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%.

Step-by-Step Explanation:

Let the cost price (production cost) of the item be $CP$, the list price (marked price) be $LP$, and the selling price be $SP$.

From the given problem, the customer pays Rs. 1920 to purchase the item.
Therefore, the selling price is:

$SP=\text{Rs. 1920}$

We are given that the retailer sets the initial list price at 100% greater than its production cost:

$LP=CP+100\%\text{of}CP=2\times CP$

The price reduction (discount) offered during the sale is given by:

$\text{Discount}=LP-SP=LP-1920$

The gain (profit) obtained from the sale is given by:

$\text{Gain}=SP-CP=1920-CP$

According to the problem, the price reduction is Rs. 960 higher than the gain obtained:

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

Substitute the expressions for Discount and Gain into the equation:

$LP-1920=(1920-CP)+960$

Simplify the right side of the equation:

$LP-1920=2880-CP$

Rearrange the terms to group $LP$ and $CP$:

$LP+CP=2880+1920$

$LP+CP=4800$

Now, substitute $LP=2\times CP$ into the equation:

$2\times CP+CP=4800$

$3\times CP=4800$

$CP=\frac{4800}{3}=\text{Rs. 1600}$

Now, calculate the gain obtained on the transaction:

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

Finally, calculate the percentage gain realized on this transaction:

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

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

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