A can run one km in half a minute less time than B. In a kilometre race, B gets a start of 100 metres and loses by 100 metres. Find the time A and B take to run a kilometre—

Options

A. 5 min, `5 1/2` min
B. 2 min, `2 1/2` min
C. 3 min, `3 1/2` min
D. None of these

Answer

To solve the problem, let's denote the time taken by A to run 1 km as \( t_A \) minutes and the time taken by B to run 1 km as \( t_B \) minutes. According to the problem, we know that:

1. A can run 1 km in half a minute less time than B:
\[ t_A = t_B - 0.5 \]

2. In a 1 km race, B gets a start of 100 meters and loses by 100 meters. This means that when A finishes 1 km, B has run only 900 meters.

### Step 1: Express the distances in terms of time
The speed of A can be expressed as:
\[ \text{Speed of A} = \frac{1000 \text{ meters}}{t_A \text{ minutes}} = \frac{1000}{t_A} \]

The speed of B can be expressed as:
\[ \text{Speed of B} = \frac{1000 \text{ meters}}{t_B \text{ minutes}} = \frac{1000}{t_B} \]

### Step 2: Set up the equation based on the race
When A finishes 1 km (which takes \( t_A \) minutes), B has run 900 meters. Therefore, we can express the distance B has run in terms of his speed and the time taken by A:
\[ \text{Distance B runs} = \text{Speed of B} \times t_A = \frac{1000}{t_B} \times t_A \]
Setting this equal to 900 meters gives us:
\[ \frac{1000}{t_B} \times t_A = 900 \]

### Step 3: Substitute \( t_A \) in the equation
We can substitute \( t_A \) from the first equation into this equation:
\[ \frac{1000}{t_B} \times (t_B - 0.5) = 900 \]

### Step 4: Simplify the equation
Expanding this gives:
\[ 1000 - \frac{500}{t_B} = 900 \]
Rearranging gives:
\[ 100 - \frac{500}{t_B} = 0 \]
\[ \frac{500}{t_B} = 100 \]
Thus, we can solve for \( t_B \):
\[ t_B = \frac{500}{100} = 5 \text{ minutes} \]

### Step 5: Find \( t_A \)
Now we can find \( t_A \):
\[ t_A = t_B - 0.5 = 5 - 0.5 = 4.5 \text{ minutes} \]

### Final Answer
- Time taken by A to run 1 km: **4.5 minutes**
- Time taken by B to run 1 km: **5 minutes**