Two runner start running together for a certain distance, one at 8 km/h and another at 5 km/h. The former arrives one and half an hour, before the latter. The distance (in km) is:

Options

A. 12
B. 20
C. 25
D. 36

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the problem
We have two runners:
- Runner 1 (R1) runs at a speed of 8 km/h.
- Runner 2 (R2) runs at a speed of 5 km/h.
They both start together but R1 finishes 1.5 hours before R2.

### Step 2: Define the variables
Let the time taken by Runner 2 to complete the distance be T hours. Therefore, the time taken by Runner 1 will be \( T - 1.5 \) hours.

### Step 3: Write the distance equations
Since both runners cover the same distance (D), we can express the distance in terms of speed and time:
- For Runner 1:
\[ D = \text{Speed of R1} \times \text{Time of R1} = 8 \times (T - 1.5) \]
- For Runner 2:
\[ D = \text{Speed of R2} \times \text{Time of R2} = 5 \times T \]

### Step 4: Set the equations equal to each other
Since both expressions equal the same distance D, we can set them equal:
\[ 8(T - 1.5) = 5T \]

### Step 5: Solve for T
Expanding the left side:
\[ 8T - 12 = 5T \]
Now, isolate T:
\[ 8T - 5T = 12 \]
\[ 3T = 12 \]
\[ T = 4 \text{ hours} \]

### Step 6: Find the time taken by Runner 1
Now, we can find the time taken by Runner 1:
\[ \text{Time of R1} = T - 1.5 = 4 - 1.5 = 2.5 \text{ hours} \]

### Step 7: Calculate the distance
Now we can calculate the distance using either runner's speed and time. We'll use Runner 2:
\[ D = \text{Speed of R2} \times \text{Time of R2} = 5 \times 4 = 20 \text{ km} \]

### Conclusion
The distance covered by both runners is **20 km**.

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