If the symbol @ represents either addition or multiplication, which op

Question

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

Bunuel · Accepted Answer

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

(1) a@b=b@a for all numbers a and b.

\(\text{@}\) can] be addition (as a + b = b + a), as well as, multiplication (as a*b = b*a). Not sufficient.

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

If @ represents addition, then the left hand side would be \(a@(b - c) = a + b - c\), which would not equal the right hand side \((a@b) - (a@c) = (a + b) - (a + c) = b - c\). Therefore, \(\text{@}\) 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.

Similar problems:
https://gmatclub.com/forum/symbols-8956 ... 20addition
https://gmatclub.com/forum/operations-1 ... iplication
https://gmatclub.com/forum/symbol-probl ... ol#p788840
https://gmatclub.com/forum/ds-question- ... lit=symbol
https://gmatclub.com/forum/just-800-lev ... lit=symbol

Hope it helps.

kpali · Answer

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

Bunuel · Answer

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

All DS Functions and Custom Characters questions: search.php?search_id=tag&tag_id=40
All PS Functions and Custom Characters questions: search.php?search_id=tag&tag_id=61

TheMechanic · Answer

in statement B.. Why dont we consider the case of 0 where both +, x will work there by yielding E as answer? Is it because the Question says a,b,c are Numbers and hence we dont consider the case of 0? Which implies to consider 0 only when the it is mentioned to consider all integers/ or just integers. Therefore, if question mentions numbers it will be false to assume integers?

Bunuel please verify. Thanks.

Bunuel please verify. Thanks.

MathRevolution · Answer

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.

Alecita · Answer

The symbol riangle can be + or * 1) a riangle b = b riangle a for all numbers a and b If a=1 and b=1 1+1 = 1+1 = 2 1*1 = 1*1 = 1 If a=2 and b=1 2+1 = 1+2 = 3 2*1 = 1*2 = 2 These give us 2 different answers + or * INSUFFICIENT 2) a riangle (b-c) = (a riangle b) - (a riangle c) for all numbers a, b, and c. If a=1 b=1 c=1 1 + (1 - 1) = (1+1) - (1+1) 1 + 0 = 2 - 2 1=0 It can not be addition + 1 * (1 - 1) = (1*1) - (1*1) 1* 0 = 1 - 1 0 = 0 It has to be multiplication * Only one answer. SUFFICIENT Answer B Hope it helps! Thanks, Alecita

Juanacho · Answer

If in statement 2 we consider number 0 , △ can be "+" or "*" . Can someone explain please?  Bunuel

Bunuel · Answer

The statement (2) says that the equation must hold true for ALL numbers, not just specific ones. If  	ext{@}  were addition, the equation wouldn't work for all numbers. However, with multiplication, it does work for ALL numbers, including 0.

GMATinsight · Answer

Question: symbol @ represents either addition or multiplication, which one?

Statement 1  : a@b = b@a for all numbers a and b
This is true for  to  be + or x both hence
 NOT SUFFICIENT

Statement 2  : a@(b – c) = (a@b) – (a@c) for all numbers a, b, and c
Now, this relationship won't be true if  gets  replaced by +
therefore this has to be true only if  SIGN  is replaced by x hence
 SUFFICIENT

Answer: Option B

If the symbol @ represents either addition or multiplication, which op

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

If the symbol @ represents either addition or multiplication, which op

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards