Two cars A and B travel from one city to another city, at speeds of 60 km/hr and 108 km/hr respectively. If car B takes 2 hours lesser time than car A for the journey, then what is the distance (in km) between the two cities ?

Options

A. 240
B. 270
C. 300
D. 330

Answer

To solve the problem step by step, we will use the relationship between distance, speed, and time.

### Step-by-Step Solution:

1. **Define the Variables**:
Let the distance between the two cities be \( D \) km.
Let the time taken by car A be \( T_A \) hours.
Let the time taken by car B be \( T_B \) hours.

2. **Set Up the Speed Equations**:
The speed of car A is 60 km/hr, and the speed of car B is 108 km/hr.
Using the formula \( \text{Distance} = \text{Speed} \times \text{Time} \), we can write:
\[ D = 60 \times T_A \quad \text{(1)} \]
\[ D = 108 \times T_B \quad \text{(2)} \]

3. **Relate the Times**:
According to the problem, car B takes 2 hours less than car A:
\[ T_B = T_A - 2 \quad \text{(3)} \]

4. **Substitute Equation (3) into Equation (2)**:
Substitute \( T_B \) from equation (3) into equation (2):
\[ D = 108 \times (T_A - 2) \]

5. **Set the Two Distance Equations Equal**:
Since both equations represent the same distance \( D \), we can set them equal to each other:
\[ 60 \times T_A = 108 \times (T_A - 2) \]

6. **Expand and Rearrange the Equation**:
Expanding the right side gives:
\[ 60 T_A = 108 T_A - 216 \]
Rearranging this equation results in:
\[ 108 T_A - 60 T_A = 216 \]
\[ 48 T_A = 216 \]

7. **Solve for \( T_A \)**:
Dividing both sides by 48:
\[ T_A = \frac{216}{48} = 4.5 \text{ hours} \]

8. **Find \( T_B \)**:
Using equation (3) to find \( T_B \):
\[ T_B = T_A - 2 = 4.5 - 2 = 2.5 \text{ hours} \]

9. **Calculate the Distance \( D \)**:
Now, use either equation (1) or (2) to find the distance \( D \). Using equation (1):
\[ D = 60 \times T_A = 60 \times 4.5 = 270 \text{ km} \]

### Final Answer:
The distance between the two cities is **270 km**.