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 25m . By how many metres will A beat C in a 400 m race?

Options

A. 56m
B. 60 m
C. 58 m
D. 54 m

Answer

To solve the problem step by step, we will first determine the speeds of A, B, and C based on the information given in the question.

### Step 1: Determine the ratio of speeds of 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).

So, we can set up the ratio of their speeds:
\[ \frac{A}{B} = \frac{600}{540} \]

Simplifying this ratio:
\[ \frac{A}{B} = \frac{600 \div 60}{540 \div 60} = \frac{10}{9} \]

### Step 2: Determine the ratio of speeds of B and C
In a race of 500 meters, B beats C by 25 meters. This means when B finishes 500 meters, C has only run 475 meters (500 - 25).

So, we can set up the ratio of their speeds:
\[ \frac{B}{C} = \frac{500}{475} \]

Simplifying this ratio:
\[ \frac{B}{C} = \frac{500 \div 25}{475 \div 25} = \frac{20}{19} \]

### Step 3: Determine the combined ratio of A and C
Now, we need to find the ratio of A to C. We can do this by multiplying the two ratios we found:
\[ \frac{A}{C} = \frac{A}{B} \times \frac{B}{C} = \frac{10}{9} \times \frac{20}{19} \]

Calculating this:
\[ \frac{A}{C} = \frac{10 \times 20}{9 \times 19} = \frac{200}{171} \]

### Step 4: Calculate how far C runs when A runs 400 meters
We know that in a race of 400 meters, we want to find out how far C has run when A runs 400 meters. Using the ratio we found:
\[ \frac{A}{C} = \frac{200}{171} \]

Let \( C \) be the distance C runs when A runs 400 meters:
\[ \frac{400}{C} = \frac{200}{171} \]

Cross-multiplying gives:
\[ 200C = 400 \times 171 \]
\[ C = \frac{400 \times 171}{200} = \frac{68400}{200} = 342 \]

### Step 5: Determine by how many meters A beats C
Now we can find out by how many meters A beats C in a 400-meter race:
\[ \text{Distance A beats C} = 400 - 342 = 58 \text{ meters} \]

### Final Answer
A beats C by **58 meters** in a 400-meter race.

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