If mn ≠ 0, is m > n? (1) 1/m < 1/n (2) m^2 > n^2

Hi Brent,

in statement one 1)

case a ) if M=2 and N=1 . then how come M > N ? could you please clarify

Calling the master of Algebra!

All hail Bunuel !!
Please help us with a logical (algebraic) solution.

If mn ≠ 0, is m > n?

(1) 1/m < 1/n.

Two cases:
If m and n have the same sign (so if both are negative, or both are positive), then when cross multiplying we'll get n < m. Answer: YES.
If m and n have different signs signs (so if n is positive and m is negative), then when cross multiplying we'll get n > m. Answer: NO.
Not sufficient.

(2) m^2 > n^2.
Take the square root from both sides: |m| > |n|. This implies that m is further from 0 than n is. Clearly insufficient to say which is greater.

(1)+(2) The second statement still allows m and n to have the same sign (m = 2 and n = 1) as well as m and n to have different signs (m = -2 and n = 1) thus we can still have two different answers: n < m and n > m. Not sufficient.

Answer: E.

Hi..

Algebraic approach..
Given that neither of m and n is 0, they can be negative, positive..

Let's see the statements..
1) \(\frac{1}{m}<\frac{1}{n}.....\frac{1}{n}-\frac{1}{m}.....\frac{m-n}{mn}>0\)
So two cases..
mn>0... m-n>0, m>n.....
That is if both n and m are of SAME sign, m>n
mn<0.. m-n<0, m<n...
That is if both are of different sign, m<n..
Insufficient

2) \(m^2>n^2.....m^2-n^2>0......(m-n)(m+n)>0\)..
Two cases..
m+n>0...m-n>0, m>n...
That is if both are POSITIVE, or one is ATLEAST positive, m>n
m+n<0... m-n<0..
So if both are NEGATIVE m<n...
Again different possibilities
Insufficient

Combined..
Find the common in both statements..
A) when both are NEGATIVE, m<n
B) when both are of different sign, m>n
Again insufficient

E

Just for info

Ofcourse when both are positive ans is different from each statement, so not possible..
It seems Bunuel has already replied in a crisp manner, this is on same line slightly in detail

Great question.

The bottom line is => If you don't know the signs of the inequality then do not change the sides.
i.e => If 1/m<1/n
then it does not mean n<m unless we know the signs for the involved numbers.



As for this question -> Here is an easy way out
Taking two examples => (100,1) and (-100,1) => Push the E option.

Number plugging was faster IMO!

1) \(\frac{1}{m}\) < \(\frac{1}{n}\)
m = 2, n = 1 => Yes
m = -2, n = 1 => No
Insufficient.

2) \(m^2\) > \(n^2\)
m = 2, n = 1 => Yes
m = -2, n = 1 => No
Insufficient.

1+2)
Nothing new. Insufficient.
E is the answer

Clearly its E

1) 1/m<1/n
(m-n)/mn >0
numerator and denominator both have to be simultaneously positive or negative to be the statement to be true. A,D ruled out

2) m^2>n^2
m<-n and m>n and therefore B is out

combine them

put m=-2 n=1 and also m=2 and n=1
both the condition will satisfy

clearly we cant say if m>n or not. C is out

E is answer

If \(mn\neq{0}\) is m>n?

1. \(\frac{1}{m} < \frac{1}{n}\)
2. \(m^2> n^2\)

Dear Gnpth,

I'm happy to respond. :-) This is brilliant question!

Statement #1: \(\frac{1}{m} < \frac{1}{n}\)
I think this is the trickier of the two statements.

Here positive/negative sign is crucial.
Case I: if m = 5 and n = 2, then: \(\frac{1}{5} < \frac{1}{2}\) and m > n
Case II: if m = -5 and n = 2, then: \(-\frac{1}{5} < \frac{1}{2}\) but m < n

Two different choices consistent with the statement produce two different answers. Thus, this statement, alone and by itself, is insufficient.

Statement #2: \(m^2> n^2\)[/quote]
Clearly, positive/negative signs make a difference here. We can use the same two choices.
Case I: if m = 5 and n = 2, then: \(5^2> 2^2\) and and m > n
Case II: if m = -5 and n = 2, then: \((-5)^2> 2^2\) but m < n

Again, two different choices consistent with the statement produce two different answers. Thus, this statement, alone and by itself, is insufficient.

Combining the statements produces no additional restraints, and both pairs still can be used with the combination. Thus, everything is insufficient.

OA = (E)

A truly wonderful question!

Does all this make sense?
Mike :-)

Official Explanation:
The constraint in the question stem indicates that neither m nor n equals zero.
(1) INSUFFICIENT: You can solve algebraically/theoretically or you can Test Cases. If you solve algebraically, be careful:
you have to account for multiplying the inequality by a negative.
If m and n are both positive, then \(m > n\).
If m and n are both negative, the sign flips twice, so \(m > n\) again.
If only one is negative, then the sign flips once and \(m < n\). In this case, m must be the negative number, since any positive
is greater than any negative.
Alternatively, Test Cases:
If m = 3 and n = 2, then statement (1) is true and the answer to the question is Yes, \(m > n\).
If m = −3 and n = 2, then statement (1) is true and the answer to the question is No, m is not greater than n.

(2) INSUFFICIENT: This statement indicates nothing about the signs of the two variables. Either one could be positive or
negative.

(1) AND (2) INSUFFICIENT. If you are solving algebraically, test the scenarios that you devised for statement (1).
If m and n are both positive, then \(m > n\) and \(m^2 > n^2\). Both statements allow this scenario.
If m and n are both negative, then \(m > n\) but and \(m^2\) is not greater than \(n^2\). Discard this scenario, since it makes statement
(2) false.
If m is negative and n is positive, then \(m < n\). It could also be true that \(m^2 > n^2\), as long as m's magnitude is larger than
n's. If you're not sure Test Cases (see below).
Alternatively, test cases. Start by testing whether the cases you already tried for statement (1) also apply to statement (2).
If m = 3 and n = 2, then \(m > n\) and m² > n². Both statements allow this scenario.
If m = −3 and n = 2, then \(m < n\) and m² > n². Both statements allow this scenario.
Because there are scenarios in which \(m > n\) and \(m < n\), both statements together are still insufficient to answer the question. If
you forgot to account for the positive and negative cases, you may end up with (A) or (D) as your (incorrect) answer.
The correct answer is (E).

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