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If # represents one of the operations +, -, x and /,

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If # represents one of the operations +, -, x and /,

by sachin_yadav » Wed Oct 08, 2014 9:03 am
If # represents one of the operations +, -, x and /, is (a#b)(c#d)=d(b#a)#c(b#a) for all numbers a, b, c and d?

(1). a(b#c)=ab#ac
(2). c#d not equal to d#c


OA is B

Please help in solving this questions

Thanks & Regards
Sachin
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by ceilidh.erickson » Wed Oct 08, 2014 11:50 am
In order to understand the question, we must first determine what each operation would do in this context.

If # is addition:
(a + b)(c + d) = ac + ad + bc + bd
and
d(b + a) + c(b + a) = bd + ad + bc + ac
These would be equal.

If # is subtraction,
(a - b)(c - d) = ac - ad - bc + bd
and
d(b - a) + c(b - a) = bd - ad + bc - ac
The last two have different signs, so these are not equal.

If # is multiplication,
(ab)(cd) = abcd
and
d(ba)c(ba) = (a^2)(b^2)cd
These are not equal.

If # is division,
(a/b)(c/d) = ac/bd
and
d(b/a)/c(b/a) = d/b
Not equal.

So the question is really asking... does # represent addition?

1) a(b#c)=ab#ac

This would be true if # was addition or subtraction.

(2) c#d not equal to d#c

This tells us that # cannot represent addition, otherwise they would be equal. So we know that the answer to the question is definitively "no." Sufficient.
Ceilidh Erickson
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by Brent@GMATPrepNow » Wed Oct 08, 2014 3:17 pm
I'd like to point out how much easier Ceilidh made that question by taking the target question, Is (a#b)(c#d)=d(b#a)#c(b#a)?, rephrasing it to get Does # represent addition?.
This is an important strategy that can help us solve Data Sufficiency questions faster and easier, so be sure to watch for opportunities to rephrase the target question.

Aside: we have a free video covering this strategy: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

Also, here are some additional Data Sufficiency questions that you can practice the strategy with:
https://www.beatthegmat.com/pls-help-x-2 ... 79970.html
https://www.beatthegmat.com/good-ds-t270904.html
https://www.beatthegmat.com/gmat-focus-ds-3-t277927.html
https://www.beatthegmat.com/ds-query-t276597.html
https://www.beatthegmat.com/powers-and-i ... 77249.html
https://www.beatthegmat.com/mgmat-the-po ... 77596.html
https://www.beatthegmat.com/calculate-th ... 99347.html
https://www.beatthegmat.com/coordiante-p ... 72785.html
https://www.beatthegmat.com/og-13-proble ... 76424.html

Cheers,
Brent

For even more information on rephrasing the target question, you can read this article I wrote for BTG: https://www.beatthegmat.com/mba/2014/06/ ... t-question
Brent Hanneson - Creator of GMATPrepNow.com
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by sachin_yadav » Fri Oct 10, 2014 9:41 am
Ceilidh, thanks for your reply, and Brent, thanks for the questions. I have practiced all of them and they were all really good.

But i tried to work on the statement (1) of the posted question. From my working it seems to be sufficient. Please correct me if i am wrong.

Statement (1) a(b#c)=ab#ac

Case (2) subtraction
If # is subtraction,
(a - b)(c - d) = ac - ad - bc + bd
and
d(b - a) + c(b - a) = bd - ad + bc - ac
Why there is + sign in the middle ? Isn't that supposed to be - (minus) here ?

this is what i got when i put - sign here :-

ac - ad - bc + bd = bd - ad - bc + ac

Statement (1) looks sufficient.
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by Matt@VeritasPrep » Sun Oct 12, 2014 9:52 pm
Sachin, I'm getting the same thing you're getting.

If # represents addition, we have

(a+b)(c+d) = d(b+a) + c(b+a)

or

ac + bc + ad + bd = bd + ad + bc + ac. This works for any {a,b,c,d}.

If # represents subtraction, we have

(a-b)(c-d) = d(b-a) - c(b-a)

or

ac - bc - ad + bd = bd - ad - bc + ac. This works for any {a,b,c,d}.

The issue with S1 is that we don't know the specific value of a that we're given in S1. For instance, if a = 0 and # = *, we would have 0(b*c) = 0*b*0*c, which is true, but which would tell us # = * and give us an answer of NO to the original question.

That said, the question is a bit ambiguous: it isn't entirely clear that S1 is a specific equation rather than a relation true over # for all {a,b,c}. Where's this question from?
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