Two trains are on the same track a distance \(100\ \text{km}\) apart heading towards one another, each at a speed of \(50\ \text{km/h}\). A fly starting out at the front of one train, flies towards the other at a speed of \(75\ \text{km/h}\). Upon reaching the other train, the fly turns around and continues towards the first train. It continues flying back and forth till the two trains meet and it gets squashed. The time taken (in hours) for the fly to make \(7\) 'rounds' is \(\color{blue}{\dfrac{p}{q}}\), where \(\color{blue}{p}\) and \(\color{blue}{q}\) are positive coprime integers. Calculate \(\large\color{blue}{p+q}\).

**Note :** A round is defined as moving from one train to another.

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