If m and n are negative, is m/n less than 1?

Hi Mateo,

Stem says m and n are neg or m<0 and n<0. It does not state that m and n are integers. It asks us whether m/n<1, which means whether |m|<|n|.

1. m*n<1

This can happen in 3 cases

I. |m|<1 and |n|<1 (eg |-0.25| and |-0.5| or |-0.5| and |-0.25|)---> presenting 2 cases |m|/|n|>1 or |m|/|n|<1

II. |m|<1 and |n|>1 (eg |-0.25| and |-1|)---> |m|/|n|>1

III. |m|>1 and |n|<1 (eg |-1| and |-0.25|)---> |m|/|n|<1

Hence Insufficient.

2. m-n>n
---> m-2n>0
This can only happen when |m|<2*|n|-----> |m|/|n|<2, which does not necessarily mean <1. Hence insufficient

Combining statements 1 and 2, we cannot get a solution since, |m|/|n| can be <2 but >1, or < 2 and <1.

Hence, E.

Hi bunuel,

I could not understand statement 2 ; m>2n you have taken m=-3 and n =-2 this makes -3>-4 which will be 3<4 true and in second case you have taken m=-3 and n=-4 this will make -3>-8 3<8this is also true ..i could not understand Pleas explain

Given \(m<0\) and \(n<0\). Question: is \(m>n\)?

(2) \(m-n>n\) --> \(m>2n\).
If \(m=-3\) and \(n=-2\), (these values satisfy both the stem and statement: both are negative and \(-3>2*(-2)\)), then as \(m=-3<-2=n\) the answer is NO;
If \(m=-3\) and \(n=-4\), (these values satisfy both the stem and statement: both are negative and \(-3>2*(-4)\)), then as \(m=-3>-4=n\) the answer is YES. Not sufficient.

Hope it's clear.

I have a random question on this problem.

For the second statement, you're given that m - n > n, so why can't you just divide the whole thing out by n, in which case you get an expression with m/n? I can intuitively sense it's wrong but I can't place my finger on why.

It's not wrong at all, in fact it's a valid algebraic way to deal with statement (2).

Statement (2): \(m-n>n\) --> \(m>2n\) --> divide both parts by \(n\) (note that as given that \(n\) is negative then we should switch sign): \(\frac{m}{n}<2\) --> but this is not sufficient to say whether \(\frac{m}{n}<1\).

Ah, thanks Bunuel. The sign conversion was what tripped me up. Forgot to read the part that said they were negative numbers; should pay more attention.

@Murali: Good trick!

Bunuel, I was wondering about the -.2 and -1 and the -1.1 and -.6. Did you use these particular figures for a purpose or were they just easy numbers you knew configured into a number less than 1. Allowing you to make M as big as possible then M as small as possible?

Question asks is \(\frac{m}{n}<1\)? or is \(m>n\)?

From (1) we have: \(0<mn<1\), which is useless (whether alone or combined with 2) to answer which variable is greater.

From (2) we have: \(\frac{m}{n}<2\), so \(\frac{m}{n}\) is less than 2 but we can not say whether it's less than 1, so again this statement is useless to answer the question.

So even not testing the numbers we could say that the answer is E. But to show this, to demonstrate that the answer is E I just picked 2 sets of numbers, first set m>n and the second m<n, also with little trial and error it's not hard to get the numbers to satisfy stem and the statements (remember on DS questions when plugging numbers, goal is to prove that the statement is not sufficient. So we should try to get a YES answer with one chosen number(s) and a NO with another).

Question: If m and n are negative, is m/n less than 1?

(1) mn < 1
(2) m-n>n


Explanation: If both variables are negative, m/n is less than one only if
m > n, as in the case m=-2 and n=-3, making m
n = -2/-3 = 2/3. Thus, the question is equivalent to: Is m > n?
Statement (1) is insufficient; if the product is less than 1, one or both of the numbers must be less than one, but there’s no way to determine which of the two is larger. Statement (2) can be simplified by adding n to both sides:
m > 2n. That’s also insufficient: if m = 2, n could be either 1.5 or 3, giving a "yes" and a "no" answer.
Taken together, the statements are still insufficient. Statement (2) will still
be true if m = 1/2 and n is 3/8 or 3/4; both of those scenarios will make (1) true, as well. Since you can generate a "yes" answer and a "no" answer within the
constraints of both statements, the correct choice is (E).

Query: I am very confused, the explanation makes no sense to me, since if m>2n then surely m>n. I also don't follow their logic saying that if m is 2 then n could be 1.5 or 3, in both cases m is not >2n, for m>2n when m is 2 then n must be < 1, so m>n enough to satisfy the inequality the question is looking for.

any thoughts?

I don't understand that explanation either; it makes no sense. It discusses what might happen when m=2, but we know in advance that m is negative, so it's pointless to consider the possibility that m=2. Perhaps you've left out the negative signs in transcribing the solution, but it's awfully confusing in any case.

The majority of GMAT inequality questions test your understanding of inequalities and negative numbers. The most important thing to understand is that your normal ideas about inequalities are reversed for negative numbers. For example, 6 is greater than 3, but -6 is less than -3. That might seem obvious, but when we introduce letters, some surprising things can turn out to be true. For example, when n is positive, then certainly 2n is greater than n. We can picture that on the number line:

------------------0------n------2n----

But, when n is negative, 2n is most definitely less than n, since 2n will be further to the left of zero than n is. On the number line:

---2n-----n------0--------

So, when you say:

Query: I am very confused, the explanation makes no sense to me, since if m>2n then surely m>n.

that is not necessarily the case for negative numbers. It might be that n = -3, and so 2n = -6. Then if m is, say, -4, then m > 2n is true, but m > n is false. Or, rather than pick numbers, looking at the second number line above, you can see that m can be somewhere between 2n and n, in which case 2n < m < n will be true.

A further consequence of the fact that inequalities are 'reversed' for negative numbers is that we must *reverse* an inequality if we multiply or divide on both sides by a negative number. It's often obvious we need to do that; if we have, say

-x > 2

then multiplying by -1 on both sides, we must reverse the inequality:

x < -2

This can get tricky, however, when we multiply both sides of an inequality by an *unknown*. If you see the inequality:

m/n < 1

and have no information about m and n, you cannot multiply by n on both sides to get m < n, because you don't know whether n is positive or negative. If n is positive, then certainly m < n must be true, but if n is negative, we must reverse the inequality when we multiply by n on both sides: m > n will be true. In the question above, we know in advance that n is negative, so we can indeed rephrase the question 'Is m/n < 1' by multiplying on both sides by n, but we must reverse the inequality when we do: the question becomes 'Is m > n?'

The explanation quoted above uses number picking to arrive at this rephrasing, which is a very haphazard way to approach this extremely common situation. You can proceed much more quickly and reliably if you become comfortable with manipulating inequalities as you would equations, with the one caveat that you must be extremely careful when you multiply or divide on both sides of an inequality by a letter (or any other unknown expression): you must know whether that letter is positive or negative in order to proceed.

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