Original: The first conflict between the British and the Marathas was the First Anglo-Maratha War which began in 1774 and resulted in the 1782 treaty of Salbai, by which the island of Salsette, adjacent to Bombay island, was ceded to the British, while Bharuch was ceded to the Maratha ruler Scindia.
Simplified: The first battle between the British and the Marathas was the First Anglo-Maratha War which began in 1774 and resulted in the 1782 treaty of Salbai.