In a race of 800 m, A can beat B by 40 m. In a race of 500 m, B can beat C by 50 m. In a race of 200 m, By what margin will A beat C.

Answer

To solve the problem step by step, we will analyze the information given about the races between A, B, and C.

### Step 1: Understand the first race between A and B
In the first race of 800 m, A beats B by 40 m. This means:
- When A finishes 800 m, B has run 800 m - 40 m = 760 m.

### Step 2: Calculate the ratio of speeds of A and B
Since both A and B run for the same amount of time, we can set up the ratio of their speeds based on the distances they covered:
- Speed of A : Speed of B = Distance covered by A : Distance covered by B
- Speed of A : Speed of B = 800 : 760
- Simplifying this gives us:
- Speed of A : Speed of B = 20 : 19

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

### Step 4: Calculate the ratio of speeds of B and C
Again, since both B and C run for the same amount of time, we can set up the ratio of their speeds based on the distances they covered:
- Speed of B : Speed of C = Distance covered by B : Distance covered by C
- Speed of B : Speed of C = 500 : 450
- Simplifying this gives us:
- Speed of B : Speed of C = 10 : 9

### Step 5: Set the speeds of B to a common value
To compare the speeds of A, B, and C, we need to express the speeds of B in a common way. We can set the speed of B to a common multiple:
- Let Speed of B = 190 (common multiple for both ratios).
- From the ratio of A and B (20:19), we can find Speed of A:
- Speed of A = (20/19) * Speed of B = (20/19) * 190 = 200.
- From the ratio of B and C (10:9), we can find Speed of C:
- Speed of C = (9/10) * Speed of B = (9/10) * 190 = 171.

### Step 6: Find the ratio of speeds of A and C
Now we have:
- Speed of A = 200
- Speed of C = 171

### Step 7: Calculate the margin by which A beats C in a 200 m race
In a race of 200 m, we need to find out how far C runs when A runs 200 m:
- The time taken by A to run 200 m = Distance / Speed = 200 / 200 = 1 unit of time.
- In this time, C will cover:
- Distance covered by C = Speed of C * Time = 171 * 1 = 171 m.

### Step 8: Calculate the margin of victory
The margin by which A beats C is:
- Margin = Distance covered by A - Distance covered by C = 200 m - 171 m = 29 m.

### Final Answer
A beats C by **29 meters**.

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