Question

Determine the number of positive integers n that satisfy:

1/2 < n/n+1 < 99/101

I don't know how to solve this besides plugging in random numbers, which would take all day. Any other suggestions for a faster way to solve it?

Thank you!
The smallest number that satisfies
1/2 < (n/n+1) is 2/3
The largest number that satisfies
n/(n+1) < 99/101 is 49/50.
Here's proof of that:
49/50 = 0.980000
99/101 = 0.980198
50/51 = 0.98039

So all n/(n+1) numbers in the series
2/3, 3/4, 4/5 ... 49/50 satisfy the inequality.
There are therefore 48 numbers n that satisfy the condition.
maybe the largest is 99/100

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