Problem No. 6. Clock hands revisited

How many times in a day the clock hands will form the same angle as the angle at 7.30?

Student tip

For a hint use Problem No. 3.

Solution

From the solution of Problem No. 3, we have that the minute hand needs \(16 \frac{4}{11}\) minutes to go from the position when the clock hands form the angle of \(45^o\) (the angle between the clock hands at 7.30) and the minute hand is before the hour hand, to a position when they form the same angle and the minute hand is after the hour hand. In a similar way, as in Problem No. 3, it can be derived that the minute hand needs \(49 \frac{1}{11}\) minutes more to go to a next position when it is before the hour hand forming the angle of \(45^o\) with the hour hand. This means that the minute hand needs \(65 \frac{5}{11}\) minutes to go from the position when it forms the angle of \(45^o\) with the hour hand and is before the hour hand to the next position when it forms the angle of \(45^o\) with the hour hand and is before the hour hand, and for this time, we count 2 different positions (not counting the last one) when the angle between the clock hands is \(45^o\). So, in 12 hours there are exactly \(11 \cdot 2 = 22\) different positions when the angle between the clock hands is \(45^o\), since $$11 \cdot \big(65 \frac{5}{11}\big) \; \text{min} = 12 \;  \text{hours}.$$ Consequently, in a day, 44 times the angle between the clock hands is the same as the angle at 7.30.

Can you find the times when the clock hands form the same angle as the angle at 7.30? One of them is 7.30. What about the others?