The distance between two towns is 20 km less by road than by rail. A train takes three hours for the journey, a car four hours. If the average speed of the car is 15 km/hr less than that of the train, what is the average speeds of the car ?

Options

A. 40 km/hr
B. 35 km/hr
C. 25 km/hr
D. 50 km/hr

Answer

To solve the problem step by step, we will define the variables and set up equations based on the information given in the question.

### Step 1: Define the Variables
Let:
- \( x \) = distance by car (in km)
- \( y \) = average speed of the car (in km/hr)

From the problem, we know:
- The distance by train = \( x + 20 \) km (since it is 20 km more than by car)
- The time taken by the car = 4 hours
- The time taken by the train = 3 hours
- The average speed of the train = \( y + 15 \) km/hr (since it is 15 km/hr more than the car)

### Step 2: Set Up the Equations
Using the formula for speed, which is \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \), we can set up the following equations:

1. For the car:
\[ \text{Speed of the car} = \frac{x}{4} = y \quad \text{(Equation 1)} \]

2. For the train:
\[ \text{Speed of the train} = \frac{x + 20}{3} = y + 15 \quad \text{(Equation 2)} \]

### Step 3: Solve Equation 1 for \( x \)
From Equation 1:
\[ x = 4y \]

### Step 4: Substitute \( x \) in Equation 2
Now substitute \( x = 4y \) into Equation 2:
\[ \frac{4y + 20}{3} = y + 15 \]

### Step 5: Clear the Fraction
Multiply both sides by 3 to eliminate the fraction:
\[ 4y + 20 = 3(y + 15) \]
\[ 4y + 20 = 3y + 45 \]

### Step 6: Rearrange the Equation
Now, rearranging the equation to isolate \( y \):
\[ 4y - 3y = 45 - 20 \]
\[ y = 25 \]

### Step 7: Conclusion
The average speed of the car is \( y = 25 \) km/hr.

### Step 8: Find the Distance \( x \)
Using \( y \) to find \( x \):
\[ x = 4y = 4 \times 25 = 100 \text{ km} \]

### Final Answer
The average speed of the car is **25 km/hr**.