A can run a kilometre in 4 minutes 50 seconds and B in 5 minutes. How many metres start can A give B in a km race, so that the race may end in a dead heat ?

Options

A. 30 m
B. `(16) 2/3` m
C. `(33) 1/3` m
D. None of these

Answer

To solve the problem, we need to determine how many meters of a head start A can give B in a 1 km race so that they finish at the same time.

### Step-by-Step Solution:

1. **Convert the running times into seconds:**
- A's time to run 1 km = 4 minutes 50 seconds.
- Convert this to seconds:
\[ 4 \text{ minutes} = 4 \times 60 = 240 \text{ seconds} \]
\[ 50 \text{ seconds} = 50 \text{ seconds} \]
\[ \text{Total time for A} = 240 + 50 = 290 \text{ seconds} \]
- B's time to run 1 km = 5 minutes.
- Convert this to seconds:
\[ 5 \text{ minutes} = 5 \times 60 = 300 \text{ seconds} \]

2. **Calculate the speeds of A and B:**
- Speed of A:
\[ \text{Speed of A} = \frac{1000 \text{ meters}}{290 \text{ seconds}} \approx 3.448 \text{ m/s} \]
- Speed of B:
\[ \text{Speed of B} = \frac{1000 \text{ meters}}{300 \text{ seconds}} \approx 3.333 \text{ m/s} \]

3. **Determine the time taken by A to run 1 km:**
- A takes 290 seconds to run 1 km.

4. **Calculate the distance B can cover in the same time:**
- In 290 seconds, the distance covered by B is:
\[ \text{Distance covered by B} = \text{Speed of B} \times \text{Time} = 3.333 \text{ m/s} \times 290 \text{ seconds} \approx 966.67 \text{ meters} \]

5. **Calculate the head start for B:**
- The head start that A can give B is the difference between 1 km and the distance B covers in 290 seconds:
\[ \text{Head start} = 1000 \text{ meters} - 966.67 \text{ meters} \approx 33.33 \text{ meters} \]

### Final Answer:
A can give B approximately **33.33 meters** start in a 1 km race for them to finish in a dead heat.