Let's break the problem down step by step.

Information Given:

• A beats B by 200 meters in a 1000-meter race.
• B beats C by 200 meters in a 1000-meter race.

This means:

• When A runs 1000 meters, B runs only 800 meters (since A beats B by 200 meters).
• Similarly, when B runs 1000 meters, C runs only 800 meters (since B beats C by 200 meters).

Step 1: Find the time taken by A, B, and C to finish the race.

Let’s assume that A, B, and C run the race at constant speeds, and let $t_A$, $t_B$, and $t_C$ represent the times taken by A, B, and C to complete the 1000-meter race.

1. A's speed:
   Speed of A is $v_A = \frac{1000}{t_A}$.

2. B's speed:
   Since B runs 800 meters when A runs 1000 meters, we can express B’s speed relative to A’s speed: $v_B = \frac{800}{1000} \cdot v_A = 0.8 \cdot v_A$ The time taken by B to complete 1000 meters is: $t_B = \frac{1000}{v_B} = \frac{1000}{0.8 \cdot v_A} = \frac{1250}{v_A}$

3. C's speed:
   Similarly, C runs 800 meters when B runs 1000 meters, so C’s speed relative to B’s speed is: $v_C = \frac{800}{1000} \cdot v_B = 0.8 \cdot v_B = 0.8 \cdot 0.8 \cdot v_A = 0.64 \cdot v_A$ The time taken by C to complete 1000 meters is: $t_C = \frac{1000}{v_C} = \frac{1000}{0.64 \cdot v_A} = \frac{1562.5}{v_A}$

Step 2: Calculate the time differences.

• Time by which A beats B: $\text{Time difference} = t_B - t_A = \frac{1250}{v_A} - \frac{1000}{v_A} = \frac{250}{v_A}$

• Time by which A beats C: $\text{Time difference} = t_C - t_A = \frac{1562.5}{v_A} - \frac{1000}{v_A} = \frac{562.5}{v_A}$

Step 3: Find the ratio.

Now, we find the ratio of the time by which A beats C to the time by which A beats B: $\text{Ratio} = \frac{\frac{562.5}{v_A}}{\frac{250}{v_A}} = \frac{562.5}{250} = 2.25$

Final Answer:

The ratio of the number of seconds by which A beats C to the number of seconds by which A beats B is 2.25.

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Would you like further clarification or have any questions?

Here are some related questions for you to consider:

1. How would the ratio change if A beat B by 100 meters instead of 200 meters?
2. If the race distance was doubled to 2000 meters, how would the times for A, B, and C change?
3. How do you calculate the relative speeds in a race where one competitor beats another by a specific distance?
4. What is the significance of a ratio greater than 1 in a race context?
5. How can you use the concept of relative speeds to solve similar race problems?

Tip: When working with race problems involving relative distances, it's helpful to express all competitors' speeds in terms of one competitor's speed. This simplifies the calculation of time differences.