Manhattan GMAT Challenge Problem of the Week - 24 May 2011

by Manhattan Prep, May 24, 2011

Here is a new Challenge Problem! If you want to win prizes, try entering our Challenge Problem Showdown. The more people that enter our challenge, the better the prizes!

Question

For all numbers x and y, the operation # is defined by

x # y = [pmath](x + y)^3[/pmath],

whereas the operation ~ is defined by

x ~ y = [pmath](x-y)^3[/pmath].

At which of the following points in the xy plane is the value of x # y equal to the value of x ~ y?

(A) (1, 1)

(B) (1, -1)

(C) (0, 2)

(D) (-2, 0)

(E) (-2, 2)

Answer

To attack this problem, we need to deal with two strange symbol formulas. We need to identify which of the five points given in the answer choices makes the two formulas equal.

One lengthy approach is to set the formulas equal to each other and perform algebra.

x # y = x ~ y

[pmath](x + y)^3[/pmath] = [pmath](x-y)^3[/pmath]

[pmath]x^3+ 3 x^2y + 3yx^2 + y^3[/pmath] = [pmath]x^3-3{x^2}y + 3y{x^2}-y^3[/pmath]

(You dont have to know that expansion you can get it by FOILing)

[pmath]3{x^2}y +y^3[/pmath] = [pmath]-3{x^2}y -y^3[/pmath]

[pmath]6{x^2} y + 2 y^3[/pmath] = 0

[pmath]y(6x^2+2y^2) = 0[/pmath]

Either y = 0 or 6[pmath]x^2[/pmath] + 2[pmath]y^2[/pmath] = 0. The latter is only true if both x and y equal 0. So we must have at least the condition that y = 0. Comparing to the choices, we should see that only (-2, 0) works.

Alternatively, by looking at the two formulas, we might search for a quick shortcut. When would these expressions be equal? By asking ourselves that question, we might realize that if y = 0, both formulas reduce to [pmath]x^3[/pmath]. So the two formulas will be equal whenever y= 0. The only viable option among the five answer choices is (-2, 0).

Finally, we can just plug and chug the answer choices. This might take more time, but as long as our substitutions and computations are correct, we will get the right answer.

The correct answer is D.

Special Announcement: If you want to win prizes for answering our Challenge Problems, try entering our Challenge Problem Showdown. Each week, we draw a winner from all the correct answers. The winner receives a number of our our Strategy Guides. The more people enter, the better the prize. Provided the winner gives consent, we will post his or her name on our Facebook page.