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 cost price (CP) of the mixture per kg.
Given that the Selling Price (SP) of the mixture is $\text{₹}320\text{/kg}$ and the profit is $20\%$:

$$\text{SP}=\text{CP}\times \left(1+\frac{\text{Profit}\%}{100}\right)$$

$$320=\text{CP}\times \left(1+\frac{20}{100}\right)$$

$$320=\text{CP}\times \frac{120}{100}=\text{CP}\times \frac{6}{5}$$

$$\text{CP}=320\times \frac{5}{6}=\frac{1600}{6}=\text{₹}\frac{800}{3}\text{ per kg}$$

Now, let's use the rule of alligation:

* Cost of Type 1 tea ($C_1$) = $\text{₹}180$
* Cost of Type 2 tea ($C_2$) = $\text{₹}280$
* Mean Cost Price ($M$) = $\frac{800}{3}$

$$\text{Ratio }\left(\frac{\text{Quantity of Type 1}}{\text{Quantity of Type 2}}\right)=\frac{C_2-M}{M-C_1}$$

$$\text{Ratio}=\frac{280-\frac{800}{3}}{\frac{800}{3}-180}=\frac{\frac{840-800}{3}}{\frac{800-540}{3}}=\frac{40}{260}=\frac{4}{26}=\frac{2}{13}$$

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