A man travels from A to B at a speed of 40 km/h and next from B to C at 60 km/h, where B is between A and C. The ratio of distances between A B and BC is 2 : 3. He travels back from C to A at x km/h. If his average speed for the whole journey is $$61 \frac{7}{13}$$km/h, then what is the value of x?

Correct option is D

Given:

The ratio of distances between A to B and B to C is 2 : 3

Speed from A to B = 40 km/h

Speed from B to C = 60 km/h

Speed from C to A = x km/h

The average speed for the entire journey = 61$$\frac{7}{13}$$​ km/h

Formula Used:

Average Speed = $$\frac{\text{Total Distance}}{\text{Total Time}}$$​​

Solution:  

Let the distances be 2D and 3D

Total Distance(AB + BC + CA) = 10D

Total Time = $$\frac{2D}{40}+\frac{3D}{60} + \frac{5D}{x}$$​​

​$$= \frac{D}{20} + \frac{D}{20} + \frac{5D}{x}$$ 

$$= \frac{2D}{20}+\frac{5D}{x}$$

​$$\frac{D}{10}+\frac{5D}{x}$$​​

Using the formula for average speed:

​$$\frac{800}{13} = \frac{10D}{D \left( \frac{1}{10} + \frac{5}{x} \right)}$$​​

​$$\frac{800}{13} = \frac{10}{\frac{1}{10} + \frac{5}{x}}$$​​

​$$\frac{1}{10} + \frac{5}{x} = \frac{10 \times 13}{800} = \frac{130}{800} = \frac{13}{80}$$​​

​$$\frac{5}{x} = \frac{13}{80} - \frac{1}{10}$$​​

​$$\frac{5}{x} = \frac{13}{80} - \frac{8}{80} = \frac{5}{80}$$​​

​$$\frac{5}{x} = \frac{5}{80} \quad \Rightarrow \quad x = 80$$​​

The value of x is 80 km/h