A vehicle travels $$4$$ km with speed of $$3$$ km h$$^{-1}$$ and another $$4$$ km with speed of $$5$$ km h$$^{-1}$$, then its average speed is:

The vehicle travels two equal distances of 4 km each at different speeds. For equal distances, we can find the average speed by computing total distance over total time.

The time for the first 4 km at 3 km/h is $$t_1 = \frac{4}{3}$$ h, and the time for the next 4 km at 5 km/h is $$t_2 = \frac{4}{5}$$ h.

So the total distance is $$4 + 4 = 8$$ km, and the total time is

$$t_1 + t_2 = \frac{4}{3} + \frac{4}{5} = \frac{20 + 12}{15} = \frac{32}{15} \text{ h}$$

Now the average speed is

$$v_{avg} = \frac{\text{Total distance}}{\text{Total time}} = \frac{8}{\frac{32}{15}} = \frac{8 \times 15}{32} = \frac{120}{32} = 3.75 \text{ km h}^{-1}$$

So, the answer is $$3.75$$ km h$$^{-1}$$.