Introduction
Calculating the exact time when the hour and minute hands of a clock coincide can be an intriguing mathematical problem. This article explores the precise moment when the hands of a clock coincide between 4:00 and 5:00, providing a clear and detailed explanation of the underlying mathematical concepts.

Understanding the Problem

The hands of a clock coincide when they overlap, appearing as one. This overlap occurs between 4:00 and 5:00 when the minute hand is in transit and the hour hand has moved past the 4. To solve this, we need to determine the exact time when both hands align between these two hours.

Mathematical Principles

Firstly, it's crucial to understand the speeds of the hour and minute hands:

The hour hand moves by 30° per hour or 0.5° per minute. The minute hand moves by 360° per hour or 6° per minute.

At 4:00, the hour hand is at 120° (since 4 x 30°), while the minute hand is at 0°. We need to find the exact minute (X) when both hands overlap. This means that the angles moved by both hands should be equal.

Mathematical Solution

We start by stating the equations:

- Minute hand angle: (6X°)

- Hour hand angle: (X/2°)

- The hour hand starts at 120° (since it's 4:00).

Setting up the equation:

(120 frac{X}{2} 6X)

Rearranging to solve for X:

(frac{X}{2} 6X - 120)

(frac{11X}{2} 120)

(X frac{240}{11} approx 21.81818) minutes

Thus, the exact time when the hands of the clock will coincide between 4:00 and 5:00 is 4:21:49.

Verification and Real-World Application

To verify the solution, we can use the formula: ( pm 30h pm 5.5m 0 ), where h is the hour and m is the minute. Plugging in 4:00 (h 4), we get:

(120 - 5.5m 0)

(m 21.81818) minutes

This confirms our previous calculation. Therefore, the clock hands will coincide at 4:21:49, marking a fascinating intersection of time and mathematics.

Conclusion

The exact time when the clock hands overlap between 4:00 and 5:00 is 4:21:49. This calculation demonstrates the interplay between the speeds of the clock hands and their positions, providing a classical example of angle and rate-of-change problems in real-world applications.