If the symbol @ represents either addition or multiplication, which operation does it represent?

(1) a@b=b@a for all numbers a and b --> @ can be addition (a+b=b+a) as well as multiplication (a*b=b*a). Not sufficient.

(2) a@(b–c)=(a@b)–(a@c) for all numbers a, b, and c --> if @ represents addition we will have \(a@(b-c)=a+b-c\) which is not equal to \((a@b)-(a@c)=(a+b)-(a+c)=b-c\), so @ must be multiplication. Sufficient. (Just to check: \(a@(b-c)=a*(b-c)=ab-ac\) which is equal to \((a@b)-(a@c)=ab-ac\))

Answer: B.

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Hope it helps.
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Question: what is the operation @ ?
@ = + or *

As soon as you look at the questions, you will be reminded of the number properties Commutative and Distributive.

statement 1:
(1) a@b=b@a
Commutative property =>valid for both, + as well as *, =>cant say, Not Suff
(2) a@(b–c)=(a@b)–(a@c)
Distributive property=> Always valid for * -> Sufficient

Answer : B
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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.

If the symbol @ represents either addition or multiplication, which operation does it represent?

(1) a@b=b@a for all numbers a and b
(2) a@(b–c)=(a@b)–(a@c) for all numbers a, b, and c

There is one variable (@) and 2 equations are given by the 2 conditions, so there is high chance (D) will be our answer.
For condition 1, @=*,+ , so this is insufficient.
For condition 2, @=*. This is sufficient, so the answer becomes (B).

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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