A bus travels half of the distance with a speed of 48 km/hr and remaining half of the distance with a speed of 72 km/hr and reached its destination in 2.5 hrs. How much distance did the bus travel to reach its destination?

Options

A. 160 km
B. 144 km
C. 120 km
D. 121 km

Answer

To solve the problem, we need to find the total distance that the bus traveled. Let's break it down step by step.

### Step 1: Define the total distance
Let the total distance traveled by the bus be \( D \). According to the problem, the bus travels half of this distance at one speed and the other half at a different speed. Therefore, we can express the distance as:
\[ D = 2x \]
where \( x \) is half of the total distance.

### Step 2: Calculate the time taken for each half of the journey
The bus travels the first half of the distance \( x \) at a speed of 48 km/hr. The time taken to cover this distance can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
So, the time taken for the first half is:
\[ t_1 = \frac{x}{48} \]

For the second half of the distance \( x \), the bus travels at a speed of 72 km/hr. The time taken for this half is:
\[ t_2 = \frac{x}{72} \]

### Step 3: Set up the equation for total time
According to the problem, the total time taken for the journey is 2.5 hours. Therefore, we can set up the equation:
\[ t_1 + t_2 = 2.5 \]
Substituting the expressions for \( t_1 \) and \( t_2 \):
\[ \frac{x}{48} + \frac{x}{72} = 2.5 \]

### Step 4: Find a common denominator and solve for \( x \)
The least common multiple of 48 and 72 is 144. We can rewrite the equation with a common denominator:
\[ \frac{3x}{144} + \frac{2x}{144} = 2.5 \]
Combining the fractions gives:
\[ \frac{5x}{144} = 2.5 \]

To eliminate the fraction, multiply both sides by 144:
\[ 5x = 2.5 \times 144 \]
Calculating the right side:
\[ 5x = 360 \]
Now, divide both sides by 5 to solve for \( x \):
\[ x = 72 \]

### Step 5: Calculate the total distance \( D \)
Since \( D = 2x \):
\[ D = 2 \times 72 = 144 \text{ km} \]

### Final Answer
The total distance that the bus traveled to reach its destination is **144 km**.

---