In a race of 1km A beats B by 100m in a race of 300m B beats C by 50m. Then in a race of 1 km by what margin will A beat C?

Answer

To solve the problem, we need to determine by how much margin A beats C in a 1 km race, given the information about A beating B and B beating C.

### Step-by-Step Solution:

1. **Understanding A's Race Against B**:
- In a 1 km (1000 m) race, A beats B by 100 m.
- This means when A finishes 1000 m, B has only covered 900 m.
- Therefore, the ratio of A's speed to B's speed can be expressed as:
\[ \frac{A}{B} = \frac{1000}{900} = \frac{10}{9} \]

2. **Understanding B's Race Against C**:
- In a 300 m race, B beats C by 50 m.
- This means when B finishes 300 m, C has only covered 250 m.
- Therefore, the ratio of B's speed to C's speed can be expressed as:
\[ \frac{B}{C} = \frac{300}{250} = \frac{6}{5} \]

3. **Finding A's Speed Relative to C**:
- We can find the ratio of A's speed to C's speed by multiplying the two ratios we found:
\[ \frac{A}{C} = \frac{A}{B} \times \frac{B}{C} = \frac{10}{9} \times \frac{6}{5} \]
- Simplifying this:
\[ \frac{A}{C} = \frac{10 \times 6}{9 \times 5} = \frac{60}{45} = \frac{4}{3} \]

4. **Calculating the Distance Covered by C When A Covers 1000 m**:
- If A covers 1000 m, we can find out how much C covers using the ratio:
\[ \text{Distance covered by C} = \frac{3}{4} \times 1000 = 750 \text{ m} \]

5. **Finding the Margin by Which A Beats C**:
- The margin by which A beats C is the difference between the distance A covers and the distance C covers:
\[ \text{Margin} = 1000 \text{ m} - 750 \text{ m} = 250 \text{ m} \]

### Final Answer:
A beats C by **250 meters** in a 1 km race.

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