At what time, (in minutes) between 3 O'clock, and 4 O'clock, both the needles will coincide with each other?

At 3 O'clock, then minute hand is 15 min away from the hour hand. To coincide, it must gain 15 min as 55 min are gained in 60 min. 15 min are gained in (60/55) x 15 min = 16⁴/₁₁ min Hence, the hands are coincide at 16⁴/₁₁ min past 3.