Data Sufficiency

[GMAT math practice question]

$$Is\ x>0?$$

$$1)\ (x+y)^2>(x-y)^2$$
$$2)\ x+y>x-y$$

Max@Math Revolution wrote:[GMAT math practice question]

$$Is\ x>0?$$

$$1)\ (x+y)^2>(x-y)^2$$
$$2)\ x+y>x-y$$

Statement 1: (x+y)² > (x-y)²
x² + y² + 2xy > x² + y² - 2xy
2xy > -2xy
4xy > 0
xy > 0.
Implication:
x and y have the SAME SIGN.
Since x and y could both be positive or both be negative, INSUFFICIENT.

Statement 2: x+y > x-y
y > -y
2y > 0
y > 0.
No information about x.
INSUFFICIENT.

Statements combined:
Since y>0, and x and y have the same sign, x>0.
Thus, the answer to the question stem is YES.
SUFFICIENT.

The correct answer is C.

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Condition 1) tells us that
(x+y)^2 > (x-y)^2
=> x^2 + 2xy + y^2 > x^2 - 2xy + y^2
=> 2xy > -2xy
=> 4xy > 0
=> xy > 0

Condition 2) tells us that
x + y > x - y
=> y > -y
=> 2y > 0
=> y > 0.

Since xy > 0 and y > 0, we have x > 0.

Thus, both conditions 1) & 2) together are sufficient.


In general, there are many questions involving integers and statistics to which we need to apply CMT(Common Mistake Type) 4.


Condition 1):
If x = -2 and y = -1, then the answer is "yes".
If x = 1 and y = 2, then the answer is "no".

Thus, condition 1) is not sufficient.


Condition 2):
If x = -3 and y = -1, then the answer is "yes".
If x = -1 and y = -3, then the answer is "no".
Thus condition 2) is not sufficient.

Therefore, C is the answer.

Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.