Problem Solving

If mn ≠ 0, is m > n?

(1) 1/m < 1/n
(2) m^2 > n^2

Source : Manhattan Prep
OA=E

Bullzi wrote:Hello,
Is there an algebraic way of solving this question? I didn't think of testing values, started with algebra, but got stuck midway not able to reach a logical conclusion

Thanks,
Bullzi

You can do algebra with some logical reasoning mixed in.

If we're given an inequality, and we take the reciprocal of both sides, the inequality sign will flip, provided that both sides are positive or both sides are negative. If other words, if A > B, and A and B are both positive (or both negative), then 1/A < 1/B. However, one side is positive and one side is negative, the inequality sign will not flip when we take the reciprocal. (This makes sense, as the reciprocal of a negative will remain negative.)

Using that property, here's one way to evaluate statement 1.
If m and n are both positive, or both negative, then when we take the reciprocal of 1/m > 1/n, we'll get m > n, or a YES to the original question. However, if m is negative and n is positive, the reciprocal would give us m < n, or a NO. Not sufficient.

Statement 2. If we take the square root of both sides of m² > n² , we'll get |m| > |n|. All this tells us is that the distance from m to 0 is greater than the distance from n to 0. But it doesn't tell us if we're dealing with positive or negative values. If m and n are positive, yes, m will be greater. But if m is negative and n is positive, then clearly m will be not be greater.

Together: Same problem. If m is negative and n is positive, then, m < n, and we get a NO. If both are positive, then m > n and we get a YES. Not sufficient. So the answer is E.

Bullzi wrote:Hello,
Is there an algebraic way of solving this question? I didn't think of testing values, started with algebra, but got stuck midway not able to reach a logical conclusion

Thanks,
Bullzi

Question stem, rephrased:
Is m-n > 0?

Statement 1: 1/m < 1/n
1/m - 1/n < 0
(n-m)/mn < 0
(m-n)/mn > 0.
Case 1: mn>0
Here, the inequality in blue will be satisfied only if m-n>0, with the result that the answer to the question stem is YES.
Case 2: mn<0
Here, the inequality in blue will be satisfied only if m-n<0, with the result that the answer to the question stem is NO.
INSUFFICIENT.

Statement 2: m² > n²
m² - n² > 0
(m+n)(m-n) > 0.
Case 1: m+n > 0
Here, the inequality in red will be satisfied only if m-n > 0, with the result that the answer to the question stem is YES.
Case 2: m+n<0
Here, the inequality in red will be satisfied only if m-n<0, with the result that the answer to the question stem is NO.
INSUFFICIENT.

Statements combined:
(m-n)/mn > 0.
(m+n)(m-n) > 0.
Case 1: mn>0 and m+n>0
Here, the two colored inequalities will be satisfied only if m-n>0, with the result that the answer to the question stem is YES.
Case 2: mn<0 and m+n<0
Here, the two colored inequalities will be satisfied only if m-n<0, with the result that the answer to the question stem is NO.
INSUFFICIENT.

The correct answer is E.