Notice that statements I and II say the exact OPPOSITE thing.
Also, notice that either I or II appear in ALL 5 answer choices. So, one of them must be true.
Given this, let's see what happens when we assign some values to x and y.

Say x = 4 and y = 2
So, z = x/y = 4/2 = 2

Now test (x − 1)/(y − 1) by plugging in x = 4 and y = 2
We get: (4 - 1)/(2 - 1) = 3/1 = 3
Since z = 2, we can see that z < (x − 1)/(y − 1)
So, statement II is true, and statement I is false.

This means the correct answer is C or D

Now test statement III: z > (x + 1)/(y + 1)
Since z = x/y, we can rewrite III as x/y > (x + 1)/(y + 1)
Let's find out if this statement is true.
Since y is POSITIVE, we can safely multiply both sides by y to get: x > (y)(x + 1)/(y + 1)
y+1 is also POSITIVE, we can safely multiply both sides by y+1 to get: x(y + 1) > (y)(x + 1)
Expand both sides: xy + x > xy + y
Subtract xy from both sides to get: x > y
Is this true? YES! It's given in the question
So statement III is true

Answer: C