A clock shows 3:15. The minute hand is exactly 15 minutes ahead of the hour hand. What time will it be when the minute hand is 15 minutes behind the hour hand?
The Moving Hands Riddle
Hint:
The key to solving this is recognizing how the minute hand and the hour hand move relative to each other. When the time is 3:15, the minute hand is 15 minutes ahead of the hour hand, meaning the minute hand has moved more than the hour hand.
Now, you need to figure out what time it will be when the minute hand is 15 minutes behind the hour hand. Focus on how both hands move over time—what will happen as the hands continue to move forward?
The time starts at 3:15, and at this moment, the minute hand is 15 minutes ahead of the hour hand.
You need to find the time when the minute hand is 15 minutes behind the hour hand.
Step-by-Step Breakdown:
Starting Point: At 3:15, the minute hand is at 3 (since it’s pointing to the 3 on the clock, meaning 15 minutes past the hour), and the hour hand is slightly ahead of the 3, because it’s already 15 minutes into the hour. At this point, the minute hand is ahead of the hour hand by exactly 15 minutes.
What happens over time?: As time passes, the minute hand moves faster than the hour hand. The minute hand moves 360 degrees in 60 minutes, while the hour hand moves only 30 degrees in 60 minutes (since there are 12 hours on a clock).
Finding the time when the minute hand is 15 minutes behind: The minute hand moves faster, so we want to know how long it will take for the minute hand to get 15 minutes behind the hour hand. We can think of this as a problem where the minute hand and hour hand are “catching up” with each other.The minute hand moves 360 degrees every hour, and the hour hand moves 30 degrees every hour. The difference in their speeds is:
360∘−30∘=330∘
The minute hand is gaining 330 degrees per hour on the hour hand, so we want to find when it will be 15 minutes behind.
Finding the time: Since 15 minutes corresponds to a 90-degree difference on the clock (because the full circle is 360 degrees and 15 minutes is 1/4 of the circle), we need to find when the minute hand is 90 degrees behind the hour hand.
To do this, we set up the equation:
90∘=330∘×wheret is the time in hours.
This is approximately 16.36 minutes(since 3/11×60=16.36).
Final Time: Add 16.36 minutes to 3:15, and you get approximately 3:31 (rounded to the nearest minute).
Conclusion:
The time will be around 3:31 when the minute hand is 15 minutes behind the hour hand.
