For a particular merchandise, the price reduction offered during a sale is Rs .960 higher than the gain obtained from its sale. If the customer pays Rs .1920 to purchase this item and the retailer had set the initial list price at 100% greater than its production cost, calculate the percentage gain realized on this transaction.

The correct option is 20%.

Let's break down the given information step-by-step to find the percentage gain realized on this transaction.

Step 1: Define the variables and note down the given values.

Selling Price (SP) = Rs. 1920
Let Cost Price (CP) = $C$
Let List Price / Marked Price (MP) = $M$
Gain (Profit) = $\mathrm{SP}-\mathrm{CP}=1920-C$
Discount (Price Reduction) = $\mathrm{MP}-\mathrm{SP}=M-1920$

Step 2: Express List Price in terms of Cost Price.

The retailer sets the initial list price 100% greater than its production cost (Cost Price).

$M=C+100\%\text{of}C=C+C=2C$

Step 3: Set up the equation using the relation between Discount and Gain.

We are given that the price reduction (discount) offered is Rs. 960 higher than the gain obtained.

$\text{Discount}=\text{Gain}+960$

Substitute the expressions for Discount, MP, and Gain into this equation:

$(M-1920)=(1920-C)+960$

Step 4: Solve for Cost Price ($C$).

Substitute $M=2C$ into the equation:

$2C-1920=2880-C$

Combine like terms:

$2C+C=2880+1920$

$3C=4800$

$C=\frac{4800}{3}=1600$

So, the Cost Price of the item is Rs. 1600.

Step 5: Calculate the Gain and Percentage Gain.

$\text{Gain}=\mathrm{SP}-\mathrm{CP}=1920-1600=320$

Now, calculate the percentage gain:

$\text{Percentage Gain}=\left(\frac{\text{Gain}}{\mathrm{CP}}\right)\times 100$

$\text{Percentage Gain}=\left(\frac{320}{1600}\right)\times 100=\frac{320}{16}=20\%$

Thus, the percentage gain realized on this transaction is 20%.