A, B, C walk 1 km in 5 minutes, 8 minutes and 10 minutes respectively. C starts walking from a point, at a certain time, B starts from the same point 1 minutes later and A starts from the same point 2 minutes later than C. Then A meets B and C at times.

Options

A. `5/3` min, 2 min
B. 1 min, 2 min
C. 2 min, 3 min
D. `4/3` min, 3 min

Answer

To solve the problem step by step, we need to determine the speeds of A, B, and C, and then calculate the times at which A meets B and C.

### Step 1: Calculate the speeds of A, B, and C
- **Speed of A**: A walks 1 km in 5 minutes.
\[ \text{Speed of A} = \frac{1000 \text{ meters}}{5 \text{ minutes}} = 200 \text{ meters per minute} \]

- **Speed of B**: B walks 1 km in 8 minutes.
\[ \text{Speed of B} = \frac{1000 \text{ meters}}{8 \text{ minutes}} = 125 \text{ meters per minute} \]

- **Speed of C**: C walks 1 km in 10 minutes.
\[ \text{Speed of C} = \frac{1000 \text{ meters}}{10 \text{ minutes}} = 100 \text{ meters per minute} \]

### Step 2: Determine the distances covered before A starts walking
- C starts walking first. After 2 minutes (when A starts), C will have covered:
\[ \text{Distance covered by C in 2 minutes} = \text{Speed of C} \times \text{Time} = 100 \text{ m/min} \times 2 \text{ min} = 200 \text{ meters} \]

### Step 3: Calculate the time taken for A to meet C
- The distance between A and C when A starts is 200 meters. The relative speed of A with respect to C is:
\[ \text{Relative speed} = \text{Speed of A} - \text{Speed of C} = 200 \text{ m/min} - 100 \text{ m/min} = 100 \text{ m/min} \]
- Time taken for A to meet C:
\[ \text{Time} = \frac{\text{Distance}}{\text{Relative speed}} = \frac{200 \text{ meters}}{100 \text{ m/min}} = 2 \text{ minutes} \]

### Step 4: Determine the distance covered by B before A starts walking
- B starts walking 1 minute after C. Therefore, in that 1 minute, B will cover:
\[ \text{Distance covered by B in 1 minute} = \text{Speed of B} \times \text{Time} = 125 \text{ m/min} \times 1 \text{ min} = 125 \text{ meters} \]

### Step 5: Calculate the time taken for A to meet B
- The distance between A and B when A starts is 125 meters. The relative speed of A with respect to B is:
\[ \text{Relative speed} = \text{Speed of A} - \text{Speed of B} = 200 \text{ m/min} - 125 \text{ m/min} = 75 \text{ m/min} \]
- Time taken for A to meet B:
\[ \text{Time} = \frac{\text{Distance}}{\text{Relative speed}} = \frac{125 \text{ meters}}{75 \text{ m/min}} = \frac{125}{75} = \frac{5}{3} \text{ minutes} \]

### Summary of Results
- A meets C in **2 minutes**.
- A meets B in **\(\frac{5}{3}\) minutes**.