In what ratio must the two types of tea at $\text{₹ }180\text{/kg}$ and $\text{₹ }280\text{/kg}$ respectively be mixed so that the obtained mixture sold at $\text{₹ }320\text{/kg}$ earns a profit of $20\%$?

First, let's determine the actual Cost Price (CP) of the mixture from the given Selling Price (SP) and profit margin:
$$\text{SP of mixture}=\text{₹ }320\text{/kg}$$

$$\text{Profit}=20\%$$

$$\text{CP of mixture}=\frac{\text{SP}}{1+\frac{\text{Profit}\%}{100}}=\frac{320}{1+0.20}=\frac{320}{1.2}=\frac{3200}{12}=\text{₹ }\frac{800}{3}\text{/kg}$$

Now, let's use the rule of alligation to find the mixing ratio:

* Cost of cheaper tea ($C$) = $180$
* Cost of dearer tea ($D$) = $280$
* Mean cost ($M$) = $\frac{800}{3}$

$$\text{Ratio}=\frac{D-M}{M-C}=\frac{280-\frac{800}{3}}{\frac{800}{3}-180}$$

$$\text{Ratio}=\frac{\frac{840-800}{3}}{\frac{800-540}{3}}=\frac{40}{260}=\frac{4}{26}=\frac{2}{13}$$

Therefore, the required ratio is $2:13$.