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

Question

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

chetan2u · Answer

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

amanvermagmat · Answer

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

Hence answer is D

GittinGud · Answer

What does a and b have to do with m and n? The question is unclear.

laulau · Answer

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.

KarishmaB · Answer

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.

Answer (D)

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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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