Vimla starts for office every day at 9 am and reaches exactly on time if she drives at her usual speed of 40 km/hr. She is late by 6 minutes if she drives at 35 km/hr. One day, she covers two-thirds of her distance to office in one-thirds of her usual time to reach office, and then stops for 8 minutes. The speed, in km/hr, at which she should drive the remaining distance to reach office exactly on time is

The correct option is 28.

Let us solve this step-by-step to find the required speed for the remaining journey.

Step 1: Determine the distance to the office and the usual time taken.
Let the distance to Vimla's office be represented by $d$ (in km), and her usual time taken to reach the office be $t$ (in hours).
Using the relation:

$\text{Distance}=\text{Speed}\times \text{Time}$

At her usual speed of 40 km/hr, the distance is:

$$d=40\times t$$

If she drives at 35 km/hr, she is late by 6 minutes.
Since 6 minutes is equal to:

$\frac{6}{60}=0.1\text{ hours}$

The time taken at 35 km/hr is $t+0.1$ hours.
Therefore, we can write the distance equation for this case as:

$$d=35\times \left(t+0.1\right)$$

Since the distance to the office remains the same, we can equate the two expressions for $d$:

$$40t=35\left(t+0.1\right)$$

Now, let us solve for $t$:

$$40t=35t+3.5$$

Subtracting $35t$ from both sides gives:

$$5t=3.5$$

$$t=\frac{3.5}{5}=0.7\text{ hours}$$

Converting the usual time $t$ into minutes:

$$t=0.7\times 60=42\text{ minutes}$$

Now, we can find the distance $d$:

$$d=40\times 0.7=28\text{ km}$$

Step 2: Analyze the journey on the specific day.
On this day, she covers two-thirds of her distance in one-third of her usual time.
Distance covered in the first part:

$$d_1=\frac{2}{3}\times 28=\frac{56}{3}\text{ km}$$

Remaining distance to cover:

$$d_{\text{remaining}}=28-\frac{56}{3}=\frac{28}{3}\text{ km}$$

Time spent on the first part of the journey:

$$t_1=\frac{1}{3}\times 42\text{ minutes}=14\text{ minutes}$$

She then stops for 8 minutes.
Total time elapsed so far:

$$t_{\text{elapsed}}=14\text{ minutes}+8\text{ minutes}=22\text{ minutes}$$

To reach the office exactly on time (total time of 42 minutes), the remaining time available is:

$$t_{\text{remaining}}=42\text{ minutes}-22\text{ minutes}=20\text{ minutes}$$

Converting the remaining time to hours:

$$t_{\text{remaining}}=\frac{20}{60}=\frac{1}{3}\text{ hour}$$

Step 3: Calculate the speed required for the remaining distance.
The speed required to cover the remaining distance of $\frac{28}{3}\text{ km}$ in the remaining time of $\frac{1}{3}\text{ hour}$ is:

$$\text{Required Speed}=\frac{d_{\text{remaining}}}{t_{\text{remaining}}}=\frac{\frac{28}{3}}{\frac{1}{3}}=28\text{ km/hr}$$

Thus, Vimla should drive the remaining distance at a speed of 28 km/hr to reach her office exactly on time.