On selling an article for $\text{₹}1,700$, a shopkeeper loses $15\%$. To gain $20\%$, at what price must he sell that article?

Let the Cost Price ($\text{CP}$) of the article be $100\%$.
1. Analyze the loss condition:
Selling at a $15\%$ loss means the selling price is $100\%-15\%=85\%$ of $\text{CP}$.
Given:

$$85\%\text{ of CP}=\text{₹}1,700$$

$$\text{CP}=\frac{1700}{85}\times 100=20\times 100=\text{₹}2,000$$

2. **Calculate the new selling price for a $20\%$ gain:**
To gain $20\%$, the new selling price (${\text{SP}}_{\text{new}}$) must be $120\%$ of $\text{CP}$:

$${\text{SP}}_{\text{new}}=120\%\text{ of }2000=\frac{120}{100}\times 2000=120\times 20=\text{₹}2,400$$

Therefore, he must sell the article for $\text{₹}2,400$.