A and B are 25 km apart. If they travel in opposite directions, they meet after one hour. If they travel in the same direction, they meet after 5 hours. If A travels faster than B, then the speed of A is

Options

A. 10 km/hr
B. 12.5 km/hr
C. 15 km/hr
D. 20 km/hr

Answer

To solve the problem step by step, let's denote the speed of A as \( v_A \) and the speed of B as \( v_B \).

### Step 1: Understanding the scenario
A and B are 25 km apart. When they travel in opposite directions, they meet after 1 hour. This means the total distance covered by both A and B together in that hour is 25 km.

### Step 2: Setting up the equation for opposite directions
When A and B travel in opposite directions, their speeds add up. Therefore, we can write the equation:
\[ v_A + v_B = 25 \quad \text{(1)} \]

### Step 3: Setting up the equation for the same direction
When A and B travel in the same direction, they meet after 5 hours. This means that the distance covered by A relative to B in that time is also 25 km. The relative speed when they travel in the same direction is \( v_A - v_B \). Thus, we can write:
\[ (v_A - v_B) \times 5 = 25 \]
Dividing both sides by 5 gives:
\[ v_A - v_B = 5 \quad \text{(2)} \]

### Step 4: Solving the equations
Now we have a system of two equations:
1. \( v_A + v_B = 25 \)
2. \( v_A - v_B = 5 \)

We can solve these equations simultaneously.

### Step 5: Adding the equations
Adding equations (1) and (2):
\[ (v_A + v_B) + (v_A - v_B) = 25 + 5 \]
This simplifies to:
\[ 2v_A = 30 \]
Dividing both sides by 2 gives:
\[ v_A = 15 \quad \text{(3)} \]

### Step 6: Finding the speed of B
Now, we can substitute \( v_A \) back into equation (1) to find \( v_B \):
\[ 15 + v_B = 25 \]
Subtracting 15 from both sides gives:
\[ v_B = 10 \quad \text{(4)} \]

### Conclusion
Thus, the speed of A is \( 15 \) km/h.