Bunuel wrote:
If x and y are positive integers such that x < y, which of the following expressions must be less than 1?
I. \(\sqrt{\frac{y}{x}}\)
II. \(\frac{x^2 - 100}{y^2 - 100}\)
III. \(\frac{x}{y}\)
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) II and III only
As there are specific rules that govern if a fraction is larger or smaller than 1, we'll use them.
This is a Precise approach.
A fraction is smaller than 1 if its numerator is smaller than its denominator.
That is, \(\frac{smaller}{larger}<1\). Since x<y and both are positive, then \(\frac{x}{y}<1\).
III is true so (A), (B), (D) are eliminated.
So all we need to know is if II is true. Since x < y and both are positive integers then x^2 < y^2 meaning that x^2 - 100 < y^2 - 100.
So, if both are positive then \(\frac{x^2 - 100}{y^2 - 100}\) is smaller than 1.
But what happens if both are negative? In this case, then the numberator is a 'smaller number' = 'larger negative' and the denominator a 'larger number' = 'smaller negative' and once we cancel out the minus sign we get a fraction whose numerator is larger than its denominator, meaning that it is larger than 1.
(C) s our answer.
The question states that x and y are positive integers, hence our answer will be (E)