In a race of 600 m, A can beat B by 60 m and in a race of 500 m, B can beat C by 50 m. By how many metres will A beat C in a race of 400 m ?

Options

A. 70 m
B. 76 m
C. `(77) 1/3` m
D. None of these

Answer

To solve the problem step by step, we will analyze the given information about the races and derive the necessary relationships.

### Step 1: Understand the first race between A and B
In a race of 600 meters, A beats B by 60 meters. This means:
- When A finishes 600 meters, B has only run 540 meters (600 - 60).

### Step 2: Calculate the speeds of A and B
Let the speed of A be \( v_A \) and the speed of B be \( v_B \). The time taken by both A and B to finish their respective distances is the same. Therefore, we can set up the equation:
\[ \frac{600}{v_A} = \frac{540}{v_B} \]
Cross-multiplying gives:
\[ 600 v_B = 540 v_A \]
Simplifying this, we find:
\[ \frac{v_A}{v_B} = \frac{600}{540} = \frac{10}{9} \]
This means A's speed is to B's speed as 10 is to 9.

### Step 3: Understand the second race between B and C
In a race of 500 meters, B beats C by 50 meters. This means:
- When B finishes 500 meters, C has only run 450 meters (500 - 50).

### Step 4: Calculate the speeds of B and C
Using similar reasoning as before, we can set up the equation:
\[ \frac{500}{v_B} = \frac{450}{v_C} \]
Cross-multiplying gives:
\[ 500 v_C = 450 v_B \]
Simplifying this, we find:
\[ \frac{v_B}{v_C} = \frac{500}{450} = \frac{10}{9} \]
This means B's speed is to C's speed as 10 is to 9.

### Step 5: Combine the ratios to find A's speed relative to C's speed
From the two ratios we have:
- \( \frac{v_A}{v_B} = \frac{10}{9} \)
- \( \frac{v_B}{v_C} = \frac{10}{9} \)

To find \( \frac{v_A}{v_C} \), we multiply the two ratios:
\[ \frac{v_A}{v_C} = \frac{v_A}{v_B} \times \frac{v_B}{v_C} = \frac{10}{9} \times \frac{10}{9} = \frac{100}{81} \]

### Step 6: Determine the distances covered in a 400-meter race
Now, we want to find out how far C runs when A runs 400 meters. Using the ratio \( \frac{v_A}{v_C} = \frac{100}{81} \):
- If A runs 400 meters, we can find the distance C runs using the ratio:
\[ \text{Distance covered by C} = \frac{81}{100} \times 400 = 324 \text{ meters} \]

### Step 7: Calculate how much A beats C
To find out by how many meters A beats C in a 400-meter race, we subtract the distance covered by C from the distance covered by A:
\[ \text{Distance A beats C} = 400 - 324 = 76 \text{ meters} \]

### Final Answer
A beats C by **76 meters** in a race of 400 meters.

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