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Intern  Joined: 17 Aug 2010
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If the symbol @ represents either addition or multiplication, which op  [#permalink]

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Difficulty:   5% (low)

Question Stats: 79% (01:10) correct 21% (01:21) wrong based on 580 sessions

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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

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Re: If the symbol @ represents either addition or multiplication, which op  [#permalink]

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zerotoinfinite2006 wrote: OA is "B", I am confused.

How come
a + ( b -c ) = (a + b) - (a + c) ?
and why not

a + b = b + a

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 $$(a@b)-(a@c)=ab-ac$$)

Similar problems:
symbol-problem-101741.html?hilit=symbol#p788840
ds-question-100955.html?hilit=symbol
just-800-level-question-99064.html?hilit=symbol

Hope it helps.
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Re: If the symbol @ represents either addition or multiplication, which op  [#permalink]

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

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Re: If the symbol @ represents either addition or multiplication, which op  [#permalink]

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zerotoinfinite2006 wrote:
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

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

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Re: If the symbol @ represents either addition or multiplication, which op  [#permalink]

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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?

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Re: If the symbol @ represents either addition or multiplication, which op  [#permalink]

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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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Re: If the symbol @ represents either addition or multiplication, which op  [#permalink]

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1
The symbol $$\triangle$$ can be + or *

1) a $$\triangle$$ b = b $$\triangle$$ 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 $$\triangle$$ (b-c) = (a $$\triangle$$ b) - (a $$\triangle$$ 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

Hope it helps!
Thanks, Alecita  Re: If the symbol @ represents either addition or multiplication, which op   [#permalink] 01 Sep 2018, 13:49
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