In a kilometre race, A can give B a start of 20 m and also in a half kilometre race C beats A by 50 m.
If A, B and C run a 2 km race, what is the difference between the distance covered by the two losers, when the winner finishes the race?

Options

A. 64 m
B. 36 m
C. 32 m
D. 58 m

Answer

To solve the problem step by step, let's break down the information given and find the required answer.

### Step 1: Understand the given information
- A can give B a start of 20 m in a 1000 m race.
- In a 500 m race, C beats A by 50 m.

### Step 2: Determine the speeds of A, B, and C
From the first piece of information:
- When A runs 1000 m, B runs 980 m (because B gets a 20 m start).
- Therefore, the ratio of speeds of A and B is:
\[ \text{Speed of A} : \text{Speed of B} = 1000 : 980 = 100 : 98 \]

From the second piece of information:
- When C runs 500 m, A runs 450 m (because C beats A by 50 m).
- Therefore, the ratio of speeds of C and A is:
\[ \text{Speed of C} : \text{Speed of A} = 500 : 450 = 10 : 9 \]

### Step 3: Express the speeds in a common format
Let the speed of A be \(9x\), then:
- Speed of B = \( \frac{98}{100} \times 9x = 8.82x \)
- Speed of C = \( \frac{10}{9} \times 9x = 10x \)

### Step 4: Calculate the time taken to complete a 2 km race
The distance of the race is 2000 m. We can calculate the time taken by each runner to complete the race:
- Time taken by A:
\[ \text{Time}_A = \frac{2000}{9x} \]
- Time taken by B:
\[ \text{Time}_B = \frac{2000}{8.82x} \]
- Time taken by C:
\[ \text{Time}_C = \frac{2000}{10x} \]

### Step 5: Determine the winner
To find out who finishes first, we need to compare the times:
- Since \(10x\) is the largest speed, C will finish first.
- Now we need to find how far A and B have run when C finishes.

### Step 6: Calculate the distances covered by A and B when C finishes
When C finishes the race:
- Time taken by C to finish:
\[ \text{Time}_C = \frac{2000}{10x} = \frac{200}{x} \]
- Distance covered by A in this time:
\[ \text{Distance}_A = 9x \times \frac{200}{x} = 1800 \text{ m} \]
- Distance covered by B in this time:
\[ \text{Distance}_B = 8.82x \times \frac{200}{x} = 1764 \text{ m} \]

### Step 7: Find the difference in distance covered by the two losers
- The distances covered by A and B are 1800 m and 1764 m respectively.
- The difference between the distances covered by A and B:
\[ \text{Difference} = 1800 - 1764 = 36 \text{ m} \]

### Final Answer
The difference between the distances covered by the two losers when the winner finishes the race is **36 m**.

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