Which of the following is the smallest given that x > y > 1 ?

Question: Which of the following is the smallest given that x > y > 1 ?

Solution:
Given that, x > y > 1,
Consider the expressions,
∴ (x/y) >1 (since x > y)

Adding the same positive number to numerator and denominator reduces the fraction when numerator > denominator.
Thus,
(x/y) > (x+1)/(y+1) > (x + 2)/(y + 2)
Also, (x + 1)/y > (x+1)/(y+1)​ (larger denominator)
and xy > x > y > 1

∴ The smallest is (x + 2)/(y + 2)​