If a train maintains an average speed of 42km per hour, it arrives at its destination at the right time. If however, the average speed is 40km per hour, it arrives 15 minutes late. Find the length of the journey?

Options

A. 210km
B. 205km
C. 209km
D. 200km

Answer

To solve the problem, we will follow these steps:

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

### Step 2: Calculate Time Taken at Different Speeds
1. When the train travels at 42 km/h, it arrives on time.
- The time taken at this speed is given by the formula:
\[ \text{Time} = \frac{D}{42} \]

2. When the train travels at 40 km/h, it arrives 15 minutes late.
- Since 15 minutes is \( \frac{15}{60} = \frac{1}{4} \) hours, the time taken at this speed is:
\[ \text{Time} = \frac{D}{40} \]
- This time is equal to the on-time journey time plus 15 minutes:
\[ \frac{D}{40} = \frac{D}{42} + \frac{1}{4} \]

### Step 3: Set Up the Equation
Now we can set up the equation based on the above relationships:
\[ \frac{D}{40} - \frac{D}{42} = \frac{1}{4} \]

### Step 4: Find a Common Denominator
To solve this equation, we first find a common denominator for the fractions on the left side. The least common multiple of 40 and 42 is 840. We rewrite the equation:
\[ \frac{21D}{840} - \frac{20D}{840} = \frac{1}{4} \]
This simplifies to:
\[ \frac{D}{840} = \frac{1}{4} \]

### Step 5: Solve for \( D \)
Now we can solve for \( D \):
\[ D = 840 \times \frac{1}{4} = 210 \]

### Step 6: Conclusion
Thus, the length of the journey is \( D = 210 \) kilometers.