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# If x and y are positive integers and 1/x + 1/y < 2, which of the

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

Official Solution:

If $$x$$ and $$y$$ are positive integers and $$\frac{1}{x}+\frac{1}{y} \lt 2$$, which of the following must be true?

A. $$x + y \gt 4$$
B. $$xy \gt 1$$
C. $$\frac{x}{y} + \frac{y}{x} \lt 1$$
D. $$(x - y)^2 \gt 0$$
E. none of the above

Given that $$x$$ and $$y$$ are positive integers, it follows that they can take on the values $$1, 2, 3, \ldots$$. However, if $$x=y=1$$, then $$\frac{1}{x}+\frac{1}{y}=2$$, which violates the condition that $$\frac{1}{x}+\frac{1}{y}1$$. This means that option B is always true.

We can demonstrate that options A, C, and D are not always true by providing counterexamples. For instance, if we set $$x=y=2$$, then we find that none of the options holds true.

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