In a race of 50 m, A wins over B by 10 m and A wins over C by 14 m. In the same race by how many metres does B win over C ?

Answer

To solve the problem step by step, we can break it down as follows:

### Step 1: Understand the race scenario
In a 50 m race:
- A wins over B by 10 m, meaning when A finishes the race (50 m), B has only run 40 m.
- A wins over C by 14 m, meaning when A finishes the race (50 m), C has only run 36 m.

### Step 2: Set up the distances covered
- When A finishes 50 m, B covers 40 m.
- When A finishes 50 m, C covers 36 m.

### Step 3: Establish the ratios of distances
- The ratio of distances covered by A and B is:
\[ \text{Distance ratio (A:B)} = 50:40 = 5:4 \]
- The ratio of distances covered by A and C is:
\[ \text{Distance ratio (A:C)} = 50:36 = 25:18 \]

### Step 4: Find the ratio of B and C
Since A is common in both ratios, we can express the ratios of B and C in terms of A:
- From the first ratio, we have:
\[ A:B = 5:4 \quad \text{(B = 4 units)} \]
- From the second ratio, we have:
\[ A:C = 25:18 \quad \text{(C = 18 units)} \]

### Step 5: Express B in terms of C
To find the ratio of B to C, we can set up the ratios:
- From \(A:B = 5:4\) and \(A:C = 25:18\), we can equate the two ratios:
\[ \frac{B}{C} = \frac{4}{x} \quad \text{where } x \text{ is the unit for C} \]

### Step 6: Calculate the ratio of B to C
To find the relationship between B and C:
- We can express B in terms of C:
\[ 4 \cdot 18 = 5 \cdot x \implies 72 = 5x \implies x = \frac{72}{5} = 14.4 \]
- Therefore, the ratio of B to C is:
\[ B:C = 4:3.6 \quad \text{(which simplifies to } 10:9\text{)} \]

### Step 7: Determine how much B wins over C
If B runs 50 m, we can find how far C runs:
- If B runs 50 m, then C runs:
\[ C \text{ distance} = \frac{9}{10} \times 50 = 45 \text{ m} \]
- Thus, the distance by which B wins over C is:
\[ \text{Distance B wins over C} = 50 - 45 = 5 \text{ m} \]

### Final Answer
B wins over C by **5 meters**.

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