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

At 3 o'clock, the minute hand is 15 min. spaces apart from the hour hand.
To be coincident, it must gain 15 min. spaces.
55 min. are gained in 60 min.
15 min. are gained in $$\left( {\frac{{60}}{{55}} \times 15} \right)$$   min.
= $$16\frac{4}{{11}}$$ min.
∴ The hands are coincident at $$16\frac{4}{{11}}$$ min. past 3