chetan2u wrote:

If \(\frac{x}{y}>1\) and x and y are integers, is x>1?

1) \(y^2+5y=6\)

2) \(x^2+x=6\)

New tricky question

Interesting question! What makes this one tricky is the

extra information in the question. My first thought on seeing it: I really need to remember that x/y is bigger than 1! I'm not trying to

figure out whether x/y is bigger or smaller than 1. Instead, that's a fact I already know, and I'll need to combine it with the facts in the statements.

I don't know whether y is positive or negative, so I can't simplify the question by multiplying by y. So, I'll leave it how it is for now.

Statement 1: This simplifies, using quadratic rules, to 'y = -6 or y = 1'. At this point, I know two facts: I know that y is one of those two numbers, -6 or 1, although I don't know which one it is. I also know that x/y > 1, but I don't know what x is.

Given those two facts, is it possible for x to be greater than 1? Is it possible for x to be less than 1? If both of those are possible, then this statement is insufficient.

Well, since y might be 1, we could say that x = 100. That follows all of the rules (y is one of those two values, and x/y > 1), and x is greater than 1.

And since y might be -6, we could say that x = -600. That follows all of the rules (y is one of those two values, and x/y > 1), and x is less than 1.

Since x could be either bigger or smaller than 1, the statement is insufficient. Eliminate A and D.

Statement 2: This simplifies using quadratic rules to 'x = -3 or x = 2'. Again, I know two facts: I know that x is one of those two numbers (although I don't know which one). I also know that x/y > 1. That's not a very interesting fact, since I have no way of figuring out what y is.

It's possible for x to be less than 1, since x could equal -3.

It's also possible for x to be greater than 1, since x could equal 2.

So, the statement doesn't tell us whether x is less than or greater than 1. Insufficient. Eliminate B.

Both statements: Now, we know three facts! Here they are:

x = -3 or 2

y = 1 or -6

x/y > 1

And we're trying to figure out:

is x > 1?

The first thing to figure out is whether x and y could be any of those numbers, or whether we can eliminate some possibilities because they break the rules. It's not possible for x to be -3 and y to be 1, since if that's true, we'd be breaking the third rule, that says x/y > 1. Similarly, it's not possible for x to be -3 and y to be -6. It's also not possible for x to be 2 and y to be -6. The only set of numbers that follows all three of our rules is

x = 2 and y = 1.

So basically, we can translate the two statements plus the question as giving us this information:

x = 2 and y = 1.

If we know that, do we know the answer to the question ('is x > 1?')? Yes, we do, since x is definitely 2 and 2 is definitely bigger than 1.

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