Determine the correct answer for the problem described below.

A merchant disposes of two items, X and Y. The merchant sells item X at a profit of 24% and sells item Y at a loss of 12%. If the ratio of the cost price of item X to that of item Y is 2:3 respectively and the total profit realized by the merchant is Rs.300, determine the difference between the selling prices of item X and item Y.

Correct Answer: Rs.400

Step-by-step Explanation:

1. Define the Cost Prices of Items X and Y:

The ratio of the cost price of item X to that of item Y is given as 2 : 3.

Let the common ratio multiplier be $k$.

Therefore, the cost price of item X, ${\mathrm{CP}}_X=2k$

And the cost price of item Y, ${\mathrm{CP}}_Y=3k$

2. Calculate Profit on X and Loss on Y:

Item X is sold at a profit of 24%:

$\text{Profit on X}=24\%\text{of}{\mathrm{CP}}_X=0.24\×2k=0.48k$

Item Y is sold at a loss of 12%:

$\text{Loss on Y}=12\%\text{of}{\mathrm{CP}}_Y=0.12\×3k=0.36k$

3. Find the Value of $k$ using Total Profit:

The overall total profit realized is Rs.300.

$\text{Total Profit}=\text{Profit on X}-\text{Loss on Y}$

$300=0.48k-0.36k$

$0.12k=300$

$k=\frac{300}{0.12}=2500$

4. Calculate the Cost Prices:

${\mathrm{CP}}_X=2\×2500=\text{Rs.}5000$

${\mathrm{CP}}_Y=3\×2500=\text{Rs.}7500$

5. Calculate the Selling Prices of Item X and Item Y:

Selling Price of item X (${\mathrm{SP}}_X$):

${\mathrm{SP}}_X={\mathrm{CP}}_X+\text{Profit on X}=5000+(0.24\×5000)=5000+1200=\text{Rs.}6200$

Selling Price of item Y (${\mathrm{SP}}_Y$):

${\mathrm{SP}}_Y={\mathrm{CP}}_Y-\text{Loss on Y}=7500-(0.12\×7500)=7500-900=\text{Rs.}6600$

6. Find the Difference Between the Selling Prices:

$\text{Difference}={\mathrm{SP}}_Y-{\mathrm{SP}}_X=6600-6200=\text{Rs.}400$

Thus, the difference between the selling prices of item X and item Y is Rs.400.