PriyankaPalit7 wrote:
Is \(xy < x^2y^2\)?
(1) xy > 0
(2) x + y = 1
I enjoyed this problem! It tests a lot of different Data Sufficiency skills in ways that aren't obvious at first.
First of all,
understand the question stem.
There are a few different ways to handle the question stem, here. You could try to simplify it with math, but there's a trick. You can't just divide both sides by xy, because you don't know whether xy is positive or negative. If it's positive, you wouldn't have to flip the inequality sign. If it's negative, you would have to flip it. Since you don't know either way, you aren't allowed to do that division.
Instead, in this situation, try subtracting a term from both sides:
Is \(xy < x^2y^2\)?
Is \(0 < x^2y^2 - xy\)?
Is \(xy(xy-1) > 0\)?
OR, you can use a "decoding" kind of approach, and try to figure out what the question was asking you in plain English. When is xy less than (xy)^2? Well, if xy is negative, the answer would definitely be "yes". Also, if xy is a large positive number, the answer would be "yes" as well, because large positive numbers get bigger when you square them. In fact, the only situation where the answer would be "no" is if xy is between 0 and 1, inclusive. So the question is really asking, "is xy between 0 and 1?"
Now, approach the statements.
Statement 1: First, suppose that you used the math approach. You now know that xy is positive, so try plugging in some positive values for xy.
If xy = 0.5, then xy(xy-1) = 0.5(0.5-1) = -0.25, which is NOT greater than 0.
If xy = 100, then xy(xy-1) = 100(99) = 9900, which IS greater than 0.
So, the statement is insufficient.
Or, suppose that you used the "decoding" approach to the question. This statement tells you that xy is positive, but it doesn't tell you whether it's between 0 and 1, so it's not sufficient.
Statement 2: Similarly, suppose that you used the math approach. You know that x + y = 1. Try some values.
x = 0, y = 1: xy(xy-1) = 0(0-1) = 0, which is NOT greater than 0.
x = 0.5, y = 0.5: xy(xy-1) = 0.25(-0.75), which is NOT greater than 0.
x = 100, y = -99: xy(xy-1) = -9900(-9901), which IS greater than 0.
Or, suppose that you "decoded". Could xy be between 0 and 1? Yes, because x and y could both be decimals between 0 and 1. Or, xy could be negative, for instance if x is negative and y is positive. So, the answer could be yes or no, and the statement is insufficient.
Statements 1 and 2 together:
Things get a little simpler at this point! x and y can't both be negative, because then they can't sum to 1. So, x and y have to both be positive. They also have to be between 0 and 1, because their sum needs to be 1. So the answer to the question is "yes" and the statements are sufficient together.
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