one minute. We're told tank 1 starts out empty and we know the combined fill rate for tank 1 is (x-y) per minute - so after one minute, there are (x-y) gallons in tank 1. Therefore, tank 2's capacity is 2(x-y).
Okay, so we know the fill rates for both tanks 1 and 2, and we also know the capacity of both tanks. (Of course, we only know variables for all of this, not real numbers.) We're asked whether tank 1 fills up before tank 2 (a yes/no question), so basically we have to figure out either how long it takes to fill each tank or else we have to have enough info to know definitively that it will take one longer than the other, even if we don't know exactly how long each will take.
Now, to calculate how long it will take to fill a particular tank, we need to take the capacity (gallons) and divide it by the fill rate (gallons / minute); this will give us minutes.
For tank 1: z / (x-y)
For tank 2: [2(x-y)] / y
And the question is whether z / (x-y) > [2(x-y)] / y
Statement 1 gives us an inequality, so let's see if we can manipulate it to equal the above. (Note: when you want to raise something to an exponent, please use the carat symbol - ^ - so people know what you're trying to write.)
zy < 2x^2 - 4xy + 2y^2 factor out a 2
zy < 2(x^2 - 2xy + y^2) notice that the stuff in the parentheses is one of our 3 common quadratics
zy < 2(x-y)(x-y) try to match it with what we're looking for above - divide both sides by y *
z < [2(x-y)(x-y)] / y again try to match it - divide both sides by (x-y) *
z / (x-y) > [2(x-y)] / y we've just matched what we're looking for - sufficient
** Note: remember when dividing an inequality by a variable - if you don't know whether that variable is positive or negative, you have to split the problem into two, because you have two possibilities (since you have to switch the sign if you divide by a negative). In this case, we're okay, because both y and (x-y) are positive.
Statement 2 gives us an inequality we can write in math terms, so do that first:
2(x-y) < z/2
We have another equation so see if you can manipulate this one to come up with what we're looking for. We can't, basically - it has some of the same parts as the above inequality that we want to find, but it doesn't have enough to match that inequality.
Stacey Koprince
Instructor
Director, Content & Curriculum
ManhattanPrep
Instructor
Director, Content & Curriculum
ManhattanPrep
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