alice travelled from town X to town Y ,and then to town Z .the total time she took to travel from town X to town Z ,when rounded to the nearest 10 minutes , was 3hrs 20min. the time she took to travel from town X to town Y ,when rounded to the nearest 10 minutes ,was 2hrs 10min.determine the upper and lower bounds of of the time alice took to travel from town Y to town Z .give your answer in minutes

To find the upper and lower bounds of the time Alice took to travel from town Y to town Z, we first need to determine the bounds for the total travel time and the travel time from town X to town Y. Then we can use these bounds to calculate the bounds for the remaining travel time from town Y to town Z.

### Step-by-Step Solution:

1. **Convert the given times to minutes:**
   - Total travel time from town X to town Z: \(3 \text{ hours } 20 \text{ minutes} = 3 \times 60 + 20 = 200 \text{ minutes}\)
   - Travel time from town X to town Y: \(2 \text{ hours } 10 \text{ minutes} = 2 \times 60 + 10 = 130 \text{ minutes}\)

2. **Determine the bounds for the total travel time (200 minutes):**
   - Rounded to the nearest 10 minutes means the lower bound is 5 minutes less, and the upper bound is 5 minutes more than the given time.
   - Lower bound for total travel time: \(200 \text{ minutes} - 5 \text{ minutes} = 195 \text{ minutes}\)
   - Upper bound for total travel time: \(200 \text{ minutes} + 5 \text{ minutes} = 205 \text{ minutes}\)

3. **Determine the bounds for the travel time from town X to town Y (130 minutes):**
   - Similarly, the lower bound is 5 minutes less, and the upper bound is 5 minutes more than the given time.
   - Lower bound for travel time from X to Y: \(130 \text{ minutes} - 5 \text{ minutes} = 125 \text{ minutes}\)
   - Upper bound for travel time from X to Y: \(130 \text{ minutes} + 5 \text{ minutes} = 135 \text{ minutes}\)

4. **Calculate the bounds for the travel time from town Y to town Z:**
   - Lower bound for travel time from Y to Z: Subtract the upper bound of travel time from X to Y from the lower bound of total travel time.
     \[
     \text{Lower bound} = 195 \text{ minutes} - 135 \text{ minutes} = 60 \text{ minutes}
     \]
   - Upper bound for travel time from Y to Z: Subtract the lower bound of travel time from X to Y from the upper bound of total travel time.
     \[
     \text{Upper bound} = 205 \text{ minutes} - 125 \text{ minutes} = 80 \text{ minutes}
     \]

### Answer:
- **Lower bound of the time Alice took to travel from town Y to town Z:** 60 minutes
- **Upper bound of the time Alice took to travel from town Y to town Z:** 80 minutes

Would you like any details or have any questions about this solution?

Here are 5 questions you might want to ask:
1. How do I determine the bounds for different travel times?
2. Can you explain the process of rounding to the nearest 10 minutes in more detail?
3. How does subtracting bounds impact the final result?
4. What if the travel times were given in a different format?
5. How would the solution change if the times were rounded to the nearest 5 minutes?

**Tip:** When working with time calculations and bounds, ensure to consistently use the same units (e.g., minutes) throughout the process for accuracy.

Explore how to calculate the upper and lower bounds of Alice's travel time from town Y to town Z based on given rounded travel times. This example illustrates key concepts in time calculation and rounding methods.

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