Two buses travel to a place at 45 km/hr and 60 km/hr respectively. If the second bus takes `5(1)/(2)` hrs less than the first for the journey, the length of the journey is

Options

A. 900 km
B. 945 km
C. 990 km
D. 1350 km

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Define Variables
Let the length of the journey be \( x \) kilometers.

### Step 2: Calculate Time Taken by Each Bus
- **Time taken by the first bus** (speed = 45 km/hr):
\[ \text{Time}_1 = \frac{x}{45} \text{ hours} \]

- **Time taken by the second bus** (speed = 60 km/hr):
\[ \text{Time}_2 = \frac{x}{60} \text{ hours} \]

### Step 3: Set Up the Equation
According to the problem, the second bus takes \( 5\frac{1}{2} \) hours less than the first bus. We can express this as:
\[ \text{Time}_1 - \text{Time}_2 = 5.5 \]
Substituting the expressions for time:
\[ \frac{x}{45} - \frac{x}{60} = 5.5 \]

### Step 4: Find a Common Denominator
The least common multiple (LCM) of 45 and 60 is 180. We will rewrite the equation using this common denominator:
\[ \frac{4x}{180} - \frac{3x}{180} = 5.5 \]
This simplifies to:
\[ \frac{(4x - 3x)}{180} = 5.5 \]
\[ \frac{x}{180} = 5.5 \]

### Step 5: Solve for \( x \)
To eliminate the fraction, multiply both sides by 180:
\[ x = 5.5 \times 180 \]
Calculating the right side:
\[ x = 990 \]

### Step 6: Conclusion
The length of the journey is \( 990 \) kilometers.

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