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: The correct option is Rs.400.

Step-by-step Explanation:

Let the cost price of item X be ${\mathrm{CP}}_X$ and the cost price of item Y be ${\mathrm{CP}}_Y$.

Given that the ratio of the cost price of item X to that of item Y is 2 : 3, we can represent their cost prices in terms of a variable x:

${\mathrm{CP}}_X=2x$

${\mathrm{CP}}_Y=3x$

1. Calculate Profit and Loss for each item:

Item X is sold at a profit of 24%:

$\text{Profit on X}=24\%\text{of}2x=\frac{24}{100}\times 2x=0.48x$

Item Y is sold at a loss of 12%:

$\text{Loss on Y}=12\%\text{of}3x=\frac{12}{100}\times 3x=0.36x$

2. Determine the total profit and find x:

The total profit realized by the merchant is the profit on item X minus the loss on item Y:

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

$300=0.48x-0.36x$

$300=0.12x$

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

3. Calculate the Selling Prices (SP) of item X and item Y:

Cost price of item X = $2x=2\times 2500=\text{Rs.}5000$

Cost price of item Y = $3x=3\times 2500=\text{Rs.}7500$

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

${\mathrm{SP}}_X={\mathrm{CP}}_X+\text{Profit on X}=5000+(0.48\times 2500)=5000+1200=\text{Rs.}6200$

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

${\mathrm{SP}}_Y={\mathrm{CP}}_Y-\text{Loss on Y}=7500-(0.36\times 2500)=7500-900=\text{Rs.}6600$

4. Calculate 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.