How many times in a day the clock hands will form the same angle 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?
Student tip
For a hint use Problem No. 3.Solution
From the solution of Problem No. 3, we have that the minute hand needs 16411 minutes to go from the position when the clock hands form the angle of 45o (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 49111 minutes more to go to a next position when it is before the hour hand forming the angle of 45o with the hour hand. This means that the minute hand needs 65511 minutes to go from the position when it forms the angle of 45o with the hour hand and is before the hour hand to the next position when it forms the angle of 45o 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 45o. So, in 12 hours there are exactly 11⋅2=22 different positions when the angle between the clock hands is 45o, since 11⋅(65511)min=12hours. 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?
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