Two people A and B start from the same point and head toward the same destination 3.5 miles away. B starts 4.5 minutes after A. A reaches the destination, walks back 1 mile, and meets B there. If A walks 1 mile in 6 minutes, how many minutes does B take to walk 1 mile?

Introduction / Context:
This question involves two walkers starting at different times and meeting after one of them turns back from the destination. It tests your understanding of relative motion with staggered start times and meeting points that are not at the end of the path. Careful tracking of distances and times is essential in this type of time and distance problem.

Given Data / Assumptions:
A and B start from the same point toward a destination 3.5 miles away.
A starts first and walks at a constant rate of 1 mile every 6 minutes.
B starts 4.5 minutes after A, from the same starting point.
A reaches the destination, turns back immediately, and walks 1 mile back toward the starting point.

Given Data / Assumptions:
They meet at the point where A has walked 1 mile back from the destination, that is 2.5 miles from the start.
We need to determine the time B takes to walk 1 mile, expressed in minutes per mile.

Concept / Approach:
The main idea is to compare distances from the starting point at the meeting time. A has travelled a total of 4.5 miles by the time they meet (3.5 miles going plus 1 mile returning). At his known speed, this gives the total time until the meeting. For B, the distance from the start to the meeting point is 2.5 miles, and time for B is slightly less because B started later. The ratio of B distance to B time provides B speed and thus the time per mile.

Step-by-Step Solution:
Step 1: Speed of A = 1 mile in 6 minutes, so A walks at a constant rate.Step 2: At the moment they meet, A has gone from the start to the destination (3.5 miles) and then 1 mile back, so total distance walked by A is 3.5 + 1 = 4.5 miles.Step 3: Time taken by A to walk 4.5 miles at 1 mile in 6 minutes = 4.5 * 6 = 27 minutes.Step 4: Therefore, the meeting happens 27 minutes after A starts.Step 5: B starts 4.5 minutes after A, so B walks for 27 − 4.5 = 22.5 minutes until the meeting.Step 6: The meeting point is 1 mile back from the destination, that is 3.5 − 1 = 2.5 miles from the start.Step 7: So B covers 2.5 miles in 22.5 minutes.Step 8: Time taken by B per mile = 22.5 / 2.5 minutes.Step 9: 22.5 / 2.5 = 9 minutes per mile.

Verification / Alternative check:
Using B speed of 1 mile in 9 minutes, in 22.5 minutes B covers 22.5 / 9 = 2.5 miles, exactly reaching the 2.5 mile meeting point. At the same time, A has walked 27 minutes at 1 mile every 6 minutes, which is 27 / 6 = 4.5 miles, equal to 3.5 miles forward and 1 mile back. The geometry of their positions and times perfectly matches the description in the problem, confirming our result.

Why Other Options Are Wrong:
8 minutes per mile would make B too fast; he would travel more than 2.5 miles in 22.5 minutes and pass the meeting point.
10 or 12 minutes per mile would make B slower than required; he would not reach the meeting point in 22.5 minutes.
6 minutes per mile would mean B has the same speed as A, and with the delayed start they would not meet exactly where A has walked back 1 mile from the destination.

Common Pitfalls:
Students may confuse the total distance travelled by A with the distance from the start to the meeting point. Another common mistake is to forget that B starts later, so B time is less than A time. Carefully accounting for offset start times and distances is essential in chase and meeting problems like this one.

Final Answer:
B takes 9 minutes to walk 1 mile.