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Director  Joined: 29 Nov 2012
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If x and y are positive integers, is (2 + x)/(3 + y) greater  [#permalink]

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Difficulty:   35% (medium)

Question Stats: 73% (01:50) correct 27% (02:07) wrong based on 453 sessions

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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?

Originally posted by fozzzy on 12 Mar 2013, 03:48.
Last edited by hazelnut on 04 May 2017, 04:07, edited 2 times in total.
EDITED THE QUESTION.
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Math Expert V
Joined: 02 Sep 2009
Posts: 57026
Re: If x and y are positive integers, is (2 + x)/(3 + y) greater  [#permalink]

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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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Math Expert V
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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater  [#permalink]

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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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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater  [#permalink]

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Great! Explanation and elaboration really helpful !
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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater  [#permalink]

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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?
Manager  Joined: 14 Aug 2005
Posts: 57
Re: If x and y are positive integers, is (2 + x)/(3 + y) greater  [#permalink]

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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]

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[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
Intern  Joined: 02 Jun 2014
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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater  [#permalink]

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Thanks for the elaboration, really helpful !
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Joined: 18 May 2014
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Schools: Stern '21
Re: If x and y are positive integers, is (2 + x)/(3 + y) greater  [#permalink]

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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.

I took a different approach but I'm curious if it it's fail proof or I lucked out:

First, I cross multiplied "up" to read "Is (2 +x)(3+x) > (2+y)(3+y)?"

Then I used FOIL to read "Is x^2 +5x +6 > y^2 +5y+6?" or essentially "Is x>y? " (FOILing may have been unnecessary but I was quickly able to see the x>y this way)

Statement 1:
X+Y= 3 ; NS as x can be either 1 or 2; same for y

Statement 2:
x>y; This explicitly answers my question SUFFICIENT Re: If x and y are positive integers, is (2 + x)/(3 + y) greater   [#permalink] 21 Dec 2018, 18:44
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