Problem 2: Sam’s Walk to School
Visualizing the Path
Mathematical Approach
Let the total normal time to walk from Home to School be $T = 20$ min.
Scenario 1: Normal Walk
If Sam walks continuously, he arrives 8 minutes early relative to the bell.
$$ \text{Arrival Time}_1 = \text{Bell Time} – 8 \text{ min} $$
Since the walk takes 20 minutes, let’s say he starts at $t=0$. Then $\text{Arrival Time}_1 = 20$.
This implies $\text{Bell Time} = 28$ min (relative to start).
Scenario 2: Forgotten Notebook
Sam walks to point P, walks back to Home, and then walks the full distance to School. He arrives 10 minutes late.
$$ \text{Arrival Time}_2 = \text{Bell Time} + 10 \text{ min} $$
$$ \text{Arrival Time}_2 = 28 + 10 = 38 \text{ min} $$
Analyzing the Time Difference
The total time spent in Scenario 2 is 38 minutes.
The “useful” part of the walk (Home to School eventually) takes 20 minutes (the normal time $T$).
The “wasted” time is the time spent walking from Home to P and back to Home.
$$ \text{Wasted Time} = \text{Total Time} – \text{Normal Time} $$
$$ \text{Wasted Time} = 38 \text{ min} – 20 \text{ min} = 18 \text{ min} $$
This 18 minutes represents the round trip $H \to P \to H$. Therefore, the one-way trip from Home to P takes half of this time:
$$ t_{H \to P} = \frac{18}{2} = 9 \text{ min} $$Calculating the Fraction
Since the speed is constant, the fraction of the distance covered is equal to the fraction of the total time covered.
$$ \text{Fraction} = \frac{\text{Time to P}}{\text{Total Time to School}} $$ $$ \text{Fraction} = \frac{9 \text{ min}}{20 \text{ min}} = \frac{9}{20} $$Correct Option: (b)