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a^2-b^2

by j_shreyans » Mon Sep 15, 2014 3:11 am
If ab=x and a/b=y, and ab does not equal zero, then a^2−b^2

A)x(y-1)(y+1)/y

B)xy-y/x

C)X^2-y^2

D)y(x^2-1/x^2)

E)(y+x)(y-x)/x

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by GMATGuruNY » Mon Sep 15, 2014 3:17 am
j_shreyans wrote:If ab=x and a/b=y, and ab does not equal zero, then a^2−b^2

A)x(y-1)(y+1)/y

B)xy-y/x

C)X^2-y^2

D)y(x^2-1/x^2)

E)(y+x)(y-x)/x
Let a=4 and b=2.
Then:
x = ab = 4*2 = 8.
y = a/b = 4/2 = 2.
a² - b² = 4² - 2² = 16-4 = 12. This is our target.

Now plug x=8 and y=2 into the answers to see which yields our target of 12.
Only A works
[x(y-1)(y+1)]/y = [(8(2-1)(2+1)]/2 = 12.

The correct answer is A.
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by Brent@GMATPrepNow » Mon Sep 15, 2014 6:25 am
j_shreyans wrote:If ab = x and a/b = y, and ab does not equal zero, then a² − b²

A) x(y-1)(y+1)/y

B) xy - y/x

C) x² - y²

D) y(x² - 1/x²)

E) (y+x)(y-x)/x
I think Mitch's approach (what I call the INPUT-OUTPUT approach) is best here.
That said, we can also solve the question algebraically.

If ab = x and a/b = y, then (ab)(a/b) = xy
Simplify to get: a² = xy

Likewise, if ab = x and a/b = y, then ab/(a/b) = x/y
Simplify to get: b² = x/y

So, a² − b² = xy - x/y

Check the answer choices....not there. Looks like we need to take xy - x/y and rewrite it so that it looks like one of the answer choices. This is the potential problem with algebraic solutions.

First, xy - x/y = x(y - 1/y) [factored out the x] ....still not among the answer choices.
= (x/y)(y² - 1) [factored out 1/y] ....still not among the answer choices.
= (x/y)(y+1)(y-1) [factored y² - 1] ....still not among the answer choices.
= x(y-1)(y+1)/y [rewrote the expression]....BINGO!

Answer: A

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