In a 500 m race, the ratio of speeds of two runners , P and Q is 3 : 5 .P has a start of 200 m. Who wins the race and by what distance does he win ?

Answer

To determine who wins the race between runners P and Q, we can follow these steps:

### Step 1: Understand the Problem
We know that:
- The race distance is 500 meters.
- The ratio of speeds of runners P and Q is 3:5.
- Runner P has a head start of 200 meters.

### Step 2: Define the Speeds
Let the speed of runner P be \(3x\) and the speed of runner Q be \(5x\), where \(x\) is a common factor.

### Step 3: Calculate the Distance Each Runner Needs to Cover
Since P has a head start of 200 meters, he only needs to cover:
\[ 500 \text{ m} - 200 \text{ m} = 300 \text{ m} \]
Runner Q needs to cover the full 500 meters.

### Step 4: Calculate the Time Taken by Each Runner
- **Time taken by P** to cover 300 meters:
\[ \text{Time}_P = \frac{\text{Distance}}{\text{Speed}} = \frac{300}{3x} = \frac{100}{x} \]

- **Time taken by Q** to cover 500 meters:
\[ \text{Time}_Q = \frac{500}{5x} = \frac{100}{x} \]

### Step 5: Compare the Times
From the calculations:
- Time taken by P = \(\frac{100}{x}\)
- Time taken by Q = \(\frac{100}{x}\)

Both runners take the same amount of time to finish the race.

### Step 6: Conclusion
Since both runners finish at the same time, neither wins the race. They finish together.

### Final Answer
**Neither runner wins; they finish at the same time.**

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