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)