Question

A businessman bought some items for 7500. He kept 5 items for himself and sold the remaining items at a profit of 30 per item. From the amount received in this deal, he could buy 4 more items. Find the original price of each item.

Answer 1

Hint: First, let the number of items be $x$ and cost of one item be $y$. Then, $xy = 7500$. Use the given condition to form an equation. Then, substitute the value of $y = \dfrac{{7500}}{x}$ in the equation and find the value of $x$. Divide the total cost by the number of items to find the price of each item.

Complete step by step Answer:

We are given that cost of the items that the businessman bought is Rs.7500.
We have to find the price of each item.
For that, we need to find the number of items bought.
Let the number of items is $x$ and the cost of one item be $y$.
Then we know that the product of the cost of one item and the number of items is equal to the total cost of all the items.
Hence, the total cost of the items is $xy$ which is equal to 7500.
We can write this as $xy = 7500$
We are given that the men kept 5 items for himself therefore the number of items sold is equaled to 5 less than the total number of items.
That is the number of items sold is $x - 5$
Next, he sold all the remaining items at a profit of 30 per item.
This implies that the selling price of each item is $y + 30$.
Hence, the total amount received after selling the remaining items is $\left( {x - 5} \right)\left( {y + 30} \right)$.
We are also given that the amount that he received equals the cost price of four items. The money left will be calculated by subtracting the cost price from the revenue he generated.
That is, the money left with him is $\left( {x - 5} \right)\left( {y + 30} \right) - 7500$
Also, the cost of one item is $y$.
Therefore, $\left( {x - 5} \right)\left( {y + 30} \right) - 7500 = 4y$
On solving the above equation we get,
\[xy - 5y + 30x - 150 - 7500 = 4y\]
We will for the solve this equation by using substitution method.
We have $xy = 7500$.
Then, $y = \dfrac{{7500}}{x}$
On substituting for the value of $y$, we get,
\[\left( {\dfrac{{7500}}{x}} \right)x - 5\left( {\dfrac{{7500}}{x}} \right) + 30x - 150 - 7500 = 4\left( {\dfrac{{7500}}{x}} \right)\]
\[7500 - \dfrac{{37500}}{x} + 30x - 150 - 7500 = 4\left( {\dfrac{{7500}}{x}} \right)\]
\[30{x^2} + 150x - 67500 = 0\]
Dividing the equation throughout by 30.
$
  {x^2} + 5x - 2250 = 0 \\
  {x^2} - 50x + 45x - 2250 = 0 \\
  x\left( {x - 50} \right) + 45\left( {x - 50} \right) = 0 \\
  \left( {x + 45} \right)\left( {x - 50} \right) = 0 \\
  x = - 45,50 \\
$
The number of items can never be negative.
Hence the number of items is 50.
Then, divide the total cost by 50 to find the price of one item.
$
  y = \dfrac{{7500}}{{50}} \\
  y = 250 \\
$
Then, the price of one item is Rs. 250

Note: Here, we have substituted the value of $y = \dfrac{{7500}}{x}$, we can also substitute the value of $x = \dfrac{{7500}}{y}$. The amount received in this deal, he could buy 4 more items, means that he is having the amount of sum of previous items and four more items. Also, formulate the equations correctly according to the given conditions.