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Manager  Joined: 10 Jul 2009
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Location: Ukraine, Kyiv
x and y are positive integers. If 1/x + 1/y < 2, which of  [#permalink]

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Question Stats: 56% (01:55) correct 44% (02:07) wrong based on 123 sessions

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x and y are positive integers. If 1/x + 1/y < 2, which of the following must be true?

(A) x + y > 4
(B) xy>1
(C) x/y + y/x < 1
(D) (x - y)^2 > 0
(E) None of the above

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Re: X and Y. which is true?  [#permalink]

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barakhaiev wrote:
X and Y are positive integers. If 1/X + 1/Y < 2, which of the following must be true?

(A) X+Y>4
(B) X*Y>1
(C) X/Y+Y/X<1
(D) (X-Y)^2>0
(E) None of the above

Trying a few values makes us realize that the only relation that holds is (B). But how can we be sure that (B) holds for all acceptable values of X and Y.

1/X + 1/Y < 2 implies (1/X + 1/Y)/2 < 1
A useful property of positive numbers is AM >= GM
Arithmetic Mean >= Geometric Mean

Say, the numbers are 1/X and 1/Y
AM = (1/X + 1/Y)/2
It is given that (1/X + 1/Y)/2 < 1 so we know that AM < 1

GM = $$\sqrt{\frac{1}{X}*\frac{1}{Y}}$$

Since GM <= AM,

$$\sqrt{\frac{1}{X}*\frac{1}{Y}}$$ < 1

$$\frac{1}{XY} < 1$$ (Squaring the inequality)

$$XY > 1$$ (X and Y are positive so the inequality doesn't change)
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##### General Discussion
Math Expert V
Joined: 02 Aug 2009
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x and y are positive integers. If 1/x + 1/y < 2, which of  [#permalink]

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2
ans is b......

since x and y are both +ive int, $$x*y>1$$... only exception being x=y=1 as it is not given they are different integers...
however it is given $$\frac{1}{x} +\frac{1}{y}<2.$$. this cannot be true if x=y=1.... so one or both have to be > 1
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Re: X and Y. which is true?  [#permalink]

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Guys, can you guide me how D is not true? coz last time i checked, square of any number is greater than 0. Even if x is less than y, still, it's square would me more than 0..unless, x = y...
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Re: X and Y. which is true?  [#permalink]

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1
1

1/X + 1/Y < 2

The maximum value of 1/X is 1 because if X equals any other number greater than one it will be a fraction. The same is true with 1/Y.

So 1/X and 1/Y will always be less than 2 as long as both X and Y are not both equal to one at the same time.

Another way of putting it is:

X*Y>1
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Re: X and Y. which is true?  [#permalink]

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My take on this :

From the equation :
x + y < 2xy

=> xy > (x+y)/2

So if x and y are two different positive integers, taking the two least values as 1 and 2, we have x > 1.5 at least. Hence xy > 1.

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Re: X and Y. which is true?  [#permalink]

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Quote:
X and Y are positive integers. If 1/X + 1/Y < 2, which of the following must be true?

(A) X+Y>4
(B) X*Y>1
(C) X/Y+Y/X<1
(D) (X-Y)^2>0
(E) None of the above

Let X=1,
1+1/Y<2
1/Y<1
1<Y

Y>1 when X=1,
A --> yes and no
B --> yes
C--> yes and no
D--> yes and no

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Re: X and Y. which is true?  [#permalink]

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krishnasty wrote:
Guys, can you guide me how D is not true? coz last time i checked, square of any number is greater than 0. Even if x is less than y, still, it's square would me more than 0..unless, x = y...

Given: 1/X + 1/Y < 2
Say X = 2, Y = 2
These values satisfy the inequality: 1/2 + 1/2 < 2

But they do not satisfy (D)
(X-Y)^2>0
(2-2)^2 = 0, not greater than 0
Hence (D) must not be true for all values. There are values that satisfy the inequality but does not satisfy (D)
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Re: X and Y. which is true?  [#permalink]

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Karishma, now i need a confirmation on GMAT questions...
lets say that if two unknowns are given (like X and Y ), can we assume that these two are equals? I thought if we say x and y, they are implicitly different numbers..

VeritasPrepKarishma wrote:
krishnasty wrote:
Guys, can you guide me how D is not true? coz last time i checked, square of any number is greater than 0. Even if x is less than y, still, it's square would me more than 0..unless, x = y...

Given: 1/X + 1/Y < 2
Say X = 2, Y = 2
These values satisfy the inequality: 1/2 + 1/2 < 2

But they do not satisfy (D)
(X-Y)^2>0
(2-2)^2 = 0, not greater than 0
Hence (D) must not be true for all values. There are values that satisfy the inequality but does not satisfy (D)

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Re: X and Y. which is true?  [#permalink]

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krishnasty wrote:
Karishma, now i need a confirmation on GMAT questions...
lets say that if two unknowns are given (like X and Y ), can we assume that these two are equals? I thought if we say x and y, they are implicitly different numbers..

