Is x^2 – y^2 a positive number ?

Question

Is x^2 – y^2 a positive number?

(1) x – y is a positive number.
(2) x + y is a positive number.

DS48302.01

I chose (A) for this one. If x - y > 0 then x > y. So x2-y2 will be a +ve number.

the OA is C though x2 - y2 = (x+y)(x-y)

Bunuel · Accepted Answer

Is x^2 – y^2 a positive number?

Is x^2 - y^2 > 0? is x^2 > y^2? --> is |x| > |y|? So, the question asks whether x is further from 0 than y.

(1) x – y is a positive number --> x > y. One number is greater than another. Also, insufficient to say which one is further from 0. For example, if x = 2 and y = 1, then x is further but if x = 2 and y = -3, then y is further. Not sufficient.

(2) x + y is a positive number. The sum of two numbers is greater than 0. Clearly insufficient to say which one is further from 0.

(1)+(2) Is x^2 - y^2 > 0? Can be rephrased as "is (x - y)(x + y) > 0?" (1) says that x - y > 0 and (2) says that x + y > 0, thus their product, the product of two positive values, must be greater than 0. Sufficient.

Answer: C.

Hope it's clear.

alex13usa · Answer

Almost all data sufficiency questions (and generally most problem solving questions) benefit from the "dumb" approach. I'll elaborate on that in a second but first, as a trigger, you should ALWAYS attempt to rearrange the question they give you especially when you see exponents. When you see x^2 – y^2 you should immediately realize that it equals (x + y)(x - y).

Now for the dumb approach:
The question asks is (x + y)(x - y) > 0. The dumb approach says that you take this problem and translate it to the simplest terms possible. "Is a number times a number positive?" For this to be true, both numbers must either be positive or negative. That's why you need to both statements. And knowing number properties such as "a positive times a positive is a positive" or "an odd number to any power is odd" will save you the hassle of plugging and chugging.

BrentGMATPrepNow · Answer

The key to answering this question is the first recognize that x² – y² = (x + y)(x - y)
This allows us to rephrase our  target question  as follows:

Target question :    Is (x + y)(x - y) a positive number?

Statement 1 : x – y is a positive number 
Here we know that (x - y) is positive, but we know nothing about (x + y) 
To see what I mean, consider these two possible cases that satisfy statement 1:

Case a : x = 2 and y = 1. In this case, (x + y)(x - y) = (2 + 1)(2 - 1) = 3. So, the answer to the target question is  YES, (x + y)(x - y) IS a positive number 
 Case b : x = 1 and y = -2. In this case, (x + y)(x - y) = (1 + (-2))(1 - (-2)) = -3. So, the answer to the target question is  NO, (x + y)(x - y) is NOT a positive number 
Since we cannot answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : x + y is a positive number. 
Here we know that (x + y) is positive, but we know nothing about (x - y) 
Consider these two possible cases that satisfy statement 2:

Case a : x = 2 and y = 1. In this case, (x + y)(x - y) = (2 + 1)(2 - 1) = 3. So, the answer to the target question is  YES, x² – y² IS a positive number 
 Case b : x = -1 and y = 2. In this case, (x + y)(x - y) = ((-1) + 2)((-1) - 2) = -3. So, the answer to the target question is  NO, x² – y² is NOT a positive number 
Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined    
Statement 1 tells us that (x - y) is POSITIVE
Statement 2 tells us that (x + y) is POSITIVE
So, (x + y)(x - y) = (POSITIVE)(POSITIVE) = POSITIVE
The answer to the target question is  YES, x² – y² IS a positive number 
Since we can answer the  target question  with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

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Is x^2 – y^2 a positive number ?

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Is x^2 – y^2 a positive number ?

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