We can just check each of the four arithmetic operations to see when each Statement will be true. If r = s, we can replace 's' with 'r', so for Statement 1:

r+r = 2r
r-r = 0
r*r = r^2
r/r = 1

and it's only when the operation is subtraction that we get 0 as a result, and the operation therefore must be subtraction. So Statement 1 is sufficient.

For Statement 2:

r + s = s + r is always true
r - s does not equal s - r if r and s are different (one of them is positive, the other negative)
r*s = s*r is always true
r/s does not equal s/r if r and s are different positive numbers (one of them is less than 1, the other greater than 1)

so we don't know if the operation is subtraction or division, and Statement 2 is not sufficient.

AbeNeedsAnswers wrote:If r and s are positive numbers and θ is one of the operations, +, −, ×, or ÷, which operation is θ ?

(1) If r = s, then r θ s = 0.
(2) If r ≠ s, then r θ s ≠ s θ r.

Given: r and s are positive numbers and θ is one of the operations, +, −, ×, or ÷

Target question: Which operation is θ ?

Statement 1: If r = s, then r θ s = 0
Keep in mind that r and s are POSITIVE
So, we have POSITIVE θ POSITIVE = 0

POSITIVE + POSITIVE ≠ 0 So, θ cannot represent addition
POSITIVE - POSITIVE = 0 So, θ COULD represent subtraction
POSITIVE × POSITIVE ≠ 0 So, θ cannot represent multiplication
POSITIVE ÷ POSITIVE ≠ 0 So, θ cannot represent division
The answer to the target question is θ represents subtraction
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: If r ≠ s, then r θ s ≠ s θ r
Let's test some values for each possible operation.
1 ≠ 2, yet 1 + 2 = 2 + 1. So, θ cannot be addition
1 ≠ 2, yet 1 x 2 = 2 x 1. So, θ cannot be multiplication

If r and s are DIFFERENT positive numbers, then r - s will never equal s - r. So, θ COULD represent subtraction
If r and s are DIFFERENT positive numbers, then r ÷ s will never equal s ÷ r. So, θ COULD represent division
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

Hi All,

We're told that R and S are POSITIVE NUMBERS and θ is one of the operations add, subtract, multiply or divide. We're asked which operation is θ. This question comes down to some basic Arithmetic and TESTing VALUES.

(1) If R =S, then R θ S = 0.

With the information in Fact 1, we have to check the 4 possible operations to see which one(s) fit the given equation.
IF....
R=S=1....
1 + 1 = 2 this is NOT a match
1 - 1 = 0 this IS a match
(1)(1) = 1 this is NOT a match
1/1 = 1 this is NOT a match

The ONLY operation that fits is SUBTRACTION, so that is the answer to the question.
Fact 1 is SUFFICIENT

(2) If R ≠ S, then R θ S ≠ S θ R

Just as we did with Fact 1, we have to check the 4 possible operations to see which one(s) fit the given equation (meaning that the two calculations do NOT equal one another).
IF....
R=1 and S=2....
1 + 2 = 3 IS equal to 2 + 1 = 3 this is NOT a match
1 - 2 = - 1 DOES NOT equal 2 - 1 = 1 this IS a match for the given information
(1)(2) = 2 IS equal to (2)(1) = 2 this is NOT a match
1/2 = 1/2 DOES NOT equal 2/1 = 2 this IS a match for the given information.

Thus, the operation could be either SUBTRACTION OR DIVISION.
Fact 2 is INSUFFICIENT

Final Answer: A

GMAT assassins aren't born, they're made,
Rich