After 9^ 0 clock at what time between 9 p.m. and 10 p.m. will the...

At 9 O'clock, the minute hand is $9 \times 5=45$ minute - spaces behind the hour hand. Therefore, the minute hand will have to gain $45-30=10$ minute space over the hour hand. ∵ Gain of 55 minute spaces equals 60 minutes. ∴ Gain of 15 minute spaces equals $=\frac{60}{55} \times 15=\frac{180}{11}=16 \frac{4}{11}$ Therefore, hour and minute hands of a clock point in opposite direction after 9 O'clock at $16 \frac{4}{11}$ minutes past 9.