Short Answer

Problems on successive coincidences are solved by understanding when the hour and minute hands of a clock meet again after one overlap. We calculate the time gap between two consecutive overlaps using the relative speed of both hands.

The time between successive coincidences is about 65 5/11 minutes. By using this fixed interval, we can find the time of any next or previous coincidence. This helps in solving clock-based reasoning questions quickly and correctly.

Detailed Explanation:

Successive coincidence in clocks

Meaning of successive coincidence

Successive coincidence means the repeated meeting of the hour and minute hands after regular intervals. When both hands overlap once, they do not stay together. After some time, the faster minute hand again catches the slower hour hand.

This repeated meeting is called successive coincidence. It happens continuously throughout the day at fixed intervals due to the constant movement of both hands.

Understanding this concept is important because many clock problems are based on finding next or previous meeting times.

Movement of clock hands

Hour hand movement

The hour hand moves slowly and completes one full circle in 12 hours. It moves 30 degrees per hour and 0.5 degrees per minute.

Because of its slow speed, it keeps moving gradually and is always behind the minute hand.

Minute hand movement

The minute hand moves faster and completes one full circle in 60 minutes. It moves 6 degrees per minute.

Due to this speed, it keeps overtaking the hour hand again and again, leading to repeated coincidences.

Time gap between coincidences

Relative speed concept

To solve successive coincidence problems, we first find the relative speed between the two hands.

• Minute hand speed = 6 degrees per minute
• Hour hand speed = 0.5 degrees per minute
• Relative speed = 6 − 0.5 = 5.5 degrees per minute

This means the minute hand gains 5.5 degrees every minute over the hour hand.

Formula for time gap

The time taken for one full coincidence cycle is:

• Time = 360 ÷ 5.5
• Time = 360 ÷ (11/2)
• Time = 360 × 2 / 11
• Time = 720 / 11 minutes
• Time = 65 5/11 minutes

So, successive coincidences happen every 65 5/11 minutes.

Steps to solve problems

Step 1: Identify given information

First, read the question carefully. Identify whether it asks for next coincidence, previous coincidence, or number of coincidences in a time period.

Step 2: Use time gap

Use the fixed time gap of 65 5/11 minutes between two overlaps. This is the key value used in all successive coincidence problems.

Step 3: Apply multiplication or division

If we need the nth coincidence, multiply:

• Time = n × 65 5/11 minutes

If we need how many coincidences in a time period, divide total time by 65 5/11 minutes.

Step 4: Convert into hours if needed

Sometimes the answer is required in hours and minutes. So, convert the result accordingly for better understanding.

Pattern of successive coincidences

Regular intervals

Successive coincidences happen at fixed intervals throughout the day. This regular pattern makes it easier to predict future or past meeting times.

In a 12-hour cycle, there are 11 coincidences, and they are evenly spaced based on the time gap.

Continuous motion

The reason behind successive coincidences is the continuous motion of both hands. The minute hand always moves faster and keeps catching the hour hand again and again.

This continuous catching process creates a repeating pattern.

Importance in reasoning

Logical thinking

These problems improve logical thinking by showing how motion and time are connected. Students learn how one moving object repeatedly catches another.

Exam relevance

Successive coincidence questions are common in competitive exams. They test understanding of time, speed, and relative motion.

Knowing the fixed time gap helps solve such questions quickly.

Concept clarity

This topic builds strong understanding of clock behavior. It connects speed difference with repeated events in a circular system.

Conclusion

Successive coincidence problems are solved using the fixed time gap of 65 5/11 minutes between two overlaps of clock hands. This happens due to the faster movement of the minute hand compared to the hour hand. Understanding this concept helps in solving clock reasoning problems easily and accurately.