At what time between 4 o clock and 5 o clock will the hands of a clock be together

Answer

To find the time between 4 o'clock and 5 o'clock when the hands of the clock are together, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Movement of the Hands**:
- The minute hand moves at a speed of 6 degrees per minute (360 degrees in 60 minutes).
- The hour hand moves at a speed of 0.5 degrees per minute (30 degrees in 60 minutes).

2. **Calculate the Initial Positions**:
- At 4 o'clock, the hour hand is at 120 degrees (4 hours × 30 degrees).
- The minute hand is at 0 degrees.

3. **Set Up the Equation**:
- Let \( x \) be the number of minutes after 4 o'clock when the hands are together.
- The position of the minute hand after \( x \) minutes is \( 6x \) degrees.
- The position of the hour hand after \( x \) minutes is \( 120 + 0.5x \) degrees.

4. **Equate the Positions**:
- For the hands to be together, we set the positions equal:
\[ 6x = 120 + 0.5x \]

5. **Solve the Equation**:
- Rearranging gives:
\[ 6x - 0.5x = 120 \]
\[ 5.5x = 120 \]
\[ x = \frac{120}{5.5} = \frac{240}{11} \approx 21.82 \text{ minutes} \]

6. **Convert to Minutes and Seconds**:
- \( \frac{240}{11} \) minutes is approximately 21 minutes and \( \frac{240 \mod 11}{11} \times 60 \) seconds.
- \( 240 \mod 11 = 9 \), so \( \frac{9}{11} \times 60 \approx 49.09 \) seconds.

7. **Final Time**:
- Therefore, the hands of the clock will be together at approximately 4:21 and 49 seconds.

### Summary:
The hands of the clock will be together at approximately **4:21:49**.