In a 1000 m race Ameesha gives a headstart of 100 m to Bipasha and beats her by 200 m. In the same race Ameesha gives a headstart of 100 m to Celina and beats her by 300 m. By how many metres would Bipasha beat Celina in a 50 m race?

Options

A. 6.66 m
B. 7.143 m
C. 8 m
D. none of these

Answer

To solve the problem, we need to analyze the information given about the races between Ameesha, Bipasha, and Celina.

### Step 1: Understand the race conditions
- In a 1000 m race, Ameesha gives a head start of 100 m to Bipasha. This means that when Ameesha runs 1000 m, Bipasha runs only 900 m.
- Ameesha beats Bipasha by 200 m, which means when Ameesha finishes the race (1000 m), Bipasha is at 800 m (1000 m - 200 m).

### Step 2: Calculate the speeds of Ameesha and Bipasha
Let the speed of Ameesha be \( v_A \) and the speed of Bipasha be \( v_B \).
- The time taken by Ameesha to finish the race is \( \frac{1000}{v_A} \).
- In that same time, Bipasha runs 800 m, so:
\[ \frac{1000}{v_A} = \frac{800}{v_B} \]
Rearranging gives:
\[ v_B = \frac{800 \cdot v_A}{1000} = 0.8 v_A \]

### Step 3: Analyze the race between Ameesha and Celina
- In the same race, Ameesha gives a head start of 100 m to Celina. This means Celina runs only 900 m when Ameesha runs 1000 m.
- Ameesha beats Celina by 300 m, which means when Ameesha finishes the race (1000 m), Celina is at 700 m (1000 m - 300 m).

### Step 4: Calculate the speeds of Ameesha and Celina
Let the speed of Celina be \( v_C \).
- The time taken by Ameesha to finish the race is still \( \frac{1000}{v_A} \).
- In that same time, Celina runs 700 m, so:
\[ \frac{1000}{v_A} = \frac{700}{v_C} \]
Rearranging gives:
\[ v_C = \frac{700 \cdot v_A}{1000} = 0.7 v_A \]

### Step 5: Compare the speeds of Bipasha and Celina
From the previous steps, we have:
- \( v_B = 0.8 v_A \)
- \( v_C = 0.7 v_A \)

### Step 6: Calculate the relative speed in a 50 m race
To find out how many meters Bipasha would beat Celina in a 50 m race, we need to find the time taken by both in that distance.

- Time taken by Bipasha to run 50 m:
\[ t_B = \frac{50}{v_B} = \frac{50}{0.8 v_A} = \frac{62.5}{v_A} \]

- Time taken by Celina to run 50 m:
\[ t_C = \frac{50}{v_C} = \frac{50}{0.7 v_A} = \frac{71.43}{v_A} \]

### Step 7: Calculate the distance Celina runs in the time Bipasha finishes 50 m
In the time \( t_B \), the distance Celina runs is:
\[ \text{Distance} = v_C \cdot t_B = 0.7 v_A \cdot \frac{62.5}{v_A} = 43.75 \text{ m} \]

### Step 8: Calculate the difference
Thus, in a 50 m race, Bipasha finishes while Celina runs 43.75 m. Therefore, the distance by which Bipasha beats Celina is:
\[ 50 \text{ m} - 43.75 \text{ m} = 6.25 \text{ m} \]

### Final Answer
Bipasha would beat Celina by **6.25 meters** in a 50 m race.

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