Bunuel wrote:
Given that x is a negative number and 0 < y < 1, which of the following is the greatest?
(A) x^2
(B) (xy)^2
(C) (x/y)^2
(D) x^2/y
(E) x^2*y
Let’s check the answer choices by using strategic and convenient numbers for x and y. So let’s let x = -2 and y = 1/2.
A) x^2 = (-2)^2 = 4
B) (xy)^2 = [(-2)(1/2)]^2 = (-1)^2 = 1
C) (x/y)^2 = [(-2)/(1/2)]^2 = (-4)^2 = 16
D) x^2/y = (-2)^2/(1/2) = 4/(1/2) = 8
E) x^2*y = (-2)^2*(1/2) = 4*(1/2) = 2
Alternate Solution:
Since 0 < y < 1, |x|y < |x| and thus, (xy)^2 < x^2; so A > B.
Since |x| < |x|/y, x^2 < (x/y)^2 and thus, C > A.
Notice that (x/y)^2 = (x^2/y)*(1/y). Since (1/y) > 1, (x/y)^2 > x^2/y and thus, C > D.
Finally, since 0 < y < 1, x^2*y < x^2 < (x/y)^2. Thus, C > E.
Answer: C
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