Calculate the duration between the following pairs of times: 9:10 a.m. to 10:05 a.m., 11:50 a.m. to 12:30 p.m., and 12:20 a.m. to 1:00 p.m.

to 10:00 a.m.

$$10 \text{ hours } 00 \text{ minutes} - 9 \text{ hours } 10 \text{ minutes}$$ $$= (9 \text{ hours } 60 \text{ minutes}) - (9 \text{ hours } 10 \text{ minutes})$$ $$= 50 \text{ minutes}$$

to 10:05 a.m.

$$50 \text{ minutes} + 5 \text{ minutes} = 55 \text{ minutes}$$

The duration is $\boxed{\text{55 minutes}}$.

2. Duration from 11:50 a.m. to 12:30 p.m.

to 12:00 p.m.

$$12 \text{ hours } 00 \text{ minutes} - 11 \text{ hours } 50 \text{ minutes}$$ $$= (11 \text{ hours } 60 \text{ minutes}) - (11 \text{ hours } 50 \text{ minutes})$$ $$= 10 \text{ minutes}$$

to 12:30 p.m.

$$10 \text{ minutes} + 30 \text{ minutes} = 40 \text{ minutes}$$

The duration is $\boxed{\text{40 minutes}}$.

3. Duration from 12:20 a.m. to 1:00 p.m.

$$12:20 \text{ a.m.} = 00:20$$ $$1:00 \text{ p.m.} = 13:00$$

$$13 \text{ hours } 00 \text{ minutes} - 00 \text{ hours } 20 \text{ minutes}$$

To subtract, borrow 1 hour (60 minutes) from the hours.

$$= (12 \text{ hours } 60 \text{ minutes}) - (00 \text{ hours } 20 \text{ minutes})$$ $$= (12-0) \text{ hours } (60-20) \text{ minutes}$$ $$= 12 \text{ hours } 40 \text{ minutes}$$

The duration is $\boxed{\text{12 hours 40 minutes}}$.