A farmer travelled from his village to city to buy seeds. He went by bicycle at a speed of 20 km/hr and came back at a speed of 5 km/hr. If the farmer took 4 hours and 45 minutes to complete his journey, find the distance between village and city?

Options

A. 15 km
B. 19 km
C. 23 km
D. 25 km

Answer

To solve the problem step by step, we will first define the variables and then use the information given to set up equations.

### Step 1: Define the Variables
Let the distance between the village and the city be \( x \) kilometers.

### Step 2: Define the Speeds
The speed of the farmer going to the city is \( 20 \) km/hr, and the speed coming back is \( 5 \) km/hr.

### Step 3: Convert Total Time to Hours
The total time taken for the journey is \( 4 \) hours and \( 45 \) minutes. We need to convert this into hours:
\[ 4 \text{ hours } + 45 \text{ minutes } = 4 + \frac{45}{60} = 4 + 0.75 = 4.75 \text{ hours} \]
Thus, the total time in hours is \( 4.75 \) hours.

### Step 4: Calculate Time for Each Leg of the Journey
The time taken to go to the city (T1) and the time taken to return (T2) can be calculated using the formula:
\[ T = \frac{\text{Distance}}{\text{Speed}} \]
So, for the journey to the city:
\[ T1 = \frac{x}{20} \]
And for the return journey:
\[ T2 = \frac{x}{5} \]

### Step 5: Set Up the Equation for Total Time
The total time for the journey is the sum of the time going and the time returning:
\[ T1 + T2 = 4.75 \]
Substituting the expressions for \( T1 \) and \( T2 \):
\[ \frac{x}{20} + \frac{x}{5} = 4.75 \]

### Step 6: Find a Common Denominator
The common denominator for \( 20 \) and \( 5 \) is \( 20 \):
\[ \frac{x}{20} + \frac{4x}{20} = 4.75 \]
This simplifies to:
\[ \frac{5x}{20} = 4.75 \]

### Step 7: Simplify the Equation
Now, simplify the left side:
\[ \frac{x}{4} = 4.75 \]

### Step 8: Solve for \( x \)
Multiply both sides by \( 4 \) to solve for \( x \):
\[ x = 4.75 \times 4 = 19 \]

### Conclusion
The distance between the village and the city is \( 19 \) kilometers.