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

Explanation:-

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. $$\eqalign{ & 15\,{\text{min}}{\text{.}}\,{\text{are}}\,{\text{gained}}\,{\text{in}}\,\left( {\frac{{60}}{{55}} \times 15} \right)\,{\text{min}} \cr & = 16\frac{4}{{11}}\,{\text{min}} \cr & \therefore {\text{The}}\,{\text{hands}}\,{\text{are}}\,{\text{coincident}}\,{\text{at}}\,16\frac{4}{{11}}\,{\text{min}}{\text{.}}\,{\text{past}}\,3 \cr}$$