A merchant buys two items for $Rs.7500$. One item he sells at a profit of $16%$ and the other item at $14%$ loss. In the deal, the merchant makes neither any profit nor any loss. What is the difference between the selling prices of both the items?

Hint: It is given that the merchant makes neither any profit nor any loss, it means $16\% $ of the first item is equal to the $14\% $ of the second item because the profit equal to loss makes neither profit nor loss.

Complete step-by-step answer:
First, assume that the cost price of the first item is Rs.$x$, then the cost of the second item is Rs.$\left( {7500 - x} \right)$.
According to the question that the merchant makes neither any profit nor any loss, then the profit on the first item is equal to the loss in the second item.
The above relation in the equation format is expressed as:
$16\% {\text{ of }}x{\text{ = 14\% of }}\left( {7500 - x} \right)$
$\dfrac{{16}}{{100}} \times x = \dfrac{{14}}{{100}} \times (7500 - x)$
Simplify the above equation and then solve for$x$.
$16x = 14 \times (7500 - x)$
$16x = 105000 - 14x$ .
$16x + 14x = 105000$
$
  30x = 105000 \\
  x = \dfrac{{105000}}{{30}} \\
$
$x = 3500$
Therefore, the cost of the first item is Rs.$3500$ and the cost of the second item is $\left( {7500 - 3500 = {\text{Rs}}.4000} \right)$.
According to the question selling price of the first item with 16% of profit:
${\text{Selling price = }}\left( {\dfrac{{100 + {\text{Profit}}\% }}{{100}} \times {\text{Cost Price}}} \right)$
${\text{Selling price = }}\left( {\dfrac{{100 + 16}}{{100}} \times {\text{3500}}} \right)$
${\text{Selling price = }}\left( {\dfrac{{116}}{{100}} \times {\text{3500}}} \right)$
Simplify the above expression to find the selling price of the first item.
${\text{Selling price = }}\left( {116 \times {\text{35}}} \right) = {\text{Rs}}.4060$
And the selling price of the second item with 14% of loss:
${\text{Selling price = }}\left( {\dfrac{{100 - loss\% }}{{100}} \times {\text{Cost Price}}} \right)$
$
  {\text{Selling price = }}\left( {\dfrac{{100 - 14}}{{100}} \times \left( {7500 - 3500} \right)} \right) \\
  {\text{Selling price = }}\left( {\dfrac{{100 - 14}}{{100}} \times 4000} \right) \\
  {\text{Selling price = }}\left( {\dfrac{{86}}{{100}} \times 4000} \right) \\
$
Simplify the above expression to find the selling price of the second item.
${\text{Selling price = }}\left( {86 \times 40} \right) = {\text{Rs}}.3440$
Therefore, the difference between the selling prices of both items is given as:
$ = $ The selling price of 1st item$ - $ the selling price of 2nd item
$ = $ ${\text{ 4060}} - {\text{3440}}$
$ = 620$
Therefore, the difference between the selling prices of both items is Rs. 620.

Note: First assume that the cost of the first item be $x$, then the cost of the second item is given as $\left( {7500 - x} \right)$ because the merchant buys both the item at the cost of Rs.$7500$.