A shopkeeper acquired two distinct items, X and Y, for identical prices. Subsequently, he traded item X at a profit of 20% and item Y at a loss of 10%. If his total net gain from the entire transaction amounted to Rs. 40, determine the purchase price of each item.

The correct answer is Rs. 400.

Step-by-step Explanation:

Step 1: Define variables and cost prices.

Let the cost price (purchase price) of each item, X and Y, be $CP$.

Since both items were purchased for identical prices, the total cost price for both items combined is:

$\text{Total CP}=CP+CP=2\times CP$

Step 2: Calculate the profit and loss on individual items.

1. Item X was sold at a profit of 20%:

$\text{Profit on Item X}=20\%\text{ of }CP=0.20\times CP$

2. Item Y was sold at a loss of 10%:

$\text{Loss on Item Y}=10\%\text{ of }CP=0.10\times CP$

Step 3: Determine the net gain from the entire transaction.

The net gain is the total profit gained from selling item X minus the loss incurred from selling item Y:

$\text{Net Gain}=\text{Profit on Item X}-\text{Loss on Item Y}$

$\text{Net Gain}=0.20\times CP-0.10\times CP=0.10\times CP$

Step 4: Solve for the purchase price ($CP$).

We are given that the total net gain from the transaction is Rs. 40. Therefore:

$0.10\times CP=40$

$CP=\frac{40}{0.10}$

$CP=400$

Thus, the purchase price of each item is Rs. 400.