GMATPrep Algebra

[erjamit]

Hi,
Pls help with this.

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)

A) I
B) II
C) I,II
D) I,III
E) I,II,III

I couldn't figure out why III is true. Can someon explain how to derive III.

OA E

Thanks
Amit

[malolakrupa]

I think the answer is C .

[erjamit]

As already told OA is E.

Pls explain why the III option is correct.

[parallel_chase]

III. (a*b)*c = a*(b*c)


Let (a*b) = y = a+b-2ab

y*c = y + c -2yc = a+b-ab +c - 2c(a+b-2ab) = a+b+c-2ab-2ac-2bc+4abc

Let (b*c) = x = b+c-2bc

a*x = a+x-2ax = a+b+c-2bc-2a(b+c-2bc) = a+b+c-2ab-2ac-2bc+4abc


Hence the answer is E.

Hope this helps.

[erjamit]

(a*b)*c = (a + b - ab)*c
= a + b - ab + c - (a + b - ab)c


a*(b*c) = a*(b+c-bc)
= a + b +c - bc - a(b+c-bc)

What is wrong with my approach?

Thanks
Amit

[parallel_chase]

(a*b)*c = (a + b - ab)*c
= a + b - ab + c - (a + b - ab)c

absolutely nothing is wrong with your approach, If you expand,

a+b-ab+c-ac-bc+abc -------I

a*(b*c) = a*(b+c-bc)
= a + b +c - bc - a(b+c-bc)

a+b+c -bc-ab-ac+abc -------II

If you compare I and II both are equal.

In fact i took the a*b = a+b-2ab instead of a+b-ab

But it doesnt matter as you can see one can still get E as the answer.

[erjamit]

This is what nerves do to you. ......

I just solved till this part and then glanced at both options and eliminated it.

Thanks mate.

Amit

[aj5105]

imo, one of better ways to solve these kind of problems is use numbers.

[olpre4]

1) What is being tested in this problem?

2) Operationally, how did you get

From (a*b)*c = (a + b - ab)*c

To = a + b - ab + c - (a + b - ab)c

[niraj_a]

you could try picking numbers abd testing this out too....