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If x and y are positive integers, is (2 + x)/(3 + y) greater
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Updated on: 04 May 2017, 03:07
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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?
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Originally posted by fozzzy on 12 Mar 2013, 02:48.
Last edited by hazelnut on 04 May 2017, 03:07, 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
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12 Mar 2013, 03:08




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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater
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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 crossmultiply. 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: \(x^2+5x<y^2+5y\) \(x^2y^2<5y5x\) \((x+y)(xy)<5(yx)\) \((x+y)(xy)<5(xy)\) Here, can we divide by (xy)? If not, how to continue?



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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater
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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
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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
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20 Mar 2013, 04:06
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 crossmultiply. 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 \(x^2y^2<5y5x\) \((x+y)(xy)<5(yx)\) \((x+y)(xy)<5(xy)\) Here, can we divide by (xy)? If not, how to continue? First of all, is (2 + x)/(3 + y) greater than (2 + y)/(3 + x)? Means is \(\frac{2 + x}{3 + y}>\frac{2 + y}{3 + x}\)? Crossmultiply: is \((2+x)(3+x)>(2+y)(3+y)\) > is \(5x+x^2>5y+y^2\)? > is \((xy)(x+y)>5(xy)\)? Here we cannot divide by xy, since we don't know whether it's positive or negative. What we can do is: \((xy)(x+y)>5(xy)\)? > \((xy)(x+y)+5(xy)>0\)? > \((xy)(x+y+5)>0\)? (1) x + y = 3. The question becomes: is \((xy)(3+5)>0\)? > is \(xy>0\)? We don't know that, thus this statement is not sufficient. (2) x > y > \(xy>0\). So, we can reduce by xy 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
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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
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08 Jun 2014, 05:59
Great! Explanation and elaboration really helpful !



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Re: If x and y are positive integers, is (2 + x)/(3 + y) greater
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21 Dec 2018, 17:44
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 crossmultiply. 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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