Two town A and B are connected by a regular bus service with a bus leaving in either direction every T minutes. A man leaving in either direction every in the direction A to B notices that a bus goes past him every 18 min in the direction of his motion, and every 6 min in the opposite direction. The period T of the bus service is

Options

A. 4.5 min
B. 9 min
C. 1.2 min
D. 24 min

Explanation

To solve the problem, we need to analyze the situation involving the buses and the man traveling between towns A and B. Let's break it down step by step.

### Step 1: Define Variables
Let:
- \( V_m \) = speed of the man (in km/min)
- \( V_b \) = speed of the bus (in km/min)
- \( T \) = time interval between buses (in minutes)

### Step 2: Understand the Given Information
The man notices:
- A bus passes him every 18 minutes in the direction of his motion (from A to B).
- A bus passes him every 6 minutes in the opposite direction (from B to A).

### Step 3: Set Up Equations
When the man is moving from A to B:
- The relative speed of the bus with respect to the man is \( V_b - V_m \).
- The time taken for the bus to pass him is 18 minutes, so:
\[ \frac{T}{V_b - V_m} = 18 \quad \text{(1)} \]

When the man is moving from B to A:
- The relative speed of the bus with respect to the man is \( V_b + V_m \).
- The time taken for the bus to pass him is 6 minutes, so:
\[ \frac{T}{V_b + V_m} = 6 \quad \text{(2)} \]

### Step 4: Rearranging the Equations
From equation (1):
\[ T = 18(V_b - V_m) \quad \text{(3)} \]

From equation (2):
\[ T = 6(V_b + V_m) \quad \text{(4)} \]

### Step 5: Equate the Two Expressions for T
Setting equations (3) and (4) equal to each other:
\[ 18(V_b - V_m) = 6(V_b + V_m) \]

### Step 6: Simplify the Equation
Expanding both sides:
\[ 18V_b - 18V_m = 6V_b + 6V_m \]
Rearranging gives:
\[ 18V_b - 6V_b = 18V_m + 6V_m \]
\[ 12V_b = 24V_m \]
\[ V_b = 2V_m \quad \text{(5)} \]

### Step 7: Substitute V_b Back into One of the Equations
Substituting equation (5) into equation (3):
\[ T = 18(2V_m - V_m) = 18V_m \]

### Step 8: Find the Value of T
If we assume the speed of the man \( V_m \) is a known value (for example, 1 km/min), we can calculate \( T \):
\[ T = 18 \times 1 = 18 \text{ minutes} \]

However, we need to ensure the value of \( V_m \) is consistent with the other information provided.

### Step 9: Final Calculation
If we assume \( V_m = 20 \) km/h, we convert it to km/min:
\[ V_m = \frac{20}{60} = \frac{1}{3} \text{ km/min} \]
Then substituting back:
\[ T = 18 \times \frac{1}{3} = 6 \text{ minutes} \]

### Conclusion
The period \( T \) of the bus service is **6 minutes**.