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# If a*b = a/b - b/a and m > n > 0, then which of following must be true

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Joined: 12 Aug 2015
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Schools: Boston U '20 (M)
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If a*b = a/b - b/a and m > n > 0, then which of following must be true  [#permalink]

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Updated on: 16 Oct 2018, 00:50
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Difficulty:

85% (hard)

Question Stats:

56% (02:21) correct 44% (02:10) wrong based on 186 sessions

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If $$a#b=\frac{a}{b}-\frac{b}{a}$$ and $$m>n>0$$, then which of following must be true?

(I) $$\frac{1}{m}#\frac{1}{n} > \frac{1}{n} #\frac{1}{m}$$

(II) $$\frac{1}{m} #\frac{1}{n} < \frac{1}{n} #\frac{1}{m}$$

(III) $$m#n<0$$

(IV) $$m#n>0$$

A) I and III
B) I and IV
C) II and III
D) II and IV
E) only IV

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Originally posted by stonecold on 19 Apr 2017, 10:24.
Last edited by Bunuel on 16 Oct 2018, 00:50, edited 2 times in total.
Renamed the topic and edited the question.
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Joined: 02 Aug 2009
Posts: 7984
Re: If a*b = a/b - b/a and m > n > 0, then which of following must be true  [#permalink]

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19 Apr 2017, 18:55
1
stonecold wrote:
If $$a*b=\frac{a}{b}-\frac{b}{a}$$ and $$m>n>0$$, then which of following must be true?

(I)$$\frac{1}{m}*\frac{1}{n} > \frac{1}{n} *\frac{1}{m}$$

(II)$$\frac{1}{m} *\frac{1}{n} < \frac{1}{n} *\frac{1}{m}$$

(III)$$m*n<0$$
(IV)$$m*n>0$$

A)I and III
B)I and IV
C)II and III
D)II and IV
E)only IV

ONLY one out of III and IV & one out of I and II.
So just check for only one of them.
Let's see between III n IV
M*n= m/n-n/m...
Now m>n, so m/n will be GREATER than 1 and n/m will be lesser than 1..
This means m/n-n/m will be GREATER than 0..
Hence IV is correct..

Also we can straight way take that when we take reciprocal of these numbers 1/n becomes greater than 1/m and hence the INEQUALITY sign will change from > to <.. II is correct
But say you want to check out..
Make it easier by taking m and n as fraction
So m =1/2 and n=1/3....m>n...
So 1/m becomes 2 and 1/n becomes 3..
Now you can check between 2*3 & 3*2

So II and IV are correct, hence D
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Re: If a*b = a/b - b/a and m > n > 0, then which of following must be true  [#permalink]

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20 May 2017, 22:38
Please note that since m>n>0, m/n >1 and n/m<1

(1/m)*(1/n) = (1/m)/(1/n) - (1/n)/(1/m) = (n/m) - (m/n)
This is <0 because n/m <1 and m/n>1

(1/n)*(1/m) = (1/n)/(1/m) - (1/m)/(1/n) = (m/n) - (n/m)
This is >0 because m/n >1 and n/m<1

So II is true

Also, m*n = m/n - n/m
This is >0 because m/n > 1 > n/m

So IV is also true

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Joined: 18 Feb 2018
Posts: 108
Re: If a*b = a/b - b/a and m > n > 0, then which of following must be true  [#permalink]

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14 Oct 2018, 05:00
What does a and b have to do with m and n? The question is unclear.
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Joined: 09 Apr 2018
Posts: 31
GPA: 4
Re: If a*b = a/b - b/a and m > n > 0, then which of following must be true  [#permalink]

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15 Oct 2018, 23:32
1
GittinGud wrote:
What does a and b have to do with m and n? The question is unclear.

It basically tells you the operations you have to perform. You can think of a and b as placeholders that are just there to demonstrate how the equation works and m, n as you actual inputs.

Hope this helps.
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Joined: 16 Oct 2010
Posts: 9706
Location: Pune, India
Re: If a*b = a/b - b/a and m > n > 0, then which of following must be true  [#permalink]

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16 Oct 2018, 00:43
stonecold wrote:
If $$a*b=\frac{a}{b}-\frac{b}{a}$$ and $$m>n>0$$, then which of following must be true?

(I) $$\frac{1}{m}*\frac{1}{n} > \frac{1}{n} *\frac{1}{m}$$

(II) $$\frac{1}{m} *\frac{1}{n} < \frac{1}{n} *\frac{1}{m}$$

(III) $$m*n<0$$

(IV) $$m*n>0$$

A) I and III
B) I and IV
C) II and III
D) II and IV
E) only IV

The format of the question is incorrect. It is not given as a function or user defined operator. You cannot use a standard operator as a user defined operator that too without explaining explicitly. We can guess that the relation given for a and b is supposed to hold for m and n too to solve it but an actual GMAT question would need to be formatted differently.

Given the question as is, I would just say that m > n > 0 so m and n are positive and hence m*n > 0. This is the ONLY statement that will be true.

If you define a new operator such as
a#b = a/b - b/a

now, you can worry about (1/m) # (1/n) and (1/n)#(1/m)

(1/m) # (1/n) = n/m - m/n = (n - m)/mn (since n < m, this is negative)
(1/n)#(1/m) = m/n - n/m = (m - n)/mn (since m > n, this is positive)

So II is correct.

m#n = m/n - n/m = (m - n)/mn (since m > n, this is positive)
So IV is correct.

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Karishma
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Re: If a*b = a/b - b/a and m > n > 0, then which of following must be true   [#permalink] 16 Oct 2018, 00:43
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