Question

A dealer sells a toy for Rs. 24 and gains as much percent as the cost price of the toy. Find the cost price of the toy.

Answer 1

Hint:
Let x be the cost price of the toy.
It is given that the percentage of profit gained by the seller is the same as the cost price.
 $\Rightarrow \% gain = \text{cost price} = x\% $
Then, find the total profit by using the formula $\text{cost price} \times \dfrac{{gain}}{{100}}$ .
Thus, form a linear equation by using Profit (P) = Selling Price (SP) – Cost Price (CP).
Hence, on solving the equation, we will get the value of x and thus we get the cost price.

Complete step by step solution:
Let x be the cost price of the toy.
It is given that the percentage of profit gained by the seller is the same as the cost price.
 $\Rightarrow \% gain = \text{cost price} = x\% $
Also, it is given that he sells that toy for the price of Rs. 24.
Now, the formula for the total gain is given by $\text{cost price} \times \dfrac{{gain}}{{100}}$.
Thus, total gain $ = x \times \dfrac{x}{{100}} = \dfrac{{{x^2}}}{{100}}$.
Also, the formula for profit is Profit (P) = Selling Price (SP) – Cost Price (CP).
 $
  \Rightarrow P = SP - CP \\
  \Rightarrow \dfrac{{{x^2}}}{{100}} = 24 - x \\
  \Rightarrow {x^2} = 100\left( {24 - x} \right) \\
  \Rightarrow {x^2} = 2400 - 100x \\
  \Rightarrow {x^2} + 100x - 2400 = 0
 $
Now, we will solve the above equation by using the method of splitting the middle term.
 $
  \Rightarrow {x^2} + 120x - 20x - 2400 = 0 \\
  \Rightarrow x\left( {x + 120} \right) - 20\left( {x + 120} \right) = 0 \\
  \Rightarrow \left( {x + 120} \right)\left( {x - 20} \right) = 0
 $
 $\Rightarrow x + 120 = 0$ or $x - 20 = 0$
 $\Rightarrow x = - 120$ or $x = 20$
The cost price of any object cannot be negative. So, we get $x = 20$ .

Thus, the cost price of the toy is Rs. 20.

Note:
Alternate method:
Let x be the cost price of the toy.
It is given that the percentage of profit gained by the seller is the same as the cost price.
 $\Rightarrow \% gain = \text{cost price} = x\% $
Also, it is given that he sells that toy for the price of Rs. 24.
Now, we will use the formula $\% gain = \dfrac{{SP - CP}}{{CP}} \times 100$
 $
  \Rightarrow x = \dfrac{{24 - x}}{x} \times 100 \\
  \Rightarrow {x^2} = \left( {24 - x} \right) \times 100 \\
  \Rightarrow {x^2} = 2400 - 100x \\
  \Rightarrow {x^2} + 100x - 2400 = 0
 $
Now, we will solve the above equation by using the method of splitting the middle term.
 $
  \Rightarrow {x^2} + 120x - 20x - 2400 = 0 \\
  \Rightarrow x\left( {x + 120} \right) - 20\left( {x + 120} \right) = 0 \\
  \Rightarrow \left( {x + 120} \right)\left( {x - 20} \right) = 0
 $
 $\Rightarrow x + 120 = 0$ or $x - 20 = 0$
 $\Rightarrow x = - 120$ or $x = 20$
The cost price of any object cannot be negative. So, we get $x = 20$.
Thus, the cost price of the toy is Rs. 20.