When Gopal walks from A to B and returns from B to A on cycle, then it takes him 46 minutes. By walking both ways, it takes him 75 minutes. If he rides cycle both ways then what will be the total time (in minutes)?

Options

A. 19
B. 17
C. 23
D. 15

Answer

To solve the problem step by step, let's define the variables and analyze the information given:

1. **Define Variables:**
- Let the distance between A and B be \( d \).
- Let the speed of walking be \( x \) (in meters per minute).
- Let the speed of cycling be \( y \) (in meters per minute).

2. **Time Taken for Walking and Cycling:**
- Time taken to walk from A to B is \( \frac{d}{x} \).
- Time taken to cycle from B to A is \( \frac{d}{y} \).

3. **Total Time for Walking One Way and Cycling Back:**
- According to the problem, when Gopal walks from A to B and returns from B to A on a cycle, the total time taken is 46 minutes:
\[ \frac{d}{x} + \frac{d}{y} = 46 \quad \text{(1)} \]

4. **Total Time for Walking Both Ways:**
- When Gopal walks both ways, the total time taken is 75 minutes:
\[ \frac{d}{x} + \frac{d}{x} = 75 \implies \frac{2d}{x} = 75 \implies \frac{d}{x} = 37.5 \quad \text{(2)} \]

5. **Substituting Equation (2) into Equation (1):**
- From equation (2), we know \( \frac{d}{x} = 37.5 \). Substitute this into equation (1):
\[ 37.5 + \frac{d}{y} = 46 \]
- Rearranging gives:
\[ \frac{d}{y} = 46 - 37.5 = 8.5 \quad \text{(3)} \]

6. **Calculating Time Taken for Cycling Both Ways:**
- If Gopal cycles both ways, the total time taken would be:
\[ \frac{d}{y} + \frac{d}{y} = 2 \cdot \frac{d}{y} \]
- From equation (3), we know \( \frac{d}{y} = 8.5 \):
\[ 2 \cdot \frac{d}{y} = 2 \cdot 8.5 = 17 \text{ minutes} \]

7. **Final Answer:**
- Therefore, the total time taken if Gopal cycles both ways is **17 minutes**.