In a race of 1200m, A can beat B by 120 m and in a race of 500 m, B can beat C by 100 m. By how many metres will A beat C in a race of 800 m?

Options

A. 216
B. 220
C. 224
D. 229

Answer

To solve the problem step by step, we will analyze the information given and derive the necessary ratios to find out how many meters A will beat C in a race of 800 meters.

### Step 1: Understand the race between A and B
In a race of 1200 meters, A beats B by 120 meters. This means when A finishes 1200 meters, B has only run 1080 meters (1200 - 120).

### Step 2: Calculate the speed ratio of A and B
The ratio of the distances traveled by A and B can be expressed as:
- Distance A : Distance B = 1200 : 1080
- Simplifying this gives us A : B = 10 : 9.

### Step 3: Understand the race between B and C
In a race of 500 meters, B beats C by 100 meters. This means when B finishes 500 meters, C has only run 400 meters (500 - 100).

### Step 4: Calculate the speed ratio of B and C
The ratio of the distances traveled by B and C can be expressed as:
- Distance B : Distance C = 500 : 400
- Simplifying this gives us B : C = 5 : 4.

### Step 5: Combine the ratios of A, B, and C
Now we need to combine the ratios of A : B and B : C. We will make the B values the same in both ratios:
- From A : B = 10 : 9, we can multiply both sides by 5 to get A : B = 50 : 45.
- From B : C = 5 : 4, we can multiply both sides by 9 to get B : C = 45 : 36.

Now we have:
- A : B = 50 : 45
- B : C = 45 : 36

### Step 6: Write the combined ratio of A, B, and C
Thus, the combined ratio of A, B, and C is:
- A : B : C = 50 : 45 : 36.

### Step 7: Determine how much A beats C
From the combined ratio, we can see that for every 50 meters A runs, C runs 36 meters. Therefore, A beats C by:
- Distance A beats C = Distance A - Distance C = 50 - 36 = 14 meters.

### Step 8: Calculate the distance A beats C in a race of 800 meters
Now, we need to find out how many meters A will beat C in a race of 800 meters. We can set up a proportion:
- If A beats C by 14 meters when A runs 50 meters, then in a race of 800 meters, A will beat C by:
\[ \text{Distance A beats C} = \left( \frac{14}{50} \right) \times 800. \]
Calculating this gives:
\[ \text{Distance A beats C} = \frac{14 \times 800}{50} = \frac{11200}{50} = 224 \text{ meters}. \]

### Final Answer
A will beat C by **224 meters** in a race of 800 meters.

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