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Math Expert V
Joined: 02 Sep 2009
Posts: 59561
If x and y are positive numbers, is (x + 1)/(y + 1) > x/y  [#permalink]

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Difficulty:   25% (medium)

Question Stats: 76% (01:18) correct 24% (01:44) wrong based on 1053 sessions

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If x and y are positive numbers, is $$\frac{x + 1}{y + 1} > \frac{x}{y}$$?

(1) x > 1
(2) x < y
GMAT Club Legend  V
Joined: 12 Sep 2015
Posts: 4121
If x and y are positive numbers, is (x + 1)/(y + 1) > x/y  [#permalink]

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Bunuel wrote:
If x and y are positive numbers, is $$\frac{x + 1}{y + 1} > \frac{x}{y}$$?

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

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
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

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Originally posted by GMATPrepNow on 26 Jun 2017, 08:10.
Last edited by GMATPrepNow on 02 Jan 2019, 15:24, edited 1 time in total.
Retired Moderator P
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Re: If x and y are positive numbers, is (x + 1)/(y + 1) > x/y  [#permalink]

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23
4
If x and y are positive numbers, is $$\frac{x + 1}{y + 1} > \frac{x}{y}$$?

x & y are POSITIVE numbers

$$\frac{x + 1}{y + 1} > \frac{x}{y}$$

Lets simplify the equation, as we are given that x & y are positive integers we can multiply the R.H.S. with the L.H.S. as there will not be any impact on the inequality SIGN of the equation.

$$(x + 1) * (y) > (y + 1) * (x)$$

$$xy + y > xy + x$$

Cancelling $$xy$$ from both the sides.

$$y > x$$

So we want to answer, Is $$y > x$$ ?

(1) $$x > 1$$

This does not tell us anything about y. Hence, Eq. (1) =====> NOT SUFFICIENT

(2) $$x < y$$

We can write it as y > x, this is what we have to prove. Hence, Eq. (2) =====> SUFFICIENT

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Intern  Joined: 27 May 2017
Posts: 12
Re: If x and y are positive numbers, is (x + 1)/(y + 1) > x/y  [#permalink]

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1
x+1/y+1>x/y
cross multiply
xy +y > xy + x
subtract xy from both sides
y > x or x<y?

Ans B.

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Manager  B
Joined: 15 Mar 2015
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Might as well share that I picked C due to not Reading that x and y - positive. This would not allow crossmultiplication. Posted from my mobile device
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If x and y are positive numbers, is (x + 1)/(y + 1) > x/y  [#permalink]

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x and y are positive numbers
$$x,y>0 ; (x+1)y>(y+1)x ; y>x$$

(1) x > 1
X can be a proper fraction or improper fraction and y can be proper or improper fraction
NS
(2) x < y
Rephrased expression
Suff

Option B
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Re: If x and y are positive numbers, is (x + 1)/(y + 1) > x/y  [#permalink]

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Bunuel wrote:
If x and y are positive numbers, is $$\frac{x + 1}{y + 1} > \frac{x}{y}$$?

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

We are given that x and y are positive numbers and need to determine whether (x + 1)/(y + 1) > x/y. Simplifying the question, we have:

Is y(x + 1) > x(y + 1)? (Note that we multiplied each side by y(y + 1), which is allowed because y > 0.)

Is yx + y > xy + x?

Is y > x?

Statement One Alone:

x > 1

Since we do not know anything about y, statement one alone is not sufficient to answer the question.

Statement Two Alone:

x < y

We see that statement two has answered the question; y is greater than x.

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Intern  S
Joined: 24 Nov 2018
Posts: 19
Location: India
Re: If x and y are positive numbers, is (x + 1)/(y + 1) > x/y  [#permalink]

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1
1
(1) x <1 -- Insufficient
(2) x< y
let's understand:

if the ratio is less than 1: After adding some positive number to the numerator and denominator, the resulting ration increases
eg 2/3
adding 10 to numerator and denominator
2+10 / 3+10 =12/13 > 2/3

if the ratio is greater than 1: After adding some positive number to the numerator and denominator, the resulting ration decreases eg. 7/5
7+20 /5 +20 = 27/25 <75

Hence B is correct

please correct me if I am wrong
VP  V
Joined: 23 Feb 2015
Posts: 1335
If x and y are positive numbers, is (x + 1)/(y + 1) > x/y  [#permalink]

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Bunuel wrote:
If x and y are positive numbers, is $$\frac{x + 1}{y + 1} > \frac{x}{y}$$?

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

Here, x and y are positive numbers (i.e., $$x>0$$ and $$y>0$$)
If this is the case then $$\frac{x}{y}$$ must be positive. If $$\frac{x}{y}$$ is positive, then $$\frac{x + 1}{y + 1}$$ must be positive too..
So, the question is:
$$\frac{x + 1}{y + 1} > \frac{x}{y}$$?
I can answer this question if I know value of x and y (whatever the value it is!).

In statement 2:
Here is given that $$x < y$$.
I don't care what about the specific value of x and y here. I know from this statement that y is greater than x. If the statement even says that x is greater than y, i still don't care because statement NEVER lies!. Without any calculation in statement 2, I can surely say that it gives a definite YES or a definite NO, but NOT the simultaneous YES and NO. So, sufficient.

In statement 1:
There is no value of y. So, insufficient.
Is my understanding wrong Bunuel, IanStewart?
Thanks__
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Intern  Joined: 02 Nov 2019
Posts: 2
Re: If x and y are positive numbers, is (x + 1)/(y + 1) > x/y  [#permalink]

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It might be a silly question, but before ruling Statement 1 out, can't we use the fee values of Y (as the question stem says Y is a positive number) ?
Use few positive values of Y and plug in the question to see if we get all Yes or all No? Re: If x and y are positive numbers, is (x + 1)/(y + 1) > x/y   [#permalink] 16 Nov 2019, 14:04
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