A bus starts at 9 am and follows a fixed route every day. One day, it traveled at a constant speed of 60 km per hour and reached its destination 3.5 hours later than its scheduled arrival time. The next day, it traveled two-thirds of its route in one-third of its total scheduled travel time, and the remaining part of the route at 40 km per hour to reach just on time. What is the scheduled arrival time of the bus?

The correct option is 7:30 pm.

Let us break down the problem step-by-step to understand why this scheduled arrival time is correct.

Step 1: Define the variables
Let the total distance of the fixed route be $D$ (in km).
Let the scheduled travel time of the bus be $T$ (in hours).
Since the bus starts at 9 am, its scheduled arrival time will be $T$ hours after 9 am.

Step 2: Analyze the first scenario
On the first day, the bus travels at a constant speed of 60 km per hour and reaches its destination 3.5 hours later than its scheduled arrival time.
This means the actual travel time is:
$T_1=T+3.5$
Using the formula $\text{Distance}=\text{Speed}\times \text{Time}$, we can write the first relation:
$D=60\times \left(T+3.5\right)$     — (Equation 1)

Step 3: Analyze the second scenario
On the second day:
- The bus travels two-thirds of the route ($\frac{2}{3}D$) in one-third of its total scheduled travel time ($\frac{1}{3}T$).
- The remaining part of the route is $D-\frac{2}{3}D=\frac{1}{3}D$.
- This remaining part ($\frac{1}{3}D$) is traveled at a speed of 40 km per hour.
- The total time taken is exactly the scheduled time $T$, so the bus reaches just on time.
The time spent on the remaining part of the route is the total time minus the time spent on the first part:
$T_{\text{remaining}}=T-\frac{1}{3}T=\frac{2}{3}T$
Since the speed during the remaining part is 40 km per hour, we have:
$\text{Distance of remaining part}=\text{Speed}\times T_{\text{remaining}}$
$\frac{1}{3}D=40\times \left(\frac{2}{3}T\right)$
We can multiply both sides of this equation by 3 to simplify:
$D=40\times 2T$
$D=80T$     — (Equation 2)

Step 4: Solve for the scheduled travel time ($T$)
Equating the expressions for $D$ from Equation 1 and Equation 2:
$80T=60\left(T+3.5\right)$
Divide both sides by 20:
$4T=3\left(T+3.5\right)$
Expand the right side:
$4T=3T+10.5$
Subtract $3T$ from both sides:
$T=10.5\text{hours}$

Step 5: Determine the scheduled arrival time
The scheduled travel duration is 10.5 hours (or 10 hours and 30 minutes).
Since the bus starts at 9:00 am:
- 9:00 am + 3 hours = 12:00 pm (noon)
- 12:00 pm + 7 hours = 7:00 pm
- 7:00 pm + 30 minutes = 7:30 pm
Therefore, the scheduled arrival time of the bus is 7:30 pm.