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by Param800 » Wed Jan 09, 2013 4:49 am
Can someone show me a short cut method to solve the below question ?

Q. If x and y are positive integers, is (2+x)/(3+y) greater than (2+y)/(3+x) ?

(1) x+y =3

(2) x>y

OA B

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Source: — Data Sufficiency |

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by Brent@GMATPrepNow » Wed Jan 09, 2013 5:18 am
Param800 wrote:Can someone show me a short cut method to solve the below question ?

Q. If x and y are positive integers, is (2+x)/(3+y) greater than (2+y)/(3+x) ?

(1) x+y =3

(2) x>y

OA B

Thank You
Target question: Is (2+x)/(3+y) greater than (2+y)/(3+x) ?

In other words, is (2+x)/(3+y) > (2+y)/(3+x)?
If x and y are both positive, then the denominators on both sides of the inequality are positive. This means we can simplify the target question by multiplying both sides by (3+y)(3+x) to get:
Is (2+x)(3+x) > (2+y)(3+y)?

To further simplify the target question, we'll first expand both sides to get:
Is 6 + 5x + x^2 > 6 + 5y + y^2
Simplify to get: Is 5x + x^2 > 5y + y^2
Notice that this inequality will true if x > y

Rephrased target question: Is x > y ?

Statement 1: x+y =3
There are several pairs of values that meet this condition. Here are two:
Case a: x=2 and y=1, in which case x is greater than y
Case b: x=1 and y=2, in which case x is not greater than y
Since we cannot answer the rephrased target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x>y
This definitely helps us answer our rephrased target question with certainty.
So, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent
Last edited by Brent@GMATPrepNow on Wed Jan 09, 2013 10:08 am, edited 1 time in total.
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by GMATGuruNY » Wed Jan 09, 2013 9:53 am
Param800 wrote:Can someone show me a short cut method to solve the below question ?

Q. If x and y are positive integers, is (2+x)/(3+y) greater than (2+y)/(3+x) ?

(1) x+y =3

(2) x>y

OA B

Thank You
Since x>0 and y>0, 2+x, 3+y, 2+y, and 3+x are all POSITIVE.
This means we can rephrase the question stem by cross-multipying:
(2+x) / (3+y) > (2+y) / (3+x) ?

(2+x)(3+x) > (2+y)(3+y)

6 + 5x + x² > 6 + 5y + y²

5x + x² > 5y + y² ?

The lefthand side will be greater than the righthand side if x > y.

Question rephrased: Is x > y?

Statement 1: x+y = 3
If x=2 and y=1, then x>y.
If x=1 and y=2, then x<y.
INSUFFICIENT.

Statement 2: x>y
SUFFICIENT.

The correct answer is C.
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by Brent@GMATPrepNow » Wed Jan 09, 2013 10:08 am
Ahhh...I never saw the / symbol. I thought it was straight multiplication. Bad me!

I've edited my post accordingly.

Cheers,
Brent
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