What do the hands of a clock coincide between 5 and 6?

Options

A. 0.22916666666667
B. 0.22726851851852
C. 0.23074074074074
D. 0.22842592592593

Answer

To find out when the hands of a clock coincide between 5 and 6, we can use the formula for the position of the minute hand and the hour hand.

### Step-by-Step Solution:

1. **Understand the Positions of the Hands**:
At 5:00, the hour hand is at 150 degrees (since each hour represents 30 degrees, and 5 hours = 5 * 30 = 150 degrees). The minute hand is at 0 degrees.

2. **Set Up the Formula**:
The formula to find the time (in minutes) when the hands coincide is:
\[ m = \frac{60h - 11m}{11} \]
where \( h \) is the hour and \( m \) is the minutes past the hour.

3. **Substituting Values**:
Here, \( h = 5 \). We need to find \( m \) when the hands coincide:
\[ m = \frac{60 \times 5 - 11m}{11} \]

4. **Rearranging the Equation**:
Multiply both sides by 11 to eliminate the fraction:
\[ 11m = 300 - 11m \]
Now, combine like terms:
\[ 11m + 11m = 300 \]
\[ 22m = 300 \]

5. **Solving for m**:
Divide both sides by 22:
\[ m = \frac{300}{22} = \frac{150}{11} \]

6. **Convert to Minutes and Seconds**:
To convert \( \frac{150}{11} \) minutes into minutes and seconds:
- \( 150 \div 11 = 13 \) (whole minutes)
- \( 150 \mod 11 = 7 \) (remainder)
- Therefore, \( \frac{150}{11} \) minutes = 13 minutes and \( \frac{7 \times 60}{11} \) seconds.

7. **Final Calculation**:
Calculate the seconds:
\[ \frac{7 \times 60}{11} = \frac{420}{11} \approx 38.18 \text{ seconds} \]
Thus, the hands of the clock coincide at approximately 13 minutes and 38 seconds past 5.

### Conclusion:
The hands of the clock coincide between 5 and 6 at approximately **5:13 and 38 seconds**.