The difference between a gain of 15% and a loss of 9% is ₹ 768. What is the cost price of the item under consideration here?

Let the Cost Price (CP) of the item be $100\%$.
1. A gain of $15\%$ implies a selling price of:

$${\text{SP}}_1=100\%+15\%=115\%\text{ of CP}$$

2. A loss of $9\%$ implies a selling price of:

$${\text{SP}}_2=100\%-9\%=91\%\text{ of CP}$$

The absolute difference between these two selling prices in terms of percentage is:

$$\text{Difference}=115\%-91\%=24\%\text{ of CP}$$

We are given that this monetary difference is $\text{₹ }768$:

$$24\%\text{ of CP}=768$$

$$1\%\text{ of CP}=\frac{768}{24}=32$$

Now find the full cost price ($100\%$):

$$\text{CP}=32\times 100=3200$$

Thus, the cost price of the item is $3200$.