A beats B by 100 m in a race of 1200 m and B baets C by 200 m in a race of 1600 m Approximately by how many metres can A beat C in a race of 9600 m?

Options

A. 1600 m
B. 1800 m
C. 1900 m
D. 2400 m

Answer

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

### Step-by-step Solution:

1. **Understanding the Race Between A and B:**
- A beats B by 100 meters in a race of 1200 meters.
- This means when A finishes 1200 meters, B has run only 1100 meters.
- Therefore, the ratio of distances run by A and B is:
\[ \text{Distance ratio (A:B)} = \frac{1200}{1100} = \frac{12}{11} \]

2. **Understanding the Race Between B and C:**
- B beats C by 200 meters in a race of 1600 meters.
- This means when B finishes 1600 meters, C has run only 1400 meters.
- Therefore, the ratio of distances run by B and C is:
\[ \text{Distance ratio (B:C)} = \frac{1600}{1400} = \frac{16}{14} = \frac{8}{7} \]

3. **Finding the Ratio of A to C:**
- We know the ratios A:B and B:C. We can find A:C by multiplying these ratios:
\[ \text{Distance ratio (A:C)} = \text{Distance ratio (A:B)} \times \text{Distance ratio (B:C)} = \frac{12}{11} \times \frac{8}{7} \]
- Calculating this gives:
\[ \text{Distance ratio (A:C)} = \frac{12 \times 8}{11 \times 7} = \frac{96}{77} \]

4. **Calculating the Distances in a 9600 Meter Race:**
- In a race of 9600 meters, we need to find out how far C runs when A runs 9600 meters.
- Using the ratio A:C, we can set up the equation:
\[ A : C = 96 : 77 \]
- If A runs 9600 meters, we can find the distance C runs using the ratio:
\[ C = \frac{77}{96} \times 9600 \]
- Calculating this gives:
\[ C = \frac{77 \times 9600}{96} = 77 \times 100 = 7700 \text{ meters} \]

5. **Finding the Distance A beats C:**
- Now, to find how many meters A beats C in a 9600 meter race:
\[ \text{Distance A beats C} = 9600 - 7700 = 1900 \text{ meters} \]

### Final Answer:
A beats C by approximately **1900 meters** in a race of 9600 meters.