In a 500 metres race, B gives A a start of 160 metres. The ratio of the speeds of A and B is 2 : 3. Who wins and by how much ?

Options

A. `6 2/3` m
B. 8 m
C. `8 1/3` m
D. `6 1/3` m

Answer

To solve the problem, we need to determine who wins the race between A and B and by how much.

### Step-by-Step Solution:

1. **Understanding the Race Setup:**
- The total distance of the race is 500 meters.
- A starts 160 meters ahead, which means A only has to run 500 - 160 = 340 meters.
- B has to run the full 500 meters.

2. **Speed Ratio:**
- The ratio of the speeds of A and B is given as 2:3.
- Let's denote the speed of A as 2x and the speed of B as 3x.

3. **Time Taken to Complete the Race:**
- The time taken by A to complete 340 meters can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
- For A:
\[ \text{Time taken by A} = \frac{340}{2x} = \frac{170}{x} \]
- For B:
\[ \text{Time taken by B} = \frac{500}{3x} \]

4. **Comparing the Times:**
- We need to compare the times taken by A and B:
\[ \text{Time taken by A} = \frac{170}{x} \]
\[ \text{Time taken by B} = \frac{500}{3x} \]
- To compare, we can find a common denominator or simply cross-multiply:
\[ 170 \cdot 3x \quad \text{and} \quad 500 \cdot x \]
\[ 510x \quad \text{and} \quad 500x \]
- Since \(510x > 500x\), it follows that:
\[ \frac{170}{x} < \frac{500}{3x} \]
- This means A takes less time than B.

5. **Conclusion:**
- A wins the race because he takes less time to complete his distance.
- To find out by how much A wins, we can calculate the time difference:
\[ \text{Time difference} = \frac{500}{3x} - \frac{170}{x} \]
- Finding a common denominator (which is \(3x\)):
\[ = \frac{500 - 510}{3x} = \frac{-10}{3x} \]
- This indicates that A finishes the race before B by a certain time, but we need to convert this time difference into distance.

6. **Distance B Runs in Time A Finishes:**
- We can find out how far B runs in the time it takes A to finish:
\[ \text{Distance B runs in time A finishes} = \text{Speed of B} \times \text{Time taken by A} \]
\[ = 3x \times \frac{170}{x} = 510 \text{ meters} \]
- Since B has to run 500 meters, the distance by which A wins is:
\[ 510 - 500 = 10 \text{ meters} \]

### Final Answer:
A wins the race by 10 meters.