For any numbers a and b, a#b=a + b - ab. If a#b=0, which of

Some function (#) is defined for all numbers \(a\) and \(b\) as \(a\#b= a + b - ab\).

Now, since given that \(a\#b=0\), then \(a + b - ab=0\);

Express a in terms of b: \(a=\frac{b}{b-1}\)

If \(b=1\) then the given expression is undefined so \(b\) cannot equal to 1.

Answer: B.

Or:

\(a + b - ab=0\);

\((a-1)(1-b)+1=0\);

\((a-1)(1-b)=-1\).

If \(b=1\), then \((a-1)(1-b)=0\) not -1, so \(b\) cannot equal to 1.

Answer: B.

Hope it's clear.
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Can I do it this way?

I will just plug the number in to see if we can solve for a.

a#b=a+b-ab and a#b=0

=> a+b-ab=0

A. b=2 -> a=2
B. b=1 -> a+1-a=0 -> 1=0???
C. b=0 -> a=0
D. b=-1 -> a=1/2
E. b=-3/2 -> a= 3/5

So it's clear that b cannot be 1!

Yes, it's a valid approach.

Check other function questions in our Special Questions Directory:

Operations/functions defining algebraic/arithmetic expressions
Symbols Representing Arithmetic Operation
Rounding Functions
Various Functions

Hope it helps.
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Hi All,

This is an example of a Symbolism question (and you'll likely see 1 on Test Day). The idea is that you'll be given a "made up" math symbol, told what it "means" mathematically and then asked to solve some minor equation.

Here, we're told to substitute values in for A and B so that the equation….

A + B - AB = 0

We're asked which of the following answers CANNOT be the value of B? So 4 of the answers are POSSIBLE and one is IMPOSSIBLE. There are a couple of ways to approach this prompt. You could work mathematically or you can TEST THE ANSWERS. I'm going to use the answers to my advantage and find the 4 that are possible solutions and the one the creates an impossible situation:

If B = 2, then we'd have...
A + 2 - 2A = 0
2 = A
So B COULD be 2

If B = 1, then we'd have…
A + 1 - A = 0
1 = 0???????
B CANNOT equal 1

At this point, we could stop. I'll show you why the other answers are possible though:

If B = 0, then we'd have…
A + 0 - 0 = 0
0 = A
So B COULD be 0

If B = -1, then we'd have…
A -1 -(-1)(A) = 0
2A = 1
1/2 = A
So B COULD be -1

If B = -2/3, then we'd have…
A - 2/3 -(-2/3)(A) = 0
A + 2A/3 = 2/3
5A/3 = 2/3
A = 6/15
So B COULD be -2/3

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Given:
a # b = a + b – ab.
and a # b = 0

i.e. a + b – ab = 0

(Adding 1 and subtracting 1 simultaneously in the left part of the equation)

i.e. a + b - ab -1 +1 = 0

i.e. a(1-b) -1 (1-b) = -1

i.e. (a-1)*(1-b) = -1

i.e. (a-1)*(b-1) = 1

i.e. either a = 0 and b = 0
or a = 2 and b = 2
or (a, b) = (3, 3/2) in any order etc.

Point to be Noted:
for \((a-1)*(b-1) = 1\)to be true none of the parts of equation can be zero i.e.
(a-1)≠0 as well as (b-1)≠0
i.e. a and b can NOT be 1

Answer: Option B

Rich,

Thanks for the TEST the Answers method. I was trying to find answer through solving the function and took time whereas your method was very effective.
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Hi akadiyan,

Since the GMAT is a big 'critical thinking test', you'll find that almost every question that you'll face can be approached using more than one approach. In that way, the Exam actually 'rewards' strong thinkers - if you know multiple ways to get to the correct answer, and choose the most efficient method for the question that's in front of you, then you can get to the solution with less work and in a shorter amount of time. By extension, it then becomes easier to score at a higher level overall.

Many Test Takers focus solely on questions that they get wrong - and that makes a certain amount of sense - but there's some significant potential benefits to reviewing questions that you answered correctly (since there might be faster methods that you could use or other 'efficiencies' that can help you to improve your Scores AND your pacing).

GMAT assassins aren't born, they're made,
Rich

We are given that a#b=a + b - ab and that a#b=0. Thus we have

a + b - ab = 0

a + b = ab

Now let’s look at the choices:

A. If b = 2, then

a + 2 = a(2)

2 = a

So b can be 2.

B. If b = 1, then

a + 1 = a(1)

We see that a + 1 (1 more than a) can’t be equal to a itself. So b CANNOT be 1, and that is the number we are looking for.

Alternate Solution:

We are given that a#b=a + b - ab and that a#b=0. Thus we have

a + b - ab = 0

a + b = ab

b = ab - a

b = a(b - 1)

We observe here that b = 1 results in the inconsistent result of 1 = 0. Thus, the only value that b cannot assume is 1.

Answer: B
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An alternate way to think about could be (and I may be wrong)

a + b - ab = 0

If b = 1
then
a + 1 - a => 0 + 1 ≠ 0

Fair enough?

Hi D4kshGargas,

Yes - your approach IS valid (the approach that you used involves TESTing THE ANSWERS 'against' the prompt) - and you'll likely be able to use it a couple of times in the Quant section on Test Day (I go into more detail with that approach in my post that's earlier in the thread). One of the great aspects of the GMAT is that most questions are written so that they can be approached in more than way - and there are often strategic shortcuts that you can use to avoid having to do long-winded math. Keep an eye out for those opportunities; they can help you speed up and increase your Scores!

GMAT assassins aren't born, they're made,
Rich

Since it has been given that :
a#b = a+b - ab
If a#b = 0.
a+b = ab.
This can be rewritten as :
\(\frac{1}{a}+\frac{1}{b}=\ 1\)
If one of the two terms is equal to 1. Then the other term must be equal to zero.
For the other term to be 0.
The denominator must be equal to infinity which is not defined.
Hence if either of a, b is equal to 1, then the other term is not defined.


Why can't i rearrange the formula like this?

ab=a+b
ab-b=a
b(a-1)=a
(a-1)=a/b

b=/= 0