In a 600 m race Prabhat has a start of 200 m and the ratio of speeds of prabhat and Nishith is 4:5 then the distance by which prabhat wins by:

Options

A. 100 m
B. 80 m
C. 120 m
D. none of these

Answer

To solve the problem, we need to determine the distance by which Prabhat wins the race against Nishith, given that Prabhat has a head start of 200 meters and the ratio of their speeds is 4:5.

### Step-by-Step Solution:

1. **Understanding the Race Setup**:
- The total distance of the race is 600 meters.
- Prabhat starts 200 meters ahead, meaning he only needs to run 400 meters to finish the race.

2. **Speed Ratio**:
- The ratio of the speeds of Prabhat and Nishith is given as 4:5.
- Let the speed of Prabhat be \(4x\) and the speed of Nishith be \(5x\).

3. **Time Calculation**:
- Since both runners will run for the same amount of time until one of them finishes the race, we can use the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
- For Prabhat, the time taken to run 400 meters is:
\[ \text{Time}_{Prabhat} = \frac{400}{4x} = \frac{100}{x} \]
- For Nishith, the time taken to run the full 600 meters is:
\[ \text{Time}_{Nishith} = \frac{600}{5x} = \frac{120}{x} \]

4. **Comparing Times**:
- Since both start running at the same time, we can set their times equal to find out how far Nishith runs when Prabhat finishes:
\[ \frac{100}{x} = \frac{120}{x} \]
- This means that when Prabhat finishes running 400 meters, Nishith has run for the same time but at his speed.

5. **Distance Covered by Nishith**:
- To find out how far Nishith has run when Prabhat finishes, we can calculate:
\[ \text{Distance}_{Nishith} = \text{Speed}_{Nishith} \times \text{Time}_{Prabhat} \]
- Plugging in the values:
\[ \text{Distance}_{Nishith} = 5x \times \frac{100}{x} = 500 \text{ meters} \]

6. **Calculating the Distance by which Prabhat Wins**:
- Since Nishith has run 500 meters when Prabhat finishes the race, the distance by which Prabhat wins is:
\[ \text{Distance}_{Win} = 600 - 500 = 100 \text{ meters} \]

### Final Answer:
Prabhat wins by **100 meters**.