Fundamental Concepts: 1. The Speed of the Hands
The foundation of all clock problems is understanding the speed of each hand. We calculated the speed of both hands in degrees per minute, which allows for direct comparison.
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Minute Hand Speed:
The minute hand travels a full 360° circle in 60 minutes.
Its speed is 360° / 60 minutes = 6° per minute.
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Hour Hand Speed:
The hour hand travels a full 360° circle in 12 hours.
In 1 hour, it travels 360° / 12 = 30°.
To find its speed per minute, we divide by 60: 30° / 60 minutes = 0.5° per minute.
2. Angle Between the Hands
This concept involves finding the angle at a specific time. The key is to calculate the absolute position of each hand and then find the difference.
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Method:
1. Calculate the position of the minute hand from the 12 o'clock mark:
Position = M x 6°
2. Calculate the total position of the hour hand:
Position = (H x 30°) + (M x 0.5°)
3. Find the absolute difference between the two positions.
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Formula:
|(6M) - (30H + 0.5M)| = |5.5M - 30H|
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Example Question: What is the smaller angle between the hour hand and the minute hand at 7:20?
Solution:
Minute hand position: 20 x 6 = 120°.
Hour hand position: (7 x 30) + (20 x 0.5) = 210 + 10 = 220°.
Angle: |220 - 120| = 100°.
3. Coinciding Hands (0° Angle)
This is about finding the time when the minute hand catches up to the hour hand. The key concept is relative speed.
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Relative Speed:
The minute hand gains on the hour hand at a rate of 6° - 0.5° = 5.5° per minute.
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Method:
1. Find the initial gap between the hands at the start of the hour (H x 30°).
2. Use the formula Time = Distance / Relative Speed to find the minutes needed to close the gap.
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Formula:
Time in minutes past H o'clock = 30H / 5.5 = 60H / 11
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Example Question: At what time do the hands coincide between 9:00 and 10:00?
Solution:
Initial gap at 9:00: 9 x 30 = 270°.
Time to close the gap: 270 / 5.5 = 540 / 11 = 49 1/11 minutes past 9 o'clock.
4. Opposite Hands (180° Angle)
This is a variation of the coinciding problem. The minute hand must not only catch up to the hour hand but also go an additional 180° beyond it.
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Method:
1. Find the initial gap at the start of the hour (H x 30°).
2. Add 180° to this initial gap to get the total distance the minute hand must travel.
3. Use Time = Total Distance / Relative Speed to find the minutes.
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Formula:
Time in minutes past H o'clock = (30H + 180) / 5.5 = (60H + 360) / 11
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Example Question: At what time are the hands opposite each other between 2:00 and 3:00?
Solution:
Initial gap at 2:00: 2 x 30 = 60°.
Total distance to cover: 60° + 180° = 240°.
Time: 240 / 5.5 = 480 / 11 = 43 7/11 minutes past 2 o'clock.
5. Strikes and Intervals
This is a simpler concept that relies on a single rule.
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The Rule:
The number of intervals is always one less than the number of strikes.
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Method:
1. Use the given information (e.g., "7 strikes in 30 seconds") to find the time it takes for one interval.
2. Use that value to calculate the total time for the new number of strikes.
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Example Question: A clock takes 30 seconds to strike 7 o'clock. How long will it take to strike 10 o'clock?
Solution:
7 strikes = 7 - 1 = 6 intervals.
Time per interval: 30 seconds / 6 intervals = 5 seconds per interval.
10 strikes = 10 - 1 = 9 intervals.
Total time: 9 intervals x 5 seconds per interval = 45 seconds.
