M15-02

Question

If  m  and  n  are positive numbers, is  \frac{mn}{m + n} > 1 ?

(1)  \frac{1}{m} \gt \frac{1}{n} > \frac{1}{2} 
 
(2)  m = n - 1

Bunuel · Answer

Official Solution: If m and n are positive numbers, is \frac{mn}{m + n} > 1 ? Let's simplify the question. Since both m and n are positive, the question "is \frac{mn}{m + n} > 1 ?" can be rephrased as "is \frac{m+n}{mn} < 1 ?", and further as "is \frac{1}{n} +\frac{1}{m} < 1 ?" (1) \frac{1}{m} \gt \frac{1}{n} > \frac{1}{2} Since both \frac{1}{m} and \frac{1}{n} are greater than \frac{1}{2} , their sum is greater than 1: \frac{1}{n} +\frac{1}{m} > 1 . Therefore, we have a NO answer to the question. Sufficient. (2) m = n - 1 If both m and n are large enough numbers, such as 99 and 100, then \frac{1}{n} +\frac{1}{m} < 1 , giving a YES answer to the question. However, if m = 1 and n = 2 , then \frac{1}{n} +\frac{1}{m} > 1 , giving a NO answer to the question. Not sufficient. Answer: A

londonconsultant · Answer

So obvious when it's written out - feel like a complete idiot!

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

wpt2009 · Answer

Bunuel , is the rule for taking the reciprocal of an inequality to flip the inequality sign always?

Bunuel · Answer

No, the rule for taking the reciprocal of an inequality to flip the inequality sign isn't always applied; it depends on the signs of the variables on both sides. In the original problem we were able to flip the inequality sign because m and n are both positive.

M15-02

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