Answer

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**Hint:**This problem deals with distance, speed and time taken for the journey. Here given that there are two means of transport throughout the journey. So the speed for both the modes of transport would be distinctive. Hence applying the formula of distance which is given by the product of speed and the time taken to cover the distance.

**Formula used:**distance = speed $ \times $ time.

**Complete step-by-step solution:**

We know that 60 minutes = 1 hour.

Hence 30 minutes = $\dfrac{1}{2}$hour.

Given that the sanjana travels a distance of 660 km, the means of transport is both train and the car.

So given that the time taken is 13.5 hours if she travels 300 km by train and 360 km by car.

Also given that the time taken is $\left( {13.5 + \dfrac{1}{2}} \right)$hours, if she travels 360 km by train and 300 km by car.

Let the speed of the train be = $v$

The speed of the car be = $u$

The distance is equal to the product of speed and time.

$ \Rightarrow $Total distance = speed $ \times $ time

$\therefore $ Time = Total distance / time

Now expressing it mathematically as given below:

Time taken by Sanjana, if she takes the train for 300km and car for 360 km = 13.5 hrs.

$ \Rightarrow \dfrac{{300}}{v} + \dfrac{{360}}{u} = 13.5$

Time taken by Sanjana, if she takes the train for 360km and car for 300 km = 14 hrs.

$ \Rightarrow \dfrac{{360}}{v} + \dfrac{{300}}{u} = 14$

Now we have 2 equations and 2 variables, solving the equations to get the value of $v$ and $u$.

Consider the equation $\dfrac{{300}}{v} + \dfrac{{360}}{u} = 13.5$

$ \Rightarrow \dfrac{{300}}{v} = 13.5 - \dfrac{{360}}{u}$

$ \Rightarrow \dfrac{{300}}{v} = \dfrac{{13.5u - 360}}{u}$

Reciprocate the above equation :

$ \Rightarrow \dfrac{v}{{300}} = \dfrac{u}{{13.5u - 360}}$

Obtaining the expression for $v$:

$ \Rightarrow v = \dfrac{{300u}}{{13.5u - 360}}$

Now consider the equation $\dfrac{{360}}{v} + \dfrac{{300}}{u} = 14$, simplifying it:

$ \Rightarrow \dfrac{{360u + 300v}}{{uv}} = 14$

$ \Rightarrow 360u + 300v = 14uv$

$ \Rightarrow 360u = \left( {14u - 300} \right)v$

Now substitute the expression of $v = \dfrac{{300u}}{{13.5u - 360}}$ in the above equation :

$ \Rightarrow 360u = \left( {14u - 300} \right)\left( {\dfrac{{300u}}{{13.5u - 360}}} \right)$

$ \Rightarrow 360u\left( {13.5u - 360} \right) = 300u\left( {14u - 300} \right)$

In the above equation $u$ gets cancelled on both sides, as given below:

$ \Rightarrow 360\left( {13.5u - 360} \right) = 300\left( {14u - 300} \right)$

$ \Rightarrow 360\left( {13.5u} \right) - 360\left( {360} \right) = 300\left( {14u} \right) - 300\left( {300} \right)$

$ \Rightarrow 4860u - 129600 = 4200u - 90000$

$ \Rightarrow 660u = 39600$

$\therefore u = 60$

Hence the speed of the car is $u = 60$km/hr

The time taken by Sanjana if she travels 660 km by car (in hrs), is given by:

$ \Rightarrow \dfrac{{660}}{{60}} = 11$hrs.

$\therefore $The time taken if she travels 660 km by car is 11 hours.

**Option C is the correct answer.**

**Note:**Please note that this problem can be done in another way also but with a slight change that is we found the expression for speed of the train and substituted it in the other equation and obtained the speed of the car, instead of that we can obtain the expressions of speed of the train from both the equations and equate it, in order to get the value of the speed of the car. Either of the ways give us the same final answer.

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