Question

At what time, between seven o'clock and eight o'clock, will the hands of a clock be in the same straight line, but not together?

Answer 1

Figure (a) shows the positions of hands of the clock at seven o'clock and Figure (b) shows the positions of hands of the clock when both the hands are in opposite direction in a straight line (i.e., the angle between the two hands is 180${}^o$).
The angle between the two hands at seven o'clock is 150${}^o$.
The two hands will be in a straight line, when the angle between them is 180${}^o$, i.e., when the angle increases by another 30${}^o$.
The angle changes at the rate of 5.5${}^o$ per minute.
So, the time taken to change the angle by 30${}^o$ is $\frac{1}{5.5}\times 30=5\frac{5}{11}$ minutes.
Therefore, the hands are in the same straight line (but not together), at $5\frac{5}{11}$ minutes past 7.