If x and y are positive numbers, is (x + 1)/(y + 1) > x/y?

[BTGModeratorVI]

If x and y are positive numbers, is (x + 1)/(y + 1) > x/y?

(1) x > 1
(2) x < y

Answer: B
Source: Official guide

[Jay@ManhattanReview]

If x and y are positive numbers, is (x + 1)/(y + 1) > x/y?

(1) x > 1
(2) x < y

Answer: B
Source: Official guide

Note the following.

1. If x/y < 1, where x and y are positive numbers, then (x + a) / (y + a) > x/y.

This is because since x < y and an equal positive quantity 'a' is being added to them, the proportionate increment on x, i.e., the numerator is more than that on y, the denominator.

2. If x/y > 1, where x and y are positive numbers, then (x + a) / (y + a) < x/y.

This is because since x > y and an equal positive quantity 'a' is being added to them, the proportionate increment on x, i.e., the numerator is less than that on y, the denominator.

Let's take each statement one by one.

(1) x > 1

We do not have any information about y. x/y can be greater or less than y. Insufficient.

(2) x < y

Since x < y, we have x/y < 1. Seeing (x + 1)/(y + 1), we find that a = 1 is added to x and y. So, (x + 1)/(y + 1) > x/y. Sufficient.

Correct answer: B

Hope this helps!

-Jay
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[Brent@GMATPrepNow]

If x and y are positive numbers, is (x + 1)/(y + 1) > x/y?

(1) x > 1
(2) x < y

Answer: B
Source: Official guide

Target question: Is (x + 1)/(y + 1) > x/y ?

Given: x and y are positive numbers

This is a great candidate for rephrasing the target question.

We have the inequality: (x + 1)/(y + 1) > x/y
Since y is positive, we can multiply both sides of the inequality by y
Likewise, since y is positive, we know that y+1 is positive, which means we can multiply both sides of the inequality by (y+1)
When perform both of these multiplications we get: (x + 1)(y) > (x)(y + 1)
Expand to get: xy + y > xy + x
Subtract xy from both sides to get: y > x
So, we can now ask...
REPHRASED target question: Is y > x ?

Statement 1: x > 1
There's no information about y, so there's no way to determine whether or not y > x
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x < y
Perfect!!!
Since we can answer the REPHRASED target question with certainty, statement 2 is SUFFICIENT

Answer: B

[deloitte247]

$$\frac{\left(x+1\right)}{\left(y+1\right)}>\frac{x}{y}\ \ \ \ \ \left(cross\ multiply\right)$$
$$y\left(x+1\right)>x\left(y+1\right)$$
$$yx+y\ >\ xy+x$$
$$xy+y\ >\ xy+x$$
$$xy+y-xy>xy+x-xy$$
$$y>x$$

Target question => $$is\ y>x\ ?$$

Statement 1 => x > 1
If x = 3 and y = 1 then x > y but if x = 2 and y = 5 then y > x. Since there is no specific information about the value of y, we cannot the definite answer
Statement 1 is NOT SUFFICIENT

Statement 2 => x < y
This definitely means that y > x. Statement 2 is SUFFICIENT

Since statement 2 alone is SUFFICIENT
Answer = B