Data Sufficiency

Is y>0?
1) x^2<y^2
2) x+y=1

*An answer will be posted in 2 days.

Max@Math Revolution wrote:Is y>0?
1) x^2<y^2
2) x+y=1

*An answer will be posted in 2 days.

(1) Not sufficient. This implies only that the absolute value of y is greater than the absolute value of x - it tells us nothing about the sign of either variable. This also implies y cannot be zero, because the inequality could not hold if y were zero (at best y^2 would be equal to x^2, it could never be greater than x^2).

(2) Not sufficient. x and y can be any value, positive, negative, or zero, as long as the sum of x and y is 1.

Combined (1) & (2). From statement (2) we know that at least one of x or y must be positive in order for their sum to be 1. From statement (1), we know that the absolute value of y must be larger than the absolute value of x (and we also proved y cannot be zero). With that said, if x is negative, y must be positive. If x is zero, y must be positive (namely, 1). Most importantly, if x is positive, y still cannot be negative - because if y were negative, we would be adding a negative number with a greater absolute value to a positive number of lesser absolute value, and the sum would have to be negative. Therefore, in all possible cases, y must be positive. Hence, we can say with certainty that y > 0. So the answer must be C.

Max@Math Revolution wrote:Is y>0?
1) x^2<y^2
2) x+y=1

Statement 1:
It's possible that x=0 and y=1, in which case y>0.
It's possible that x=0 and y=-1, in which case y<0.
INSUFFICIENT.

Statement 2:
It's possible that x=0 and y=1, in which case y>0.
It's possible that x=2 and y=-1, in which case y<0.
INSUFFICIENT.

Statements combined:
Statement 2 implies that x = 1-y.

Substituting x = 1-y into x² < y², we get:
(1-y)² < y²
1² + y² - 2y < y²
1 < 2y
1/2 < y
y > 1/2.

Thus, y > 0.
SUFFICIENT.

The correct answer is C.

There are 2 variables in the original condition, Hence, since we need 2 equations, the correct answer is C.

- Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

There are 2 variables in the original condition, Hence, since we need 2 equations, the correct answer is C.

The line of reasoning is typically valid for a DS with two distinct linear equations and two variables.
But if a DS problem is constrained to positive integers, one equation with two variables can be sufficient: https://www.beatthegmat.com/to-find-the- ... 73004.html).
Here, Statement 1 is not an equation but an inequality -- and a NONLINEAR inequality at that.
As a result, we cannot conclude that the two statements are sufficient simply because they offer two variables and two distinct relationships with these variables.
Consider the following variation:

Is y > 0?

1) x² > y²
2) x+y = 1

Statement 1:
It's possible that x=2 and y=1, in which case y>0.
It's possible that x=2 and y=-1, in which case y<0.
INSUFFICIENT.

Statement 2:
It's possible that x=0 and y=1, in which case y>0.
It's possible that x=2 and y=-1, in which case y<0.
INSUFFICIENT.

Statements combined:
Statement 2 implies that x = 1-y.

Substituting x = 1-y into x² > y², we get:
(1-y)² > y²
1² + y² - 2y > y²
1 > 2y
1/2 > y
y < 1/2.

If y=1/4, then the answer to the question stem is YES.
If y=-1, then the answer to the question stem is NO.
INSUFFICIENT.

Virtually the same set of facts, yet the correct answer is not C but E.