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

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If x and y are positive integers, is (2 + x)/(3 + y) greater [#permalink] New post 12 Mar 2013, 02:48
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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

Can we cross multiply in this question? Suppose it wasn't given positive then we move the variables to one side and solve?
[Reveal] Spoiler: OA

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Last edited by Bunuel on 12 Mar 2013, 02:58, edited 1 time in total.
EDITED THE QUESTION.
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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater [#permalink] New post 12 Mar 2013, 03:08
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If x and y are positive integers, is (2 + x)/(3 + y) greater than (2 + y)/(3 + x)?

Is \frac{2 + x}{3 + y}>\frac{2 + y}{3 + x}? Since both denominators are positive we can safely cross-multiply. Though we can solve the question without doing that.

(1) x + y = 3. If x=1 and y=2, then the answer is NO (3/5<4/4) but if x=2 and y=1, then the answer is YES (4/4>3/5). Not sufficient.

(2) x > y. This implies that the numerator of LHS is more than the numerator of RHS, and the denominator of LHS is less than the denominator of RHS, which means that LHS > RHS. Sufficient.

Answer: B.

Hope it's clear.
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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater [#permalink] New post 17 Mar 2013, 07:36
Bunuel wrote:
If x and y are positive integers, is (2 + x)/(3 + y) greater than (2 + y)/(3 + x)?

Is \frac{2 + x}{3 + y}>\frac{2 + y}{3 + x}? Since both denominators are positive we can safely cross-multiply. Though we can solve the question without doing that.

(1) x + y = 3. If x=1 and y=2, then the answer is NO (3/5<4/4) but if x=2 and y=1, then the answer is YES (4/4>3/5). Not sufficient.

(2) x > y. This implies that the numerator of LHS is more than the numerator of RHS, and the denominator of LHS is less than the denominator of RHS, which means that LHS > RHS. Sufficient.

Answer: B.

Hope it's clear.


Dear Bunuel,

Can you please elaborate on an Algebra approach?

So far I'd go like this:

Since we know that the denominators are positive, we can cross multiply:

x^2+5x<y^2+5y

x^2-y^2<5y-5x

(x+y)(x-y)<5(y-x)

(x+y)(x-y)<-5(x-y)

Here, can we divide by (x-y)? If not, how to continue?
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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater [#permalink] New post 18 Mar 2013, 21:57
Not sure if you would really like to take the algebra approach.

The question is pretty much clear about the usage of positive integer. So lets take a small set of positive integers {1,2,3,4}

Now, for us to get 2+x > 3+y we need to have only x>y; thats the only condition which can help us solve the equation. Since, that's given in B! Hence, B is the answer
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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater [#permalink] New post 20 Mar 2013, 02:02
[quote="fozzzy"]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


x,y +ve intigers

from 1

x,y are in fact 1,2 but we dont know which is which...insuff

from 2

if x>y then : numerator 2+x >2+y ( numerator of each side) and 3+y<3+x (denominator of each side), thus larger numerator/smaller denominator is surely > smaller numerator/ larger denominator ..hope this makes sense
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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater [#permalink] New post 20 Mar 2013, 04:06
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LGOdream wrote:
Bunuel wrote:
If x and y are positive integers, is (2 + x)/(3 + y) greater than (2 + y)/(3 + x)?

Is \frac{2 + x}{3 + y}>\frac{2 + y}{3 + x}? Since both denominators are positive we can safely cross-multiply. Though we can solve the question without doing that.

(1) x + y = 3. If x=1 and y=2, then the answer is NO (3/5<4/4) but if x=2 and y=1, then the answer is YES (4/4>3/5). Not sufficient.

(2) x > y. This implies that the numerator of LHS is more than the numerator of RHS, and the denominator of LHS is less than the denominator of RHS, which means that LHS > RHS. Sufficient.

Answer: B.

Hope it's clear.


Dear Bunuel,

Can you please elaborate on an Algebra approach?

So far I'd go like this:

Since we know that the denominators are positive, we can cross multiply:

x^2+5x<y^2+5y

x^2-y^2<5y-5x

(x+y)(x-y)<5(y-x)

(x+y)(x-y)<-5(x-y)

Here, can we divide by (x-y)? If not, how to continue?


First of all, is (2 + x)/(3 + y) greater than (2 + y)/(3 + x)? Means is \frac{2 + x}{3 + y}>\frac{2 + y}{3 + x}?

Cross-multiply: is (2+x)(3+x)>(2+y)(3+y) --> is 5x+x^2>5y+y^2? --> is (x-y)(x+y)>-5(x-y)? Here we cannot divide by x-y, since we don't know whether it's positive or negative.

What we can do is: (x-y)(x+y)>-5(x-y)? --> (x-y)(x+y)+5(x-y)>0? --> (x-y)(x+y+5)>0?

(1) x + y = 3. The question becomes: is (x-y)(3+5)>0? --> is x-y>0? We don't know that, thus this statement is not sufficient.

(2) x > y --> x-y>0. So, we can reduce by x-y and the question becomes: is x+y+5>0? Since x and y are positive then the answer to this question is YES. Sufficient.

Answer: B.

Hope it helps.
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COLLECTION OF QUESTIONS:
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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater [#permalink] New post 08 Jun 2014, 05:37
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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater [#permalink] New post 08 Jun 2014, 05:40
Thanks for the elaboration, really helpful !
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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater [#permalink] New post 08 Jun 2014, 05:59
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Great! Explanation and elaboration really helpful !
Re: If x and y are positive integers, is (2 + x)/(3 + y) greater   [#permalink] 08 Jun 2014, 05:59
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