In a 500 m race, the ratio of the speeds of A and B is `3:4` . A has a start of 140 m A wins by : Options are (a) 10m (b) 20m (c) 30m (d) None of the above.

Options

A. 10 m
B. 20 m
C. 30 m
D. None of the above

Answer

To solve the problem step by step, we will analyze the given information and calculate how much A wins by in the race.

### Step 1: Understand the speed ratio
The ratio of the speeds of A and B is given as 3:4. This means that for every 3 units of distance A covers, B covers 4 units in the same time.

### Step 2: Determine the effective distance A runs
A has a head start of 140 meters in a 500-meter race. Therefore, the distance A actually runs is:
\[ \text{Distance A runs} = 500 \text{ m} - 140 \text{ m} = 360 \text{ m} \]

### Step 3: Calculate the distance B runs when A finishes
Since the speeds are in the ratio of 3:4, we can express the distances run by A and B in terms of a common factor. Let’s denote the distance A runs as \(3x\) and the distance B runs as \(4x\).

When A runs 360 meters:
\[ 3x = 360 \implies x = \frac{360}{3} = 120 \]
Now, we can find the distance B runs:
\[ 4x = 4 \times 120 = 480 \text{ m} \]

### Step 4: Determine how much distance B has left to run
In a 500-meter race, when A finishes running 500 meters, we need to find out how much distance B has left:
\[ \text{Distance B has left} = 500 \text{ m} - 480 \text{ m} = 20 \text{ m} \]

### Step 5: Conclusion
Since A finishes the race while B is still 20 meters away from the finish line, A wins by:
\[ \text{A wins by} = 20 \text{ m} \]

### Final Answer
A wins by 20 meters.