A man travels the first $200\text{km}$ of his journey at $80\text{km/h}$. Due to bad weather he travels the next $200\text{km}$ of his journey at $50\text{km/h}$. What is his average speed for the first $400\text{km}$ of his journey?

When a person travels two equal distances ($d_1=d_2=200\text{km}$) at two different speeds $v_1$ and $v_2$, the shortcut formula for the average speed is given by the harmonic mean:
$$\text{Average Speed}=\frac{2v_1{v}_2}{v_1+v_2}$$

Given:

* $v_1=80\text{km/h}$
* $v_2=50\text{km/h}$

Substitute these values into the formula:

$$\text{Average Speed}=\frac{2\times 80\times 50}{80+50}$$

$$\text{Average Speed}=\frac{8000}{130}=\frac{800}{13}\approx 61.538\text{km/h}$$

Therefore, his average speed for the first $400\text{km}$ is $61.53\text{km/h}$.