At what time between 4 o'clock and 5 o'clock will the hour hand and minute hand coincide?

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Accepted Answer

#### Step 1: Understand the Movement of Clock Hands
- The **minute hand** moves at **6 degrees per minute**.  
- The **hour hand** moves at **0.5 degrees per minute** (30 degrees per hour).

At **4 o'clock**, the hour hand is at **120 degrees** (4 × 30).

#### Step 2: Relative Speed of the Hands
The relative speed between the minute hand and hour hand is:  
$$
6 - 0.5 = 5.5 \, 	ext{degrees per minute}.
$$

#### Step 3: Time Taken for Coincidence
The minute hand needs to cover the initial gap of **120 degrees** to coincide with the hour hand. Using the formula:  
$$
	ext{Time} = \frac{	ext{Initial Gap}}{	ext{Relative Speed}} = \frac{120}{5.5}.
$$

Simplify:  
$$
	ext{Time} = \frac{120}{5.5} = \frac{240}{11} \, 	ext{minutes}.
$$

#### Step 4: Convert Time to Hours and Minutes
Convert $$\frac{240}{11}$$ minutes into hours, minutes, and seconds:  
- Minutes: $$\frac{240}{11} = 21 \, 	ext{minutes} \, + \, \frac{9}{11} \, 	ext{of a minute}.$$  
- Seconds: $$\frac{9}{11} 	imes 60 = 49.09 \, 	ext{seconds}.$$

Thus, the hands coincide at **4:21 $$\frac{9}{11}$$**.

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### Final Answer:
The correct answer is **(c) 4:21 $$\frac{9}{11}$$**.

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