Travelling at a speed of 54 km/h, Chaman reaches a place 3 minutes early. If he had travelled at a speed of 48 km/h, he would have been 1 minute late. What is the distance Chaman had to travel?

Options

A. 25.8 km
B. 31.5 km
C. 28.0 km
D. 28.8 km

Answer

To solve the problem, we need to find the distance that Chaman had to travel based on the given speeds and time differences. Let's break it down step by step.

### Step-by-Step Solution:

1. **Define Variables:**
Let the actual time Chaman needs to travel be \( t \) hours.

2. **Calculate Time at 54 km/h:**
If Chaman travels at 54 km/h, he reaches 3 minutes early. Therefore, the time taken at this speed is:
\[ t - \frac{3}{60} = t - \frac{1}{20} \text{ hours} \]

3. **Distance at 54 km/h:**
The distance can be calculated using the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
Thus, the distance when traveling at 54 km/h is:
\[ D = 54 \left(t - \frac{1}{20}\right) \]

4. **Calculate Time at 48 km/h:**
If he travels at 48 km/h, he is 1 minute late. Therefore, the time taken at this speed is:
\[ t + \frac{1}{60} \text{ hours} \]

5. **Distance at 48 km/h:**
Similarly, the distance when traveling at 48 km/h is:
\[ D = 48 \left(t + \frac{1}{60}\right) \]

6. **Set the Distances Equal:**
Since both expressions represent the same distance \( D \), we can set them equal to each other:
\[ 54 \left(t - \frac{1}{20}\right) = 48 \left(t + \frac{1}{60}\right) \]

7. **Expand Both Sides:**
Expanding both sides gives:
\[ 54t - \frac{54}{20} = 48t + \frac{48}{60} \]
Simplifying the fractions:
\[ 54t - 2.7 = 48t + 0.8 \]

8. **Rearranging the Equation:**
Rearranging the equation to isolate \( t \):
\[ 54t - 48t = 2.7 + 0.8 \]
\[ 6t = 3.5 \]
\[ t = \frac{3.5}{6} = \frac{7}{12} \text{ hours} \]

9. **Calculate the Distance:**
Now, we can find the distance using either speed. Let's use the speed of 54 km/h:
\[ D = 54 \left(t - \frac{1}{20}\right) = 54 \left(\frac{7}{12} - \frac{1}{20}\right) \]
To subtract the fractions, we need a common denominator (60):
\[ D = 54 \left(\frac{35}{60} - \frac{3}{60}\right) = 54 \left(\frac{32}{60}\right) = 54 \times \frac{8}{15} \]
\[ D = \frac{432}{15} = 28.8 \text{ km} \]

### Final Answer:
The distance Chaman had to travel is **28.8 km**.