A businessman sold an article at a loss of $15\%$. Had he sold the article at $\text{Rs. }100$ more, he would have gained $10\%$. At what price did he sell the article?

Let the cost price of the article be $\text{CP}$.
Initial Selling Price $({\text{SP}}_1)=100\%-15\%=85\%\text{ of CP}$
New Selling Price $({\text{SP}}_2)=100\%+10\%=110\%\text{ of CP}$

The difference between the two selling prices is given as $\text{Rs. }100$:

$${\text{SP}}_2-{\text{SP}}_1=100$$

$$(110\%-85\%)\text{ of CP}=100$$

$$25\%\text{ of CP}=100$$

$$\text{CP}=\frac{100\times 100}{25}=400$$

The question asks for the price at which he initially sold the article $({\text{SP}}_1)$:

$${\text{SP}}_1=85\%\text{ of }400=0.85\times 400=\text{Rs. }340$$