This problem involves calculating travel time based on distance and speed. We are given the speeds of two cars, A and B, traveling the same route, and the time taken by car A. We need to find the time taken by car B.

Calculating the Distance Traveled

First, we need to determine the total distance between place P and place Q. We can use the information provided for car A:

• Speed of car A ($S_A$) = $60$ kmph
• Time taken by car A ($T_A$) = $12$ hrs

The formula relating distance, speed, and time is:

Distance ($D$) = Speed ($S$) $\times$ Time ($T$)

Using the values for car A, the distance is:

$D = S_A \times T_A$
$D = 60 \text{ kmph} \times 12 \text{ hrs}$
$D = 720 \text{ km}$

So, the distance between place P and place Q is $720$ km.

Determining B's Travel Time

Now that we know the distance, we can calculate the time taken by car B to travel the same route. We have:

• Distance ($D$) = $720$ km
• Speed of car B ($S_B$) = $80$ kmph

We can rearrange the distance formula to solve for time:

Time ($T$) = Distance ($D$) / Speed ($S$)

Now, let's calculate the time taken by car B ($T_B$):

$T_B = D / S_B$
$T_B = 720 \text{ km} / 80 \text{ kmph}$
$T_B = 9 \text{ hrs}$

Therefore, car B takes $9$ hours to reach place Q.