To determine the fraction of a turn made by the minute hand of a clock between 03:20 and 03:55, we need to follow these steps:
1. Identify the initial and final times in minutes:
- The initial time is 03:20. Convert this to total minutes:
[tex]\[
3 \, \text{hours} \times 60 \, \text{minutes/hour} + 20 \, \text{minutes} = 200 \, \text{minutes}
\][/tex]
- The final time is 03:55. Convert this to total minutes:
[tex]\[
3 \, \text{hours} \times 60 \, \text{minutes/hour} + 55 \, \text{minutes} = 235 \, \text{minutes}
\][/tex]
2. Calculate the difference in minutes:
- Find the difference between the final and initial times:
[tex]\[
235 \, \text{minutes} - 200 \, \text{minutes} = 35 \, \text{minutes}
\][/tex]
3. Determine the fraction of a full turn:
- A full turn of the minute hand corresponds to 60 minutes.
- The fraction of the turn made by the minute hand is:
[tex]\[
\frac{35 \, \text{minutes}}{60 \, \text{minutes}}
\][/tex]
4. Simplify the fraction:
- To simplify the fraction, find the greatest common divisor (GCD) of the numerator and the denominator.
- The GCD of 35 and 60 is 5.
- Divide both numerator and denominator by their GCD:
[tex]\[
\frac{35}{60} = \frac{35 \div 5}{60 \div 5} = \frac{7}{12}
\][/tex]
So, the fraction of a turn made by the minute hand between 03:20 and 03:55 is:
[tex]\[
\boxed{\frac{7}{12}}
\][/tex]
