Ralph is standing along a road which heads straight east. If you go nine miles east, make a left turn, and travel seven miles north, you will find Pamela with her mountain bike. At exactly the same time that Ralph begins running eastward along the road at 6 miles per hour, Pamela begins biking in a straight line at 10 miles per hour. Pamela’s direction is chosen so that she will reach a point on the road where Ralph is running at exactly the same time Ralph reaches that same point. Let $M$ and $N$ be relatively prime positive integers such that \( \frac{M}{N} \) is the number of hours that it takes Pamela and Ralph to meet. Find $M+N$.

Let's think step by step and output the final answer within \boxed{}.
user