Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 22 Dec 2009
Posts: 22

If the operation @ is defined for all integers a and b
[#permalink]
Show Tags
18 Sep 2010, 20:16
Question Stats:
62% (02:01) correct 38% (02:03) wrong based on 1617 sessions
HideShow timer Statistics
If the operation @ is defined for all integers a and b by a@b=a+bab, which of the following statements must be true for all integers a, b and c? I. a@b = b@a II. a@0 = a III. (a@b)@c = a@(b@c) (A) I only (B) II only (C) I and II only (D) I and III only (E) I, II and III
Official Answer and Stats are available only to registered users. Register/ Login.




Math Expert
Joined: 02 Sep 2009
Posts: 50002

If the operation is defined for all integers a and b
[#permalink]
Show Tags
18 Sep 2010, 20:29




Director
Joined: 29 Nov 2012
Posts: 775

Re: If the operation @ is defined for all integers a and b
[#permalink]
Show Tags
31 Oct 2013, 05:55
jlgdr wrote: Well, can't argue that learning the definitions is in fact quite interesting and thank you for that. Nevertheless, I was really intereted in solving statement 3 quicker/faster/more efficient So, being able to recognize if operations in the different order given will yield same result without having to go through all the long distribution process. I will try to come up with a faster way but if anyone else come's up with a nice and elegant approach I'd be happy to give some nice Kudos for the collection Cheers J You can always test values that's an alternative Let a=1,b=2 and c=3 Definition => a@b=a+bab Option 3 III. (a@b)@c = a@(b@c) a@b = 1 + 2 2 = 1 1@3 = 1+3 3 = 1 B@c = 2@3 = 56 = 1 1@1 = 1  1  ( 1 * 1) =1 LHS = RHS
_________________
Click +1 Kudos if my post helped...
Amazing Free video explanation for all Quant questions from OG 13 and much more http://www.gmatquantum.com/og13th/
GMAT Prep software What if scenarios http://gmatclub.com/forum/gmatprepsoftwareanalysisandwhatifscenarios146146.html




Retired Moderator
Joined: 16 Nov 2010
Posts: 1436
Location: United States (IN)
Concentration: Strategy, Technology

Re: If the operation @ is defined for all integers a and b
[#permalink]
Show Tags
19 Apr 2011, 19:25
(I) and (II) are obviously correct. For (III) (a+bab)@c = (a + b  ab)@c = a + b  ab + c  c(a + b  ab) = a + b  ab + c  ac  ab + abc a@(b@c) = a@(b + c  bc) = a + b + c  bc  a(b + c  bc) = a + b + c  bc ab  ac + abc Answer  E
_________________
Formula of Life > Achievement/Potential = k * Happiness (where k is a constant)
GMAT Club Premium Membership  big benefits and savings



Intern
Joined: 28 Nov 2012
Posts: 5
Location: Italy
GPA: 3.87

Re: If the operation @ is defined for all integers a and b
[#permalink]
Show Tags
18 Dec 2012, 08:45
Sorry for bumping up an old thread, I have a doubt: my approach for solving the question was to assume that the operation in this case was the union between two sets, a and b, and consequently the three points were the properties of the union of sets. Is that a correct approach or might it be too risky in the actual exam? (or is it even a wrong assumption, and I got it right out of luck?)



Senior Manager
Joined: 13 Aug 2012
Posts: 431
Concentration: Marketing, Finance
GPA: 3.23

Re: If the operation @ is defined for all integers a and b
[#permalink]
Show Tags
18 Dec 2012, 21:29
I. a@b = b@a \(a+bab=b+aab\) TRUE! II. a@0 = a \(a+00 = a\) TRUE! III. (a@b)@c = a@(b@c) \(a+bab+cacbc+abc = b+cbc + a  abac+abc\) STRIKE OUT DUPLICATES ON RHS and LHS! TRUE! Answer: I,II, and III or (E)
_________________
Impossible is nothing to God.



Math Expert
Joined: 02 Sep 2009
Posts: 50002

Re: If the operation @ is defined for all integers a and b
[#permalink]
Show Tags
19 Dec 2012, 03:24



Intern
Joined: 29 Dec 2012
Posts: 3

Re: If the operation @ is defined for all integers a and b
[#permalink]
Show Tags
04 Jan 2013, 11:47
in equ. 3 :
I put three random numbers like (5,3,2) and tested it. however there's always a little chance of error.



