A can run 100 m in `15 1/3` and B in 16 seconds. If B receives 4 metres start, who wins and by what distance ?

Options

A. A wins by 1/6 m
B. B wins by 1/6 m
C. A wins by 6 m
D. B wins by 8 m

Answer

To determine who wins the race between A and B, we need to analyze their speeds and the effect of the 4-meter head start given to B.

### Step 1: Calculate A's speed
A can run 100 meters in \(15 \frac{1}{3}\) seconds.
First, convert \(15 \frac{1}{3}\) seconds into an improper fraction:
\[ 15 \frac{1}{3} = \frac{46}{3} \text{ seconds} \]
Now, calculate A's speed:
\[ \text{Speed of A} = \frac{\text{Distance}}{\text{Time}} = \frac{100 \text{ m}}{\frac{46}{3} \text{ s}} = \frac{100 \times 3}{46} = \frac{300}{46} \approx 6.52 \text{ m/s} \]

### Step 2: Calculate B's speed
B can run 100 meters in 16 seconds.
\[ \text{Speed of B} = \frac{100 \text{ m}}{16 \text{ s}} = 6.25 \text{ m/s} \]

### Step 3: Determine the effective distance B runs
Since B receives a 4-meter head start, the distance B runs in the race is:
\[ \text{Distance B runs} = 100 \text{ m} - 4 \text{ m} = 96 \text{ m} \]

### Step 4: Calculate the time taken by B to run 96 meters
Using B's speed:
\[ \text{Time taken by B} = \frac{\text{Distance}}{\text{Speed}} = \frac{96 \text{ m}}{6.25 \text{ m/s}} = 15.36 \text{ seconds} \]

### Step 5: Calculate the time taken by A to run 100 meters
Using A's speed:
\[ \text{Time taken by A} = \frac{100 \text{ m}}{6.52 \text{ m/s}} \approx 15.36 \text{ seconds} \]

### Step 6: Compare the times
Both A and B take approximately the same time to complete their respective distances. However, since B has a head start, we need to determine who finishes first.

### Step 7: Calculate the distance A covers in the time B finishes
Since B finishes in approximately 15.36 seconds, we can calculate how far A has run in that time:
\[ \text{Distance A runs in 15.36 seconds} = \text{Speed of A} \times \text{Time} = 6.52 \text{ m/s} \times 15.36 \text{ s} \approx 100 \text{ m} \]

### Conclusion
Since A finishes 100 meters in approximately 15.36 seconds and B finishes 96 meters in the same time, A wins the race.

### Step 8: Determine by how much A wins
To find out by how much A wins, we can calculate the distance A covers in the time it takes B to finish:
\[ \text{Distance A wins by} = \text{Distance A runs} - \text{Distance B runs} = 100 \text{ m} - 96 \text{ m} = 4 \text{ m} \]

Thus, A wins by 4 meters.