By selling an article for ₹$480$, a man incurs a loss of $10\%$. At what price should he sell it, so that it makes a profit of $20\%$?

Let the Cost Price of the article be $\text{CP}$.
When sold at a loss of $10\%$, the selling price is $90\%$ of $\text{CP}$.

$$0.90\times \text{CP}=480$$

$$\text{CP}=\frac{480}{0.90}=\frac{4800}{9}=\frac{1600}{3}$$

To make a profit of $20\%$, the new selling price (${\text{SP}}_{\text{new}}$) must be $120\%$ of $\text{CP}$:

$${\text{SP}}_{\text{new}}=1.20\times \text{CP}$$

$${\text{SP}}_{\text{new}}=1.20\times \frac{1600}{3}=0.40\times 1600=₹640$$

Alternatively, using the direct ratio:

$${\text{SP}}_{\text{new}}={\text{SP}}_{\text{old}}\times \frac{100+\text{Profit}\%}{100-\text{Loss}\%}$$

$${\text{SP}}_{\text{new}}=480\times \frac{120}{90}=480\times \frac{4}{3}=₹640$$