Question

While selling a pen for Rs.$24,$the loss percentage is equal to its cost price in rupees. Find the cost price of the pen.

Answer 1

Hint: According to the question we have to find the cost price of the pen while selling a pen for Rs.$24,$ the loss percentage is equal to its cost price in rupees. So, first of all we have to let the cost price of the pen be Rs.$x$
Now, we have to find the loss percentage of the pen and as given in the question that the loss percentage is equal to its cost price in rupees.
Hence, with the help of the loss percentage is equal to it’s cost price in rupees we will obtain an quadratic equation and by solving the obtained quadratic expression we can find the value of $x$ which is the cost price of the pen.

Complete step-by-step answer:
Given,
Selling price of the pen = Rs. $24$ and,
Loss percentage is equal to its cost price in rupees.
Step 1: So first of all we have to let the cost price of pen $ = Rs.x$and now we have to find the loss percentage as below as mentioned in the solution hint.
$ = \left( {x - \dfrac{x}{{100}}} \right)Rs$…………….(1)
Step 2: Now, we know that as mentioned in the question loss percentage is equal to it’s cost price in rupees so, we have to multiply the term $\dfrac{x}{{100}}$ as obtained in the step 1 with the cost price we let $Rs.x$, which is equal to the selling price as mentioned in the solution hint.
$ \Rightarrow x - \dfrac{x}{{100}} \times x = 24$…………….(2)
Step 3: Now, on solving the equation (2) as obtained in the step 2 we can find the value of x hence,
$ \Rightarrow \dfrac{{100x - {x^2}}}{{100}} = 24$
Applying cross-multiplication in the expression obtained just above,
$
   \Rightarrow 100x - {x^2} = 2400 \\
   \Rightarrow 100x - {x^2} - 2400 = 0................(3)
 $
Step 4: On multiplying with negative sign in the both sides of the expression (3) as obtained in the step 3.
$ \Rightarrow {x^2} - 100x + 2400 = 0$………………(4)
Step 5: Now, we have to solve the obtained quadratic expression (4) as obtained in step 3 which can be obtained by finding the factor of 2400.
$
   \Rightarrow {x^2} - (60 + 40)x + 2400 = 0 \\
   \Rightarrow {x^2} - 60x - 40x + 2400 = 0 \\
   \Rightarrow x(x - 60) - 40(x - 60) = 0 \\
   \Rightarrow (x - 40)(x - 60) = 0
 $
Step 6: Now, we have to solve the each of the roots as obtained in the step 5 hence,
$
   \Rightarrow \left( {x - 40} \right) = 0 \\
   \Rightarrow x = 40
 $
And same as,
$
   \Rightarrow (x - 60) = 0 \\
   \Rightarrow x = 60
$

Hence, the cost price of the pen can be 40rs or 60rs while selling a pen for Rs.$24,$the loss percentage is equal to its cost price in rupees.

Note: If it is given in the question that the cost of price of the pen is less than 50rs then we can say that the cost price of pen will be 40rs and if it is given that the cost of price of the pen is greater than 50rs then we can say that the cost price of pen will be 60rs.
A quadratic expression has only two roots so, on solving the quadratic expression only two root will be obtained which can be positive or negative both.