Intern
Joined: 27 Dec 2012
Posts: 6
Location: Bulgaria
Concentration: Marketing
GMAT Date: 01302013
GPA: 3.5

Re: If the operation @ is defined for all integers a and b
[#permalink]
Show Tags
28 Jan 2013, 22:58
subhashghosh wrote: (I) and (II) are obviously correct.
For (III)
(a+bab)@c = (a + b  ab)@c = a + b  ab + c  c(a + b  ab) = a + b  ab + c  ac  ab + abc
a@(b@c) = a@(b + c  bc) = a + b + c  bc  a(b + c  bc) = a + b + c  bc ab  ac + abc
Answer  E Been looking at this for a while and still can't figure it out.. I thought that we must ALWAYS first do the calculations in the brackets and open them, and then do the remaining calculations. as we have a@(b@c), how come do you straight come up to (a + b  ab)@c, when it's b@c in the brackets, not a@b anymore.. finding this one a bit confusing.. thanks for explaining in advance EDIT:OK, I think I get it now.. pls, have a look at my upload and let me know if I am correct.. this is the left side of the equation in (III). With the right one, we do the exact same thing, right?
Attachments
a@b.jpg [ 861.06 KiB  Viewed 73099 times ]



SVP
Joined: 06 Sep 2013
Posts: 1764
Concentration: Finance

Re: If the operation @ is defined for all integers a and b
[#permalink]
Show Tags
23 Oct 2013, 13:34
Hey there folks, sorry to bump on an old thread. Just wondering, is there a way to evaluate statement 3 faster? I believe this questions takes around 2 minutes and evaluating statement 3 takes a lot of work and is prone to errors. Just wondering if I'm doing this the correct/most efficient way Thanks guys Cheers! J



Intern
Joined: 18 May 2013
Posts: 39
Concentration: Real Estate, Finance
GPA: 3.73
WE: Analyst (Real Estate)

Re: If the operation @ is defined for all integers a and b
[#permalink]
Show Tags
27 Oct 2013, 14:13
Would it be smart to pick numbers for each of the variables? I selected 1&3 only since I got mixed up with the letter variables. Is picking number the most efficient way to approach this problem?



Intern
Joined: 03 Oct 2013
Posts: 6

Re: If the operation @ is defined for all integers a and b
[#permalink]
Show Tags
28 Oct 2013, 12:40
jlgdr wrote: Hey there folks, sorry to bump on an old thread. Just wondering, is there a way to evaluate statement 3 faster? I believe this questions takes around 2 minutes and evaluating statement 3 takes a lot of work and is prone to errors. Just wondering if I'm doing this the correct/most efficient way Thanks guys Cheers! J In this problem we have been asked to check the commutative and associative property of the given function. These properties are defined as below: Commutative: In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. Associative: Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such an expression will not change its value. If you're wondering if commutativity implies associativity in mathematics then the answer is NO. However, for simple addition and multiplication functions commutativity does imply associativity and hence in such cases option 3 need not be tested if option 1 is true. However, the only way to solve such problems which involve functions other than simple addition and multiplication would be to solve the expression completely as stated above.



SVP
Joined: 06 Sep 2013
Posts: 1764
Concentration: Finance

Re: If the operation @ is defined for all integers a and b
[#permalink]
Show Tags
31 Oct 2013, 05:39
Well, can't argue that learning the definitions is in fact quite interesting and thank you for that. Nevertheless, I was really intereted in solving statement 3 quicker/faster/more efficient So, being able to recognize if operations in the different order given will yield same result without having to go through all the long distribution process. I will try to come up with a faster way but if anyone else come's up with a nice and elegant approach I'd be happy to give some nice Kudos for the collection Cheers J



Manager
Joined: 10 Mar 2013
Posts: 209
GMAT 1: 620 Q44 V31 GMAT 2: 690 Q47 V37 GMAT 3: 610 Q47 V28 GMAT 4: 700 Q50 V34 GMAT 5: 700 Q49 V36 GMAT 6: 690 Q48 V35 GMAT 7: 750 Q49 V42 GMAT 8: 730 Q50 V39

Re: If the operation @ is defined for all integers a and b
[#permalink]
Show Tags
01 Mar 2014, 16:08
Just took this question today, and I was also wondering if there were a way to solve it more quickly than performing the heavy manipulations that are in III or guessing numbers.



