A man travels some distance at a speed of 12 km/h and returns at a speed of 8 km/h. If the total time taken by him is 3 hours and 45 minutes, then the total distance (in km) travelled is:

The correct answer is 36.

Step-by-Step Explanation:

Let the onward distance from the starting point to the destination be $d$ km.

The man travels this distance $d$ at a speed of 12 km/h, and returns the same distance $d$ at a speed of 8 km/h.

First, let's convert the total time taken from hours and minutes into hours. The total time is 3 hours and 45 minutes. Since 1 hour = 60 minutes, we convert 45 minutes as follows:
$45\text{minutes}=\frac{45}{60}\text{hours}=\frac{3}{4}\text{hours}$

Thus, the total time in hours is:
$\mathrm{Total\; Time}=3+\frac{3}{4}=\frac{15}{4}\text{hours}$

Using the relation $\mathrm{Time}=\frac{\mathrm{Distance}}{\mathrm{Speed}}$, we can write the time taken for each part of the journey:

Time taken for the onward journey:
$t_1=\frac{d}{12}$

Time taken for the return journey:
$t_2=\frac{d}{8}$

The sum of these times is equal to the total time taken:
$\frac{d}{12}+\frac{d}{8}=\frac{15}{4}$

To solve for $d$, find the Least Common Multiple (LCM) of the denominators 12 and 8, which is 24:
$\frac{2d+3d}{24}=\frac{15}{4}$

$\frac{5d}{24}=\frac{15}{4}$

Now, multiply both sides by 24 to isolate $d$:
$5d=\frac{15}{4}\times 24$

$5d=15\times 6$

$5d=90$

$d=18\text{km}$

The one-way distance $d$ is 18 km. Since the question asks for the total distance travelled (onward journey plus return journey):
$\mathrm{Total\; Distance}=2\times d=2\times 18=36\text{km}$