An athlete runs to and fro between points A and B at a speed of 10 km/h. A second athlete simultaneously runs from point B to A and back at a speed of 15 km/h. If they cross each other 12 min after the start, after how much time will they cross each other?

Options

A. 18 min
B. 24 min
C. 36 min
D. 48 min

Answer

To solve the problem step by step, we will analyze the situation of the two athletes running towards each other and determine when they will cross each other again after their initial meeting.

### Step 1: Convert time from minutes to hours
The athletes cross each other after 12 minutes. We need to convert this time into hours since the speeds are given in km/h.

\[ \text{Time in hours} = \frac{12 \text{ minutes}}{60} = \frac{1}{5} \text{ hours} \]

**Hint:** Remember that to convert minutes to hours, divide by 60.

### Step 2: Calculate the distance covered until they meet
The first athlete runs at a speed of 10 km/h and the second athlete runs at a speed of 15 km/h. The total speed when they are moving towards each other is the sum of their speeds.

\[ \text{Total speed} = 10 \text{ km/h} + 15 \text{ km/h} = 25 \text{ km/h} \]

Now, we can calculate the distance they cover together in the time until they meet.

\[ \text{Distance} = \text{Speed} \times \text{Time} = 25 \text{ km/h} \times \frac{1}{5} \text{ hours} = 5 \text{ km} \]

**Hint:** When two objects move towards each other, their speeds add up to find the relative speed.

### Step 3: Determine the distances each athlete has run
In the time it takes for them to meet:
- The first athlete (from A to B) covers:
\[ \text{Distance}_{A} = \text{Speed}_{A} \times \text{Time} = 10 \text{ km/h} \times \frac{1}{5} \text{ hours} = 2 \text{ km} \]
- The second athlete (from B to A) covers:
\[ \text{Distance}_{B} = \text{Speed}_{B} \times \text{Time} = 15 \text{ km/h} \times \frac{1}{5} \text{ hours} = 3 \text{ km} \]

**Hint:** Use the formula for distance to find how far each athlete has run in the time before they meet.

### Step 4: Calculate the remaining distance to be covered
After they meet, the first athlete has 3 km left to reach point B, and the second athlete has 2 km left to reach point A. After reaching their respective points, they will turn around and head back towards each other.

**Hint:** After meeting, consider how far each athlete has to run to reach their destination before turning back.

### Step 5: Calculate the total distance they will cover until they meet again
The total distance they will cover after meeting is:
\[ \text{Total distance} = 3 \text{ km} + 2 \text{ km} + 5 \text{ km} = 10 \text{ km} \]

**Hint:** Add the distances they need to cover after meeting to find the total distance until they meet again.

### Step 6: Calculate the time taken to meet again
Now, we need to find the time taken to cover this total distance at their combined speed.

\[ \text{Time} = \frac{\text{Total distance}}{\text{Total speed}} = \frac{10 \text{ km}}{25 \text{ km/h}} = \frac{2}{5} \text{ hours} \]

To convert this back to minutes:
\[ \frac{2}{5} \text{ hours} \times 60 = 24 \text{ minutes} \]

**Hint:** Use the formula for time and convert hours back to minutes for the final answer.

### Final Answer
The athletes will cross each other again after **24 minutes**.