In a 250 m race, the ratio of speeds of A and B is 1 : 2. If A has a start of 140 m, then A wins by how many metres?

Options

A. (A) 45 m
B. (B) 15 m
C. (C) 25 m
D. (D) 30 m

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Understand the given data
- The total distance of the race is 250 meters.
- The ratio of speeds of A and B is 1:2.
- A has a head start of 140 meters.

### Step 2: Calculate the distance A has to run
Since A has a head start of 140 meters, the distance A has to run is:
\[ \text{Distance A has to run} = 250 \text{ m} - 140 \text{ m} = 110 \text{ m} \]

### Step 3: Determine the effective distances for A and B
- A runs 110 meters.
- B runs the full distance of 250 meters.

### Step 4: Set up the ratio of speeds
Let the speed of A be \( v \) and the speed of B be \( 2v \) (since the ratio is 1:2). The time taken to complete the race can be expressed as:
\[ \text{Time taken by A} = \frac{\text{Distance}}{\text{Speed}} = \frac{110}{v} \]
\[ \text{Time taken by B} = \frac{250}{2v} = \frac{125}{v} \]

### Step 5: Compare the times
Since both A and B run their respective distances simultaneously, we can compare the times:
- Time taken by A: \( \frac{110}{v} \)
- Time taken by B: \( \frac{125}{v} \)

### Step 6: Determine who wins
From the above, we can see that:
- A takes \( \frac{110}{v} \) time to finish the race.
- B takes \( \frac{125}{v} \) time to finish the race.

Since \( \frac{110}{v} < \frac{125}{v} \), A finishes the race before B.

### Step 7: Calculate the distance B runs when A finishes
To find out how far B has run when A finishes, we substitute the time taken by A into the distance formula for B:
\[ \text{Distance B runs} = \text{Speed of B} \times \text{Time taken by A} = 2v \times \frac{110}{v} = 220 \text{ m} \]

### Step 8: Calculate the distance by which A wins
Now, we can find out how many meters A wins by:
\[ \text{Distance A wins by} = \text{Total distance} - \text{Distance B runs} = 250 \text{ m} - 220 \text{ m} = 30 \text{ m} \]

### Conclusion
A wins the race by 30 meters.

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