Two boys, A and B run at `4 1/2` and 6 km an hour respectively. A having 190 metres start, who wins and by how much the course being 1 km ?

Options

A. B wins by 60 m
B. A wins by 60 m
C. A wins by 80 m
D. B wins by 80 m

Answer

To solve the problem, we need to determine who wins the race between boys A and B, considering A's head start of 190 meters and the total distance of the race being 1 km (1000 meters).

### Step-by-Step Solution:

1. **Convert A's Speed to m/s**:
- A's speed is given as \(4 \frac{1}{2}\) km/h.
- Convert this to an improper fraction:
\[ 4 \frac{1}{2} = \frac{9}{2} \text{ km/h} \]
- To convert km/h to m/s, multiply by \(\frac{5}{18}\):
\[ \text{Speed of A} = \frac{9}{2} \times \frac{5}{18} = \frac{45}{36} = \frac{5}{4} \text{ m/s} \]

2. **Convert B's Speed to m/s**:
- B's speed is 6 km/h.
- Convert this to m/s:
\[ \text{Speed of B} = 6 \times \frac{5}{18} = \frac{30}{18} = \frac{5}{3} \text{ m/s} \]

3. **Calculate the Distance B Needs to Run**:
- Since A has a 190 meters head start, the distance B needs to cover is:
\[ \text{Distance B needs to run} = 1000 - 190 = 810 \text{ meters} \]

4. **Calculate the Time Taken by B**:
- Time taken by B to cover 810 meters:
\[ \text{Time taken by B} = \frac{\text{Distance}}{\text{Speed}} = \frac{810}{\frac{5}{3}} = 810 \times \frac{3}{5} = 486 \text{ seconds} \]

5. **Calculate the Distance A Covers in the Same Time**:
- In 486 seconds, the distance A covers:
\[ \text{Distance covered by A} = \text{Speed of A} \times \text{Time} = \frac{5}{4} \times 486 = 607.5 \text{ meters} \]

6. **Determine the Remaining Distance A Needs to Cover**:
- Since A started at 190 meters, the total distance A would have covered is:
\[ \text{Total distance A covered} = 190 + 607.5 = 797.5 \text{ meters} \]

7. **Determine Who Wins**:
- B covers 810 meters while A covers 797.5 meters.
- Therefore, B wins the race.

8. **Calculate the Margin of Victory**:
- The difference in distance covered:
\[ \text{Distance B wins by} = 810 - 797.5 = 12.5 \text{ meters} \]

### Conclusion:
B wins the race by 12.5 meters.