Question

Between 1 o'clock and 2 o'clock, bills clockworks nominally, with the minute hand moving forwards 1 minute every minute period between and o'clock in real-time. The minute hand on bill's clock moves at half of the correct speed, showing minutes progress every minutes period between. Clock and real-time, the minute hand on bill's clock moves even slower commas showing minutes progress every minutes of real-time. This pattern continues for subsequent hours.

Bill sets his clock to 1 o'clock.When the real time is also one o'clock , what time will bills clock show when the real time is 7:00

Answer 1

To analyze Bill's clock situation, we need to understand how it runs compared to real time.

1. **Clock Mechanics**:
- Bill’s clock runs at half speed starting from 1:00.
- Specifically, for every 2 real minutes that pass, Bill's clock advances by 1 minute.

2. **Calculation of Time Passed**:
- We need to explore how Bill's clock moves for each hour.

Starting from 1:00 to 2:00 (1 hour real time):
- In 60 minutes of real time, Bill's clock will show:
\[
\text{Bill's clock time} = \frac{60 \text{ minutes real time}}{2} = 30 \text{ minutes}
\]
So at 2:00 real time, Bill's clock shows 1:30.

From 2:00 to 3:00 (1 hour real time):
- In another 60 minutes of real time, Bill's clock will once again move at half speed:
\[
\text{Bill's clock time} = 30 \text{ (as calculated above)}
\]
At 3:00 real time, Bill's clock shows:
\[
1:30 + 30 \text{ minutes} = 2:00 \text{ on Bill's clock}
\]

From 3:00 to 4:00 (1 hour real time):
- Again, Bill's clock moves forward:
\[
\text{Bill's clock time} = 30 \text{ minutes}
\]
At 4:00 real time, Bill's clock shows:
\[
2:00 + 30 \text{ minutes} = 2:30 \text{ on Bill's clock}
\]

From 4:00 to 5:00 (1 hour real time):
- Bill's clock moves forward again:
\[
\text{Bill's clock time} = 30 \text{ minutes}
\]
At 5:00 real time, Bill's clock shows:
\[
2:30 + 30 \text{ minutes} = 3:00 \text{ on Bill's clock}
\]

From 5:00 to 6:00 (1 hour real time):
- Once more, Bill's clock moves forward:
\[
\text{Bill's clock time} = 30 \text{ minutes}
\]
At 6:00 real time, Bill's clock shows:
\[
3:00 + 30 \text{ minutes} = 3:30 \text{ on Bill's clock}
\]

From 6:00 to 7:00 (1 hour real time):
- Finally, Bill's clock moves forward again:
\[
\text{Bill's clock time} = 30 \text{ minutes}
\]
At 7:00 real time, Bill's clock shows:
\[
3:30 + 30 \text{ minutes} = 4:00 \text{ on Bill's clock}
\]

Therefore, when the real time is 7:00, Bill's clock shows **4:00**.