Is m>n?

[aeqitas0114]

Can anyone show me a quicker way to get the answer other than the logical approach which tends to consume much time?

===
Is m>n?

(1) m/n>1
(2) (m+n)/m > (m+n)/n

My approach is below but it takes too much time, so I hope there is a better path.

Stmnt 1:

n>0 --> m>n --> Yes
n<0 --> m<n --> No

Hence insufficient.

Stmnt 2: (i spent too much time on finding out the following combinations)

m=2, n=3 --> No
m=-2, n=-3 --> No

Hence insufficient.

Stmnt 1&2:

n>0 --> m>n --> Yes / m=2, n=3 --> No

When you look at the range of "n", two statements contradict each other.

n<0 --> m<n --> No / m=-2, n=-3 --> No

ditto.
===

Thanks in advance.

[hiteshwd]

Sure

Your approach in 1 is good.

2 can be written as:

(m+n)n > (m+n)m
=> n^2 > m^2

=> |n| > |m|


i.e. For m>=0, n > m

and for m <= 0, n < m

Not sufficient


PS: If the question is written correctly, there is some problem with the question and it is not upto GMAT standard. Please check if any of the signs are inverted.

[jlucero]

The algebraic solution to the second statement is actually a bit different than what both of you proposed, and pretty difficult to type out, so I apologize in advance if this gets difficult to read. Recall that you can deal with inequalities like equations as long as you consider whether to flip the sign or not. That allows us to algebraically set up the first statement as such:

(1) m/n > 1
A) if n is positive (don't flip the sign):
m > n

B) if n is negative (flip the sign):
m < n

Insufficient

This method makes statement 2 much easier to work with. Since there is an m and an n on either side of the equation, you need to consider four conditions:

m+n+
m+n-
m-n+
m-n-

The equation will always stay the same:

The equation:
(m+n)/m > (m+n)/n
nm + n^2 > m^2 + nm
n^2 > m^2?

It's just the number of variables that are negative that will determine how many times we need to flip the sign:

m+n+ (zero flips = n^2 > m^2)
m+n- (one flip = n^2 < m^2)
m-n+ (one flip = n^2 < m^2)
m-n- (two flips = n^2 > m^2)

Note that when m is positive and n is negative, m will always be larger. And when n is positive and m is negative, n will always be larger. So this statement is definitely insufficient, but it can lead us to these four scenarios that are helpful when combined with statement 1:

m+n+ (zero flips = n^2 > m^2) (n is larger)
m+n- (one flip = n^2 < m^2) (m is positive/larger)
m-n+ (one flip = n^2 < m^2) (n is positive/larger)
m-n- (two flips = n^2 > m^2) (n is larger when squared, but because these are both negative values, m is larger, meaning a smaller integer i.e. -5 > -8)

m+n+ = N
m+n- = M
m-n+ = N
m-n- = M

Combined with statement 1, both values need to be positive or both values need to be negative which reduces our chart to:

If m+n+ then n is larger (5/10 is not greater than 1)
If m-n- then m is larger (-5/-10 is not greater than 1)

Note that these two scenarios both give us a possibility that can't work with statement 1. So as hiteshwd mentioned, you probably swapped a sign somewhere.

Whew, hope that all makes sense :)

[aeqitas0114]

Hi hiteshwd and Joe,

Thank you very much for sharing your insights for this question. I've got quite useful tips from both of you.

Because I quoted this question exactly from the GMAT Prep, I have no idea why this question violates their standard rules building a question. Anyway, using the notion of an absolute value extracted from a squared number and the plus/minus classification, I now can solve this question much quicker and simpler.

Viva algebraic solution!!

[nt2011]

How about plugging numbers method. For E.g

Stmt 1) m / n > 1
Chose m = 8, n = 4 and m = -8 and n = -4. These values show Stmt 1 is NS

Stmt 2) (m+n)/m > (m+n)/n ==> n/m > m/n
Chose m = 4, n = 8 and m = -4 and n = -8. These values show Stmt 2 is NS

Stmt 1) and 2) says that n/m > 1 and using Stmt 2 plug in numbers shows that both stmts are NS.

This gives E as the answer. I found the above method much quicker. Please confirm the OA.

[aeqitas0114]

OA is E.

[RonPurewal]

ya, there's a pretty serious problem here.
if you simplify statement 2, you get
(m+n)/m > (m+n)/n
1 + n/m > m/n + 1
n/m > m/n
this statement is insufficient (consider n = 2 and m = 1, and then n = -2 and m = -1).
statement 1 is also insufficient, so the statements have to be combined.

this is where the issue arises.
if you combine the statements, you get n/m > m/n > 1, which is impossible: if m/n is more than 1, then n/m, which is its reciprocal, must be less than 1 (and vice versa).

no DS problem can involve an impossible situation. (in fact, no DS problem will ever have two statements that contradict each other, even if those two statements never need to be combined!) so, something must have been mis-transcribed here.

please check the original. if someone doesn't post an edited version fairly soon, we'll have to kill this thread.
thanks.