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# variables that can or cannot equal each other

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variables that can or cannot equal each other [#permalink]  12 May 2012, 12:19
I have a math related question.
Many question would give you variables and you need to test cases where variables are different or can equal each other. I found two similar problems in the 12th edition OG that contradicts itself, can someone address this question for me pls?:

OG 12th edition DS: #95:
Is abc =1?

1 ab=1
2 bc =1

I understand that if a=b=c=1 creates an answer that answer the questions as "yes", and I know how to find an answer where the answer is also "no", hence the answer is (E). I also get that given no constraints by GMAT, the point of this question is testing a situation where a=b=c.

now onto OG 12th edition DS: #125
If & represents one of the operations +, -, and x is a&(b+c) = (a&b)+(a&c) for all numbers a, b, and c?

1. a&1 is not equal to 1&a for some numbers a
2. & represents subtraction

GMAT says, the answer is (D). But I agree with this only if a,b,c are all not equal to each other and are all zero. However, no where in the text does GMAT say a b c cannot equal each other. If they are, the answer is (E) Unless, in the future anytime you see wording like "for all numbers a, b, c ", you need to to interpret this as "a b c are all unique"?

Can someone please address where in the text does GMAT say a,b,c are not equivalent?

thanks! These variable ones are driving me crazy b/c I'm not sure given non-explicit constraints whether i need to test if all variables are equivalent.
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Re: variables that can or cannot equal each other [#permalink]  13 May 2012, 08:12
Even if k,l,m are equal the solution is D IMO
Let's say k=l=m=3
Statement 1
Only subtraction is possible as 3+(3+3)=|(3+3)+(3+3)
So sufficient

Statement 2
Same as above
Sufficient
Hence D

However if k,l,m are all 0 then it becomes anomalous, but that can be eliminated as for all k,l,m is mentioned in the question...
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Re: variables that can or cannot equal each other [#permalink]  13 May 2012, 08:12
Even if k,l,m are equal the solution is D IMO
Let's say k=l=m=3
Statement 1
Only subtraction is possible as 3+(3+3)=|(3+3)+(3+3)
So sufficient

Statement 2
Same as above
Sufficient
Hence D

However if k,l,m are all 0 then it becomes anomalous, but that can be eliminated as for all k,l,m is mentioned in the question...
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Re: variables that can or cannot equal each other [#permalink]  13 May 2012, 08:35
souvik101990 wrote:
Even if k,l,m are equal the solution is D IMO
Let's say k=l=m=3
Statement 1
Only subtraction is possible as 3+(3+3)=|(3+3)+(3+3)
So sufficient

Statement 2
Same as above
Sufficient
Hence D

However if k,l,m are all 0 then it becomes anomalous, but that can be eliminated as for all k,l,m is mentioned in the question...

you're correct, I mistyped my question, the answer is (e) only if k,l,m are all zero, not if they are all equal.

But I don't understand why k,l,m cannot be zero? What does the statement "for all numbers k,l,m" mean? It doesn't mean k,l,m cannot equal each other does it?

Last edited by mdaiosu on 13 May 2012, 08:53, edited 2 times in total.
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Re: variables that can or cannot equal each other [#permalink]  13 May 2012, 08:39
For all generally means for all real numbers.
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Re: variables that can or cannot equal each other [#permalink]  13 May 2012, 08:41
souvik101990 wrote:
For all generally means for all real numbers.

right, zero is a real number. But I still don't see why you can't set k,l,m = 0 in this question.
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Re: variables that can or cannot equal each other [#permalink]  13 May 2012, 08:50
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Re: variables that can or cannot equal each other [#permalink]  13 May 2012, 08:52
souvik101990 wrote:

so you can see where my confusion is. when you compare the 2 OG questions I posted. The first case you can set it all equal to each other, then 2nd you cannot (and make them all equal zero)
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Re: variables that can or cannot equal each other [#permalink]  14 May 2012, 10:00
mdaiosu wrote:
now onto OG 12th edition DS: #125
If & represents one of the operations +, -, and x is a&(b+c) = (a&b)+(a&c) for all numbers a, b, and c?

