Understanding the Speed, Distance, and Time Problem

This question is a classic problem involving speed, distance, and time. We have two individuals, A and B, moving between two points, P and Q. The distance between P and Q is given as 9 km. We are given information about their starting times, speeds, and where they meet, and we need to find B's speed.

Step-by-Step Solution to Calculate B's Speed

1. Calculate A's Speed

A's speed is given as 1 kilometre in 10 minutes.

• Distance covered by A = 1 km
• Time taken by A = 10 minutes
• A's speed = \(\frac{\text{Distance}}{\text{Time}}\)
• A's speed = \(\frac{1 \text{ km}}{10 \text{ minutes}} = \frac{1}{10} \text{ km/minute}\)

2. Calculate Time A Takes to Reach Point Q

The distance from P to Q is 9 km. A's speed is \(\frac{1}{10}\) km/minute.

• Distance P to Q = 9 km
• A's speed = \(\frac{1}{10}\) km/minute
• Time taken by A to reach Q = \(\frac{\text{Distance}}{\text{Speed}}\)
• Time taken by A to reach Q = \(\frac{9 \text{ km}}{\frac{1}{10} \text{ km/minute}} = 9 \times 10 \text{ minutes} = 90 \text{ minutes}\)

3. Determine When and Where A Meets B

A reaches Q in 90 minutes and immediately returns. A meets B after walking 1 km from Q towards P.

• Distance A travelled from Q back towards P = 1 km
• Total distance A covered until meeting B = Distance P to Q + Distance Q back towards P
• Total distance A covered = 9 km + 1 km = 10 km

Now, let's find the total time A walked until meeting B.

• Total distance A covered = 10 km
• A's speed = \(\frac{1}{10}\) km/minute
• Total time A walked = \(\frac{\text{Total distance covered by A}}{\text{A's speed}}\)
• Total time A walked = \(\frac{10 \text{ km}}{\frac{1}{10} \text{ km/minute}} = 10 \times 10 \text{ minutes} = 100 \text{ minutes}\)

So, A meets B 100 minutes after A started walking.

4. Determine the Distance B Covered Until Meeting A

The meeting point is 1 km away from Q on the path towards P. Since B started from P (which is 9 km from Q) and walked towards Q, B's distance from P to the meeting point is:

• Meeting point distance from Q = 1 km
• Distance P to Q = 9 km
• Distance B covered = Distance P to Q - Meeting point distance from Q
• Distance B covered = 9 km - 1 km = 8 km

5. Determine the Time B Walked Until Meeting A

B started 4 minutes after A. A walked for 100 minutes until they met.

• Total time elapsed since A started = 100 minutes
• B started late by = 4 minutes
• Time B walked = Total time elapsed - B's start delay
• Time B walked = 100 minutes - 4 minutes = 96 minutes

6. Calculate B's Speed

B covered 8 km in 96 minutes.

• Distance covered by B = 8 km
• Time B walked = 96 minutes
• B's speed = \(\frac{\text{Distance covered by B}}{\text{Time B walked}}\)
• B's speed = \(\frac{8 \text{ km}}{96 \text{ minutes}}\)
• B's speed = \(\frac{8}{96} \text{ km/minute}\)

Simplifying the fraction:

• \(\frac{8}{96} = \frac{1 \times 8}{12 \times 8} = \frac{1}{12}\)
• B's speed = \(\frac{1}{12}\) km/minute

Final Answer

B's speed is \(\frac{1}{12}\) kilometres per minute.

Revision Table: Key Values

Additional Information: Relative Speed Concept

While this problem was solved by calculating the time each person walked and the distance they covered individually, it can also be approached using the concept of relative speed, especially when considering the point where they meet.

• When two objects move towards each other, their relative speed is the sum of their individual speeds. The time taken to meet is the initial distance between them divided by their relative speed.
• When two objects move in the same direction, the faster object's speed relative to the slower one is the difference between their speeds.
• In this problem, A and B are initially moving towards Q (same direction), then A turns back towards B (opposite direction). The meeting point is crucial. The total distance covered by A and B combined from the moment B starts until they meet is equal to the distance from P to Q plus the distance A travels back. This perspective can sometimes simplify problems.

In this specific case, calculating individual times and distances was straightforward because we could easily determine the exact time duration for which each person travelled until the meeting occurred.