In a 500 metre race, the ratio of speeds of two runners P and Q is 3 : 5. P has a start of 200 metre then the distance between P and Q at the finish of the race is

Options

A. P wins by 100 metre
B. Both reach at the same time
C. Q wins by 100 metre
D. Q wins by 50 metre

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the problem
We have two runners, P and Q, in a 500-meter race. The ratio of their speeds is given as 3:5, and runner P has a head start of 200 meters.

### Step 2: Determine the effective distances each runner has to cover
- Runner P starts 200 meters ahead, so he only needs to run:
\[ 500 - 200 = 300 \text{ meters} \]
- Runner Q starts from the starting line and needs to run the full distance of:
\[ 500 \text{ meters} \]

### Step 3: Set the speeds of the runners based on the given ratio
Let the speed of runner P be \(3x\) and the speed of runner Q be \(5x\), where \(x\) is a common factor.

### Step 4: Calculate the time taken by each runner to finish the race
- Time taken by runner P to finish 300 meters:
\[ \text{Time}_P = \frac{\text{Distance}}{\text{Speed}} = \frac{300}{3x} = \frac{100}{x} \]

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

### Step 5: Compare the times
Since both runners finish the race at the same time, we can conclude that they both take the same amount of time, which is \(\frac{100}{x}\).

### Step 6: Determine the distance between P and Q at the finish line
At the finish line, runner P has completed 300 meters, while runner Q has completed 500 meters. To find the distance between them:
- Distance between P and Q at the finish:
\[ \text{Distance} = \text{Distance of Q} - \text{Distance of P} = 500 - 300 = 200 \text{ meters} \]

### Final Answer
The distance between P and Q at the finish of the race is **200 meters**.

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