Amit starts from a point $A$ and walks to another point $B$ and then returns from B to A by his car and thus takes a total time of 6 hours and 45 minutes. If he had driven both ways in his car, he would have taken 2 hours less. How long would it take for him to walk both ways?

### Concept & Linear Equations in Time Problems involving two modes of transport across the same distance can be represented as simple linear equations where variables represent the time taken for a one-way trip using a specific transport mode. Let $W$ be the time to walk one way, and $C$ be the time to drive one way. ### Step-by-Step Solution 1. **Set up the equations based on given data:** Walk + Car = 6 hours 45 minutes $W + C = 6 \text{ hr } 45 \text{ min}$ --- (Equation 1) 2. **Use the second condition:** If he drives both ways, he takes 2 hours less than the first scenario. $6 \text{ hr } 45 \text{ min} - 2 \text{ hr} = 4 \text{ hr } 45 \text{ min}$. Car + Car = 4 hours 45 minutes $2C = 4 \text{ hr } 45 \text{ min}$ --- (Equation 2) 3. **Solve for the target value:** We need the time to walk both ways, which is $2W$. Multiply Equation 1 by 2: $2(W + C) = 2 \times (6 \text{ hr } 45 \text{ min})$ $2W + 2C = 13 \text{ hours } 30 \text{ minutes}$ 4. **Substitute Equation 2 into the multiplied Equation 1:** $2W + 4 \text{ hr } 45 \text{ min} = 13 \text{ hr } 30 \text{ min}$ $2W = 13 \text{ hr } 30 \text{ min} - 4 \text{ hr } 45 \text{ min}$ $2W = 12 \text{ hr } 90 \text{ min} - 4 \text{ hr } 45 \text{ min} = 8 \text{ hr } 45 \text{ min}$. ### Exam Strategy & Shortcut Use logical deduction instead of strict algebra: Replacing one "Walk" trip with a "Car" trip saves exactly 2 hours (since driving both ways saves 2 hours over walking one way). Therefore, if we go the opposite direction and replace the remaining "Car" trip in the first scenario with a "Walk" trip, it will *add* exactly 2 hours to the time. $6 \text{ hr } 45 \text{ min} + 2 \text{ hr} = 8 \text{ hr } 45 \text{ min}$. This gives the answer in seconds! ### Common Pitfall Converting all times to minutes or fractions (like $6.75$ hours) before solving. This often complicates calculations unnecessarily. Keeping hours and minutes separate is usually cleaner for simple arithmetic additions and subtractions. ### Final Answer Therefore, the correct answer is **8 hours 45 minutes**.