Answer :

To find what time it is [tex]\( \frac{53}{4} \)[/tex] hours after 11:32 PM, follow these steps:

1. Convert the Time to Add into Minutes:
- First, we need to express [tex]\( \frac{53}{4} \)[/tex] hours as minutes.
- Calculate [tex]\( \frac{53}{4} \)[/tex]:
[tex]\[
\frac{53}{4} = 13.25
\][/tex]
- Convert 13.25 hours into minutes by multiplying by 60 (since 1 hour = 60 minutes):
[tex]\[
13.25 \times 60 = 795 \text{ minutes}
\][/tex]

2. Convert the Starting Time into Minutes:
- Convert 11:32 PM to minutes since midnight:
- 11 hours = 11 × 60 minutes = 660 minutes
- Add the 32 minutes, so:
[tex]\[
660 + 32 = 692 \text{ minutes}
\][/tex]

3. Add the Minutes:
- Add the minutes from step 1 to the minutes from step 2:
[tex]\[
692 + 795 = 1487 \text{ minutes}
\][/tex]

4. Convert Total Minutes Back to Hours and Minutes:
- Calculate how many full hours there are in 1487 minutes by dividing by 60:
[tex]\[
1487 \div 60 = 24 \text{ hours with a remainder of } 47 \text{ minutes}
\][/tex]
- This indicates that 1487 minutes is 24 hours and 47 minutes.

5. Adjust for the Time of Day:
- Starting from 11:32 PM, adding 24 hours brings us back to 11:32 PM the next day.
- Add the 47 minutes to this time:
- 11:32 PM + 47 minutes becomes 12:19 AM.

Therefore, [tex]\( \frac{53}{4} \)[/tex] hours after 11:32 PM is 12:47 AM.

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