In a 1000 m race Ameesha gives a headstart of 100 m to Bipasha and beats her by 200m 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 step by step, we need to analyze the information given about the races between Ameesha, Bipasha, and Celina.

### Step 1: Understand the race dynamics
In a 1000 m race:
- Ameesha gives Bipasha a head start of 100 m and beats her by 200 m.
- Ameesha gives Celina a head start of 100 m and beats her by 300 m.

### Step 2: Calculate the distance covered by Bipasha when Ameesha runs 1000 m
When Ameesha runs 1000 m, Bipasha starts at 100 m and runs:
\[ \text{Distance covered by Bipasha} = 1000 - 100 + 200 = 1100 - 100 = 900 \text{ m} \]
So, when Ameesha runs 1000 m, Bipasha runs 700 m.

### Step 3: Calculate the distance covered by Celina when Ameesha runs 1000 m
When Ameesha runs 1000 m, Celina starts at 100 m and runs:
\[ \text{Distance covered by Celina} = 1000 - 100 + 300 = 1100 - 100 = 800 \text{ m} \]
So, when Ameesha runs 1000 m, Celina runs 700 m.

### Step 4: Determine the ratio of speeds
From the above calculations:
- When Ameesha runs 1000 m, Bipasha runs 700 m.
- When Ameesha runs 1000 m, Celina runs 400 m.

### Step 5: Calculate the speed ratio
Let’s denote the speeds of Ameesha, Bipasha, and Celina as \( V_A \), \( V_B \), and \( V_C \) respectively.
From the information:
- The ratio of speeds can be derived as follows:
\[ \frac{V_B}{V_A} = \frac{700}{1000} = \frac{7}{10} \]
\[ \frac{V_C}{V_A} = \frac{400}{1000} = \frac{4}{10} = \frac{2}{5} \]

### Step 6: Find the relative speed of Bipasha and Celina
To find out how much Bipasha beats Celina, we need to find the ratio of their speeds:
\[ \frac{V_B}{V_C} = \frac{7/10}{2/5} = \frac{7}{10} \times \frac{5}{2} = \frac{35}{20} = \frac{7}{4} \]

### Step 7: Calculate the distance Bipasha beats Celina in a 50 m race
In a 50 m race, we can set up a proportion based on their speeds:
If Bipasha runs 7 parts, Celina runs 4 parts. In a 50 m race:
- The distance covered by Celina when Bipasha covers 50 m is:
\[ \text{Distance covered by Celina} = \frac{4}{7} \times 50 = \frac{200}{7} \approx 28.57 \text{ m} \]

### Step 8: Calculate the distance Bipasha beats Celina
Thus, the distance by which Bipasha beats Celina in a 50 m race is:
\[ 50 - \frac{200}{7} = 50 - 28.57 \approx 21.43 \text{ m} \]

### Final Answer
Bipasha beats Celina by approximately 21.43 meters in a 50 m race.

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