VeritasPrepKarishma wrote:
krishnasty wrote:
Guys, can you guide me how D is not true? coz last time i checked, square of any number is greater than 0. Even if x is less than y, still, it's square would me more than 0..unless, x = y...

Given: 1/X + 1/Y < 2
Say X = 2, Y = 2
These values satisfy the inequality: 1/2 + 1/2 < 2

But they do not satisfy (D)
(X-Y)^2>0
(2-2)^2 = 0, not greater than 0
Hence (D) must not be true for all values. There are values that satisfy the inequality but does not satisfy (D)

Until and unless they mention 'distinct numbers' or 'X not equal to Y', X and Y can be equal. The equality can be a deal breaker/maker sometimes so you have to make sure that you have analyzed its effects too.
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Re: X and Y. which is true?  [#permalink]

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Thanks Karishma for the information.

Quote:
Until and unless they mention 'distinct numbers' or 'X not equal to Y', X and Y can be equal. The equality can be a deal breaker/maker sometimes so you have to make sure that you have analyzed its effects too.

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Re: X and Y. which is true?  [#permalink]

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barakhaiev wrote:
X and Y are positive integers. If 1/X + 1/Y < 2, which of the following must be true?

(A) X+Y>4
(B) X*Y>1
(C) X/Y+Y/X<1
(D) (X-Y)^2>0
(E) None of the above

Let x=2
Let y=2

$$\frac{1}{2} + \frac{1}{2}< 2$$

A) X+Y=4 OUT!
B) 2*2 > 1 HOLD!
C) 2/2 + 2/2 = 2 < 1 OUT!
D) (2-2)^2 = 0 OUT!

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x and y are positive integers. If 1/x + 1/y < 2, which of  [#permalink]

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barakhaiev wrote:
x and y are positive integers. If 1/x + 1/y < 2, which of the following must be true?

(A) x + y > 4
(B) xy>1
(C) x/y + y/x < 1
(D) (x - y)^2 > 0
(E) None of the above

My Solution:

Given : x and y are positive integers

Stem: 1/x+1/y<2---x+y<2xy (as x and y are positive we can cross multiply)

So A) x+y>4 Try x=1 & y = 2 (Not true)

B) xy>1 Try x=1 and y = 2 (Always true) ) [Note (x=1 and y =1 is not possible values because with these values stem doesn't holds true]

This is our answer as not more then one correct answer choice is possible but we can try all choices for more clarity:

C) x/y+y/x<1 Try x=1 and y = 2 (Not true)

D) (x-y)^2 Try x=2 and y=2 (Not true)

E) Can never be true

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Re: x and y are positive integers. If 1/x + 1/y < 2, which of  [#permalink]

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barakhaiev wrote:
x and y are positive integers. If 1/x + 1/y < 2, which of the following must be true?

(A) x + y > 4
(B) xy>1
(C) x/y + y/x < 1
(D) (x - y)^2 > 0
(E) None of the above

I thought it is some kind of trap here..
we can rewrite the original as:
x+y<2xy

A - x=2, y=2 -> x+y is not greater than 4, yet 1/2 + 1/2 < 2. so A is out.
B - if x and y are both positive integers, xy>1 all the times - looks good.
C - x/y +y/x <1 or x^2 + y^2 < xy - which will never be true, if x and y are positive integers.
D - x^2 + y^2 > 2xy - suppose x=2 and y=2. 4+4 = 8. 2*2*2=8. 8=8, it's not an inequality.
E - since B works, e is out.
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X and Y are positive integers. If 1/x + 1/y < 2, which of the followin  [#permalink]

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jedit wrote:
X and Y are positive integers. If 1/x + 1/y < 2, which of the following must be true?

A. X+Y>4
B. XY>1
C. X/Y + Y/X < 1
D. (x-y)^2 > 0
E. none

$$\frac{1}{x} + \frac{1}{y} < 2$$

Let x = 1 and y = 2

$$\frac{1}{1} + \frac{1}{2} = \frac{3}{2}$$

$$\frac{3}{2} < 2$$ -- Those numbers for x and y work

MUST be true?

A. X+Y>4 - NO
1 + 2 = 3, which is not greater than 4

B. XY>1 YES
Because x and y are positive integers, there is only one way XY would NOT be greater than 1: if both x and y = 1. Then XY = 1.
But x = y = 1 violates the prompt: their reciprocals summed must be less than 2; in that case, they equal 2. This choice must be true.

C. X/Y + Y/X < 1 - NO
$$\frac{1}{2} + \frac{2}{1}=\frac{5}{2}$$
$$\frac{5}{2}$$ is not less than 1

D. (x-y)^2 > 0 NO
For this option, let x=y=2.
$$(2-2)^2 = 0^2 = 0$$
0 is not greater than 0

E. none - NO - One of the answers, B, must be true.

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Re: X and Y are positive integers. If 1/x + 1/y < 2, which of the followin  [#permalink]

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