Between 5pm and 6pm, I looked at my watch mistaking the hour hand for the minute hand and the minute hand for the hour hand, I mistook the time to be 57 minutes earlier than the actual time. Find the number of minutes past 5 when I looked at my watch.

Answer

To solve the problem, we need to determine how many minutes past 5 PM the person looked at their watch, given that they mistook the hour hand for the minute hand and the minute hand for the hour hand, resulting in a perceived time that was 57 minutes earlier than the actual time.

### Step-by-Step Solution:

1. **Understanding the Time Frame**:
- The actual time is between 5 PM and 6 PM.
- Let the actual time be \( x \) minutes past 5 PM. Therefore, the actual time can be expressed as \( 5: x \).

2. **Calculating the Positions of the Hands**:
- At \( 5: x \), the position of the hour hand is:
\[ \text{Hour hand position} = 30 \times 5 + \frac{x}{2} = 150 + \frac{x}{2} \text{ degrees} \]
- The position of the minute hand is:
\[ \text{Minute hand position} = 6 \times x = 6x \text{ degrees} \]

3. **Mistaken Perception**:
- The person mistook the hour hand for the minute hand and vice versa. Therefore, they perceived the time as:
\[ \text{Perceived minute hand position} = 150 + \frac{x}{2} \]
\[ \text{Perceived hour hand position} = 6x \]

4. **Setting Up the Equation**:
- The perceived time is 57 minutes earlier than the actual time. Thus, we can write:
\[ 6x = 150 + \frac{x}{2} - 57 \]
- Simplifying this gives:
\[ 6x = 150 + \frac{x}{2} - 57 \]
\[ 6x = 93 + \frac{x}{2} \]

5. **Eliminating the Fraction**:
- Multiply the entire equation by 2 to eliminate the fraction:
\[ 12x = 186 + x \]

6. **Solving for \( x \)**:
- Rearranging gives:
\[ 12x - x = 186 \]
\[ 11x = 186 \]
\[ x = \frac{186}{11} \approx 16.91 \]

7. **Final Answer**:
- The number of minutes past 5 PM when the person looked at their watch is approximately \( 16.91 \) minutes, which can be rounded to \( 17 \) minutes.

### Conclusion:
The final answer is that the person looked at their watch approximately **17 minutes past 5 PM**.