At what time between 2 and 3 o'clock will the hands of a watch point in opposite directions?
Question
At what time between 2 and 3 o'clock will the hands of a watch point in opposite directions?
Solution
The hands of a clock point in opposite directions 11 times every 12 hours. This is because the minute hand completes 12 full cycles while the hour hand completes 1 full cycle.
To find the time between 2 and 3 o'clock when the hands point in opposite directions, we need to calculate the minute when the minute hand is on the opposite side of the hour hand.
At 2 o'clock, the hour hand is at 4 (each hour is equivalent to 5 minutes on a clock face). The minute hand is at 0.
The hands will be in opposite directions when the minute hand is 30 minutes ahead of the hour hand.
So, we need to calculate when the minute hand will be at 4 + 30 = 34.
The minute hand moves 12 times faster than the hour hand. So, in t minutes, the minute hand will move 12t places and the hour hand will move t places.
We can set up the equation 12t = 34 + t to find the time when the hands will be in opposite directions.
Solving for t gives t = 34/11 = 3.09.
So, the hands will point in opposite directions at approximately 2:03.
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