16. For any numbers a and b, a · b = a + b – ab.

[jellie]

16. For any numbers a and b, a · b = a + b - ab.
If a · b = 0, which of the following CANNOT be a value of b?
(A) 2
(B) 1
(C) 0
(D) -1
(E) 23


The correct answer is B, but could someone plz explain why?

[Guest]

If you solve for a

a+b=ab
a-ab=-b
a[1-b]=-b
a=b/b-1

if b=1
then a is indeterminate

Hope this explains the answer

[Guest]

BUt if b = 1, then doesn't that mean a has to be 0? Why is it indeterminate?

[Guest]

I think what he/she is implying is that GMAT will never allow you to have 0 in the denominator. if you have b=1, then the denominator in the equation as posted above will have a value of 0. We all know that 0 in a denominator is considered "undefined," but apparently to GMAT, 0 is impossible. the value for "a" can NOT be undefined

[RonPurewal]

BUt if b = 1, then doesn't that mean a has to be 0? Why is it indeterminate?

nope.

1/0 is either "infinity" or "undefined", depending on whether you allow discussion of infinite values. it is most definitely not zero.

-- ALL MATERIAL BELOW THIS POINT IS IRRELEVANT TO THE GMAT --

digression:
1/0 is actually not "indeterminate", in the mathematically formal sense of that word; it has a determined value of infinity.
"indeterminate" actually refers to expressions that can have any value we want them to have. one such expression is 0/0, which can be shown (via the theory of limits in calculus) to be able to take on any desired value. in other words, if i want 0/0 to be 17, or at least to approach 17, i can make that happen with appropriate expressions. i could also make it 0, infinity, -1000, √5, or any other value i want.
0^0 (zero to the zero power) is another such indeterminate expression. if you see 0^0 on the gmat, run for your life; you've done something dreadfully wrong and should start essentially from scratch.

[Rahul]

The question does not explicitly state that a and b are rational numbers. Infact it says a and b are any numbers. Only rational numbers are of the form p/q where q is not equal to 0.
Can we assume that if nothing is mentioned, all numbers are expected to be atleast rational numbers ?

The question is also slightly misleading because some people (atleast I did) might tend to mistake the '·' for the multiplication operator, when infact it is being used to refer to an undefined operator.
So when I looked at the equation a · b = a + b - ab and a · b=0, I reasoned
0 = a + b - 0
a = -b
It took me some time to find out where I had gone wrong :)

[RD]

a.b = a + b - ab; if a.b = 0, which values b cannot have

1.

a + b - ab = 0

if b = 1, above equation

a + 1 - a(1) = 0

a - a + 1 = 0

1 = 0.......not possible. So b CONNOT be 1.

2.

a + b - ab = 0

a + b = ab

dividing both sides by ab

1/b + 1/a = 1

if b = 0, 1/b is undefined/infinite

i.e inifinite = 1........which is also not possible. so b CANNOT be 0.

is it GMAT's blunder? or am I missing something here?

Am puzzled??

Ron......pls help!!

[RonPurewal]

your error is here:

dividing both sides by ab

1/b + 1/a = 1

you are not allowed to divide an equation by a VARIABLE on both sides, unless you have ascertained that the variable(s) in question is/are NONZERO.

you have most certainly not ascertained that ab is nonzero. in fact, you've done 2 things that can't possibly coexist:
(a) divided by a variable expression (ab), revealing an underlying assumption that ab is nonzero; AND THEN
(b) set b to zero, which would mean that ab IS zero.
bad.

--

NEVER divide by variables that could potentially be zero. there are some rules that you can follow that occasionally allow you to do this, but, for the most part, it's just not worth it.