A covers a distance of 1 km in 4 m 54 sec and B in 5 minutes. In a race of 1000 m, at what distance B should stand ahead of A so that both reach the last point together?

Answer

To solve the problem, we need to find out how far B should stand ahead of A so that both reach the finish line at the same time in a 1000 m race.

### Step 1: Convert the time taken by A to seconds
A covers 1 km in 4 minutes 54 seconds.
- Convert 4 minutes to seconds:
\[ 4 \text{ minutes} = 4 \times 60 = 240 \text{ seconds} \]
- Add the remaining seconds:
\[ 240 \text{ seconds} + 54 \text{ seconds} = 294 \text{ seconds} \]

### Step 2: Calculate the speed of A
Speed is calculated as distance divided by time.
- Distance covered by A = 1000 m
- Time taken by A = 294 seconds
- Speed of A:
\[ \text{Speed of A} = \frac{1000 \text{ m}}{294 \text{ s}} \approx 3.4014 \text{ m/s} \]

### Step 3: Convert the time taken by B to seconds
B covers 1 km in 5 minutes.
- Convert 5 minutes to seconds:
\[ 5 \text{ minutes} = 5 \times 60 = 300 \text{ seconds} \]

### Step 4: Calculate the speed of B
- Distance covered by B = 1000 m
- Time taken by B = 300 seconds
- Speed of B:
\[ \text{Speed of B} = \frac{1000 \text{ m}}{300 \text{ s}} \approx 3.3333 \text{ m/s} \]

### Step 5: Calculate the time taken by both A and B to finish the race
- Time taken by A to finish 1000 m:
\[ \text{Time by A} = \frac{1000 \text{ m}}{3.4014 \text{ m/s}} \approx 294 \text{ seconds} \]
- Time taken by B to finish 1000 m:
\[ \text{Time by B} = \frac{1000 \text{ m}}{3.3333 \text{ m/s}} \approx 300 \text{ seconds} \]

### Step 6: Determine the distance B should stand ahead of A
Since both should finish together, we need to find out how far A travels in the time B takes to finish the race:
- Time difference between A and B:
\[ \text{Time difference} = 300 \text{ seconds} - 294 \text{ seconds} = 6 \text{ seconds} \]
- Distance A covers in 6 seconds:
\[ \text{Distance} = \text{Speed of A} \times \text{Time difference} = 3.4014 \text{ m/s} \times 6 \text{ s} \approx 20.4084 \text{ m} \]

### Conclusion
B should stand approximately 20.41 meters ahead of A so that both reach the finish line together.

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