Two cars run to a place at the speeds of 45 km/h and 55 km/h, respectively. If the second car takes 40 minutes less than the first for the journey, then what is the length of the journey (in km)?

The correct answer is 165 (Option 165).

Step 1: Understand the given information
Let the total length of the journey be $d$ km.
Speed of the first car, $v_1=45\text{ km/h}$
Speed of the second car, $v_2=55\text{ km/h}$
Difference in time taken by the two cars = 40 minutes.

Step 2: Convert time difference into hours
Since the speeds are given in kilometers per hour (km/h), we must convert the time difference from minutes to hours:

$\text{Time difference}=\frac{40}{60}\text{ hours}=\frac{2}{3}\text{ hours}$

Step 3: Express time taken by each car in terms of distance
Using the formula $\text{Time}=\frac{\text{Distance}}{\text{Speed}}$:
Time taken by the first car, $t_1=\frac{d}{45}$ hours
Time taken by the second car, $t_2=\frac{d}{55}$ hours

Step 4: Set up the equation and solve for distance ($d$)
Since the second car travels faster, it takes less time. Therefore:

$t_1-t_2=\frac{2}{3}$

$\frac{d}{45}-\frac{d}{55}=\frac{2}{3}$

Take $d$ common on the left side:

$d\times \left(\frac{1}{45}-\frac{1}{55}\right)=\frac{2}{3}$

$d\times \left(\frac{55-45}{45\times 55}\right)=\frac{2}{3}$

$d\times \left(\frac{10}{2475}\right)=\frac{2}{3}$

Now, isolate $d$:

$d=\frac{2}{3}\times \frac{2475}{10}$

$d=\frac{2\times 825}{10}$

$d=\frac{1650}{10}=165\text{ km}$

Thus, the total length of the journey is 165 km.