1. a&1 is not equal to 1&a for some numbers a
2. & represents subtraction

GMAT says, the answer is (D). But I agree with this only if a,b,c are all not equal to each other and are all zero. However, no where in the text does GMAT say a b c cannot equal each other. If they are, the answer is (E) Unless, in the future anytime you see wording like "for all numbers a, b, c ", you need to to interpret this as "a b c are all unique"?

The variables needn't be unique until and unless it is explicitly mentioned. a, b and c can all be equal.

Question: Is a&(b+c) = (a&b)+(a&c) for all a, b and c?
This relation will hold for all values of a, b and c only when '&' is the multiplication sign.
We know a*(b+c) = a*b + a*c for all values of a, b and c.
It doesn't hold for '+' and '-'. If & = + or -, the relation could hold for some values of a, b and c e.g. a = b = c = 0 but it will not hold for ALL values of a, b and c.

So basically the question is asking: Is & = multiplication?

1. a&1 is not equal to 1&a for some numbers a
If & were equal to multiplication, a&1 = 1&a for all values of a. Since that is not the case, & is not multiplication. We can answer the question with a 'NO'. Sufficient

2. & represents subtraction
This is direct. Again sufficient.
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Re: variables that can or cannot equal each other [#permalink]  14 May 2012, 10:21
@^
the solution holds true for all values except for 0,0,0....

Posted from my mobile device
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Re: variables that can or cannot equal each other [#permalink]  14 May 2012, 10:22
@^
the solution holds true for all values except for 0,0,0....

Posted from my mobile device

Posted from my mobile device
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Re: variables that can or cannot equal each other [#permalink]  14 May 2012, 10:24
@^
the solution holds true for all values except for 0,0,0....

Posted from my mobile device

Posted from my mobile device

Posted from my mobile device
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Re: variables that can or cannot equal each other [#permalink]  14 May 2012, 13:55
VeritasPrepKarishma wrote:
mdaiosu wrote:
now onto OG 12th edition DS: #125
If & represents one of the operations +, -, and x is a&(b+c) = (a&b)+(a&c) for all numbers a, b, and c?

1. a&1 is not equal to 1&a for some numbers a
2. & represents subtraction

GMAT says, the answer is (D). But I agree with this only if a,b,c are all not equal to each other and are all zero. However, no where in the text does GMAT say a b c cannot equal each other. If they are, the answer is (E) Unless, in the future anytime you see wording like "for all numbers a, b, c ", you need to to interpret this as "a b c are all unique"?

The variables needn't be unique until and unless it is explicitly mentioned. a, b and c can all be equal.

Question: Is a&(b+c) = (a&b)+(a&c) for all a, b and c?
This relation will hold for all values of a, b and c only when '&' is the multiplication sign.
We know a*(b+c) = a*b + a*c for all values of a, b and c.
It doesn't hold for '+' and '-'. If & = + or -, the relation could hold for some values of a, b and c e.g. a = b = c = 0 but it will not hold for ALL values of a, b and c.

So basically the question is asking: Is & = multiplication?

1. a&1 is not equal to 1&a for some numbers a
If & were equal to multiplication, a&1 = 1&a for all values of a. Since that is not the case, & is not multiplication. We can answer the question with a 'NO'. Sufficient

2. & represents subtraction
This is direct. Again sufficient.

Got it, your explanation makes sense. Just to clarify, the logic process based on how GMAT asked this question is:
1. The question is: Is & = multiplication?
2. statement 1, the answer is a definite "NO". If &=multiplication, a&1 = 1&a for all values. Statement 1 says that for some numbers a&1 is not equal to 1&a, therefore & is not multiplication.

My approach is flawed b/c i interpreted the question as "what is &" ?
Re: variables that can or cannot equal each other   [#permalink] 14 May 2012, 13:55
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