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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]

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Updated on: 04 May 2017, 04:07
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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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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater  [#permalink]

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12 Mar 2013, 04:08
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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]

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20 Mar 2013, 05:06
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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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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater  [#permalink]

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08 Jun 2014, 06:59
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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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17 Mar 2013, 08:36
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?
Manager
Joined: 14 Aug 2005
Posts: 55
Re: If x and y are positive integers, is (2 + x)/(3 + y) greater  [#permalink]

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18 Mar 2013, 22:57
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]

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20 Mar 2013, 03:02
[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]

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08 Jun 2014, 06:40
Thanks for the elaboration, really helpful !
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Joined: 18 May 2014
Posts: 8
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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater  [#permalink]

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21 Dec 2018, 18:44
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
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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater  [#permalink]

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24 Dec 2019, 17:05
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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater   [#permalink] 24 Dec 2019, 17:05
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