I started on my bicycle at 7 a.m. to reach a certain place. After going a certain distance, my bicycle went out of order. Consequently, I rested for 35 minutes and came back to my house walking all the way. I reached my house at 1 p.m. If my cycling speed is 10 kmph and my walking speed is 1 kmph, then on my bicycle I covered a distance of

### Concept & Effective Moving Time When a round trip involves a rest period, subtract the rest duration from the total elapsed time to find the active travel time. Since the starting and ending points are the same, the distance cycled out equals the distance walked back. $$Time = \frac{Distance}{Speed}$$ ### Step-by-Step Solution * **Step 1: Determine total moving time.** * Total elapsed time from 7 a.m. to 1 p.m. is $6$ hours. * Rest time = $35$ minutes = $\frac{35}{60}$ hours = $\frac{7}{12}$ hours. * Actual moving time = $6 - \frac{7}{12} = \frac{72}{12} - \frac{7}{12} = \frac{65}{12}$ hours. * **Step 2: Set up the distance equation.** * Let the one-way distance be $d$ km. * Cycling speed = $10$ kmph. Walking speed = $1$ kmph. * Time cycling + Time walking = Total moving time * $\frac{d}{10} + \frac{d}{1} = \frac{65}{12}$ * **Step 3: Solve for $d$.** * $\frac{d + 10d}{10} = \frac{65}{12}$ * $\frac{11d}{10} = \frac{65}{12}$ * $11d = \frac{650}{12}$ * $d = \frac{650}{12 \times 11} = \frac{650}{132}$ * Simplify by dividing by 2: $d = \frac{325}{66}$ km. * **Step 4: Convert to a mixed fraction.** * $325 \div 66$: $66 \times 4 = 264$. * Remainder = $325 - 264 = 61$. * Thus, $d = 4 \frac{61}{66}$ km. ### Exam Strategy & Shortcut Whenever distances are equal, you can quickly find average speed: $\frac{2 \times 10 \times 1}{10 + 1} = \frac{20}{11}$ km/hr. Total distance (there and back) = Average speed $\times$ Moving time = $\frac{20}{11} \times \frac{65}{12} = \frac{1300}{132}$ = $\frac{325}{33}$ km. One-way distance is half of this: $\frac{325}{66} = 4 \frac{61}{66}$ km. ### Common Pitfall Using the total 6 hours directly without subtracting the 35 minutes of rest, which skews the entire calculation. ### Final Answer Therefore, the correct answer is **$4 \frac{61}{66}$ km**.