At what time between 3 o clock and 4 olck will the hands of a clock be in opposite directions

Answer

To find the time between 3 o'clock and 4 o'clock when the hands of a clock are in opposite directions, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Positions of the Clock Hands**:
- At 3 o'clock, the hour hand is at 3 and the minute hand is at 12. The angle between them is 90 degrees.

2. **Determine the Angle for Opposite Directions**:
- The hands of the clock are in opposite directions when they are 180 degrees apart.

3. **Use the Formula for Finding the Time**:
- The formula to find the time when the hands are in opposite directions is given by:
\[ \text{Time} = \frac{60 \times (H \times 2 - 1)}{11} \]
where \( H \) is the hour. Here, \( H = 3 \).

4. **Substituting the Values**:
- Substitute \( H = 3 \) into the formula:
\[ \text{Time} = \frac{60 \times (3 \times 2 - 1)}{11} \]
\[ = \frac{60 \times (6 - 1)}{11} \]
\[ = \frac{60 \times 5}{11} \]
\[ = \frac{300}{11} \]

5. **Calculating the Time**:
- Divide 300 by 11 to get the minutes:
\[ 300 \div 11 \approx 27.27 \text{ minutes} \]
- This means the hands will be in opposite directions approximately 27 minutes and 16 seconds past 3 o'clock.

6. **Final Answer**:
- Therefore, the time when the hands of the clock are in opposite directions between 3 o'clock and 4 o'clock is approximately **3:27:16**.