Bunuel wrote:
Is \(\frac{x + 1}{y + 1} > \frac{x}{y}\)?
(1) \(0 < x < y\)
(2) \(xy > 0\)
Target question: Is (x+1)/(y+1) > x/y ? Statement 1: 0 < x < y This tells us that y is POSITIVE, which means y+1 is also POSITIVE.
This means we can safely take the inequality
(x+1)/(y+1) > x/y and safely multiply both sides by y
When we do so, we get:
(y)(x+1)/(y+1) > x We can also multiply both sides by y+1 to get:
(y)(x+1) > (x)(y+1) Expand to get:
xy + y > xy + x Subtract xy from both sides to get:
y > x So, with the help of statement 1, our original target question
Is (x+1)/(y+1) > x/y ? becomes
Is y > x ?Since statement 1 tells us that y > x, the answer to the target question is a definitive
YESSince we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: xy > 0Let's TEST some values
There are several values of x and y that satisfy statement 2 (xy > 0). Here are two:
Case a: x = 1 and y = 1. In this case, (x+1)/(y+1) = (1+1)/(1+1) = 2/2 = 1, and x/y = 1/1 = 1. So, the answer to the target question is
NO, (x+1)/(y+1) is NOT greater than x/y ?Case b: x = -3 and y = -2. In this case, (x+1)/(y+1) = (-3 +1)/(-2 +1) = -2/-1 = 2, and x/y = -3/-2 = 3/2. So, the answer to the target question is
YES, (x+1)/(y+1) IS greater than x/y ?Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Answer: A
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