GMATPrep Question - If the operation (*) is defined for...

[Prep#1]

If the operation (*) is defined for all integers a and b by a (*) b = a + b - ab, 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)

Here was a question that I was wondering if anyone could offer a simpler method for calculating out option III?

[RonPurewal]

i suppose it's safe to assume that by 'simpler' you mean simpler than doing out all the algebra - so let's assume that you know how to do out the algebra. if you don't, then please reply accordingly.

a 'simpler' approach here is to pick three totally random numbers - say a = 3, b = 8, c = 10 - and test the statement; if the statement works with these arbitrarily chosen integers, then there's an excellent probability that it will work all the time. (notice that this is NOT A VALID METHOD OF PROOF; you may find that the statement works, but are unlucky enough to have chosen precisely the numbers that make it work by coincidence; however, that's extremely unlikely. also, notice that if the statement DOESN'T work, then you HAVE proved that it's invalid.)

try those numbers:

I. 3 + 8 - 24 = 8 + 3 - 24
-13 = -13
works

II. 3 + 0 - 0 = 3
3 = 3
works

III. first find a*b = -13 (see above), and b*c = 8 + 10 - 80 = -62.
so this says
-13 + 10 - (-130) = 3 - 62 - (-186)
127 = 127
works

looks like all three are good

[Amit]

Ron, but you are assuming "*" to be the operator "+", it can be any other operator such as -, X or /....

So what is a general approach for this

[TakingGMAT]

Here Ron is not assuming * to be +. But in the question itself it is given that a*b = a+b-ab
Sorry Ron for replying on your behalf

[RonPurewal]

Ron, but you are assuming "*" to be the operator "+", it can be any other operator such as -, X or /....

So what is a general approach for this

as pointed out by the poster above (thanks), i am making no such assumption.
all i am doing is following the exact directions provided in the question prompt, which defines the "*" operator. i am just plugging the values into the provided definition; that is all.

this is why you should really like problems involving strange made-up functions symbols; if you can FOLLOW DIRECTIONS and PLUG NUMBERS INTO THE PROVIDED DEFINITION, then you will be able to solve the problems. good stuff.

[RL]

If the operation @ is defined for all integers A and B by A@B = A+B-AB, which of the following statements must be true for all integers A, B, and C?

You can work the algebra on this to find that all three answers are true.

A@B = B@A
Expand this out and you get A + B - AB = B + A - BA, an equation that is obviously true

A @ 0 = A
Expand this one and you get A + 0 - 0A = A, equally obviously true.

(A@B)@C = A@(B@C)
This one is a bit more difficult to expand:

(A+B-AB)@C = A@(B+C-BC)
(A+B-AB) + C - C(A+B-AB) = A + (B+C-BC) - A(B+C-BC)
(A+B-AB) + C - CA - CB + ABC = A + (B+C-BC) - AB - AC +ABC

You can now see that there are the same elements on both sides, so this is also true, but certainly not obvious!

I agree with the previous post that for most people plugging in some random numbers is the easiest way to get the right answer 95% of the time on these kinds of problems. For those reaching for scores of 750+, you'll want to be able to quickly recognize associative formulas like this one. See http://en.wikipedia.org/wiki/Associativity

[JonathanSchneider]

Nice work.

[NinaP494]

Hi RL/Other MGMAT experts,

i did go through the wiki page but did not find any rule that can help me conclude III w/o algebra or number picking. Are there any rules that imply associativity and/or commutativity for certain operations?

[RonPurewal]

i did go through the wiki page but did not find any rule that can help me conclude III w/o algebra or number picking.

you're looking for a way to do this without EITHER using algebra OR picking numbers?

...couldn't tell you. from where i stand, that seems to be the whole universe of possibilities.

[RonPurewal]

Are there any rules that imply associativity and/or commutativity for certain operations?

i don't know what these things mean, so i can't specifically answer this.

what i CAN tell you, though, is that this is pretty much exactly the WRONG approach to take to this exam. you CANNOT 'memorize your way to success' on this exam.
in fact, this is pretty much the entire purpose of the whole test—to require as little knowledge as possible, and to rely as much as possible on pattern recognition, strategic thinking, mental flexibility, and universal problem-solving skills.