Manager
Joined: 10 Mar 2014
Posts: 199

Re: If the operation @ is defined for all integers a and b
[#permalink]
Show Tags
13 Jul 2014, 22:50
Bunuel wrote: cmugeria wrote: If the operation @ is defined for all integers a and b by a@b=a+bab, which of the following statements must be true for all integers a, b and c?
I. a@b = b@a II. a@0 = a III. (a@b)@c = a@(b@c)
(A) I only (B) II only (C) I and II only (D) I and III only (E) I, II and III We have that: \(a@b=a+bab\) I. \(a@b = b@a\) > \(a@b=a+bab\) and \(b@a=b+aab\) > \(a+bab=b+aab\), results match; II. \(a@0 = a\) > \(a@0=a+0a*0=0\) > \(0=0\), results match; III. \((a@b)@c = a@(b@c)\) > \((a@b)@c=a@b+c(a@b)*c=(a+bab)+c(a+bab)c=a+b+cabacbc+abc\) and \(a@(b@c)=a+b@ca*(b@c)=a+(b+cbc)(b+cbc)a=a+b+cbcab+abc\), results match. Answer: E. Hi Bunuel, Here in second statement how you are getting 0=0? Thanks.



Math Expert
Joined: 02 Sep 2009
Posts: 50002

Re: If the operation @ is defined for all integers a and b
[#permalink]
Show Tags
14 Jul 2014, 03:07
PathFinder007 wrote: Bunuel wrote: cmugeria wrote: If the operation @ is defined for all integers a and b by a@b=a+bab, which of the following statements must be true for all integers a, b and c?
I. a@b = b@a II. a@0 = a III. (a@b)@c = a@(b@c)
(A) I only (B) II only (C) I and II only (D) I and III only (E) I, II and III We have that: \(a@b=a+bab\) I. \(a@b = b@a\) > \(a@b=a+bab\) and \(b@a=b+aab\) > \(a+bab=b+aab\), results match; II. \(a@0 = a\) > \(a@0=a+0a*0=0\) > \(0=0\), results match; III. \((a@b)@c = a@(b@c)\) > \((a@b)@c=a@b+c(a@b)*c=(a+bab)+c(a+bab)c=a+b+cabacbc+abc\) and \(a@(b@c)=a+b@ca*(b@c)=a+(b+cbc)(b+cbc)a=a+b+cbcab+abc\), results match. Answer: E. Hi Bunuel, Here in second statement how you are getting 0=0? Thanks. II says: \(a@0 = a\) LHS = \(a@0=a+0a*0=a\). RHS = a a=a.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Current Student
Joined: 28 Sep 2014
Posts: 16

Re: If the operation is defined for all integers a and b
[#permalink]
Show Tags
20 Jan 2015, 23:41
I have a question regarding this problem. Don't we have to test for negative numbers while expanding the equation? This is where I got stuck and it became time consuming for me at which point I guessed and moved on. Please advise why that's not a factor.



Intern
Joined: 05 Jul 2015
Posts: 2

Re: If the operation is defined for all integers a and b
[#permalink]
Show Tags
26 Jul 2015, 07:16
Yes, tested for negative values and got stuck. For a@0 = a. If we put a is negative then it doesn't hold. Any thoughts on this? Where am I going wrong?



Current Student
Joined: 20 Mar 2014
Posts: 2633
Concentration: Finance, Strategy
GPA: 3.7
WE: Engineering (Aerospace and Defense)

Re: If the operation is defined for all integers a and b
[#permalink]
Show Tags
26 Jul 2015, 07:23
Yoshit wrote: Yes, tested for negative values and got stuck. For a@0 = a. If we put a is negative then it doesn't hold. Any thoughts on this? Where am I going wrong? It does: lets say a=4 > a@0=4@0 = 4+00*4 = 4 =a. This HAS to be true even by the underlying algebra! Thus satisfies the given equation. Can you show your steps for calculations?



Intern
Joined: 05 Jul 2015
Posts: 2

Re: If the operation is defined for all integers a and b
[#permalink]
Show Tags
26 Jul 2015, 08:17
You are right, taking negative values I jumbled up equations. Thank you for your help.




Re: If the operation is defined for all integers a and b &nbs
[#permalink]
26 Jul 2015, 08:17



Go to page
1 2
Next
[ 26 posts ]



