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Algebra Difficulty 3.8 AMC 10/12 Find the answer Slovenia

Let mm and nn be positive integers, such that 19≤m≤4919 \le m \le 49, 51≤n≤10151 \le n \le 101. What is the greatest possible value of the expression n+mn−m\frac{n+m}{n-m}?

Pick one

Solution

We have n+mn−m=n−m+2mn−m=1+2mn−m\frac{n+m}{n-m} = \frac{n-m+2m}{n-m} = 1 + 2\frac{m}{n-m}, so the value is maximal when n−m=2n-m = 2 and m=49m = 49. Then, the value is 5050.

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