Question

# While selling a pen for rupees 24 the loss in percentage is equal to its cost price. Cost price of a pen is less than rupees 50.

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Hint: Here this question is related to loss and profit. In this they have mentioned about the selling price and the cost price is equal to the loss percentage. By using the formula of loss percentage, we can find the cost price and that cost price must be less than rupees 50.

The selling price of the pen $S.P = 24$
The loss percentage is equal to its cost price
Since they mentioned loss percentage the cost price is more than the selling price and the cost price is less than the rupees 50.
The formula for the loss percentage is given by
Loss in percentage $= \dfrac{{C.P - S.P}}{{C.P}} \times 100$ --------------(1)
Where C.P means cost price and S.P means selling price.
The cost price is known. So, let us consider the cost price be $x$
From the data we have the loss in percentage is equal to the cost price
Loss in percentage = cost price-----------------(2)
Cost price = x
So we can write (1) and (2) as
Loss in percentage $= \dfrac{{x - S.P}}{x} \times 100$ --------------(3)
Loss in percentage = x -----------------(4)
From (3) and (4) we have
$x = \dfrac{{x - S.P}}{x} \times 100$
Substitute the value of selling price S. P=24
$x = \dfrac{{x - 24}}{x} \times 100$
$\Rightarrow {x^2} = (x - 24) \times 100$
On simplification we have
$\Rightarrow {x^2} = 100x - 2400 \\ \Rightarrow {x^2} - 100x + 2400 = 0 \;$
Now we have to find the roots for this quadratic equation
We use formula $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ to find roots, where a=1, b=-100 and c=2400
Substituting the values, we have
$x = \dfrac{{ - ( - 100) \pm \sqrt {{{(100)}^2} - 4(1)(2400)} }}{{2(1)}} \\ \Rightarrow x = \dfrac{{100 \pm \sqrt {10000 - 9600} }}{2} \;$
On further simplification
$\Rightarrow x = \dfrac{{100 \pm \sqrt {400} }}{2} \\ \Rightarrow x = \dfrac{{100 \pm 20}}{2} \;$
Therefore, we have
$x = \dfrac{{100 + 20}}{2} = \dfrac{{120}}{2} = 60$ or $x = \dfrac{{100 - 20}}{2} = \dfrac{{80}}{2} = 40$
Hence x=60 or x=40
Since by the data we had that the cost price should be less than 50 and we had got the cost price as 60 and 40. So we consider x=40
Therefore, the cost of a pen is 40 rupees.
So, the correct answer is “40 rupees”.

Note: If the cost price is greater than the selling price then it is a loss. Otherwise, if the selling price is greater than cost price then it is gained. We can calculate the loss in percentage by Loss in percentage $= \dfrac{{C.P - S.P}}{{C.P}} \times 100$ and gain in percentage by gain in percentage $= \dfrac{{S.P - C.P}}{{C.P}} \times 100$ . The loss and gain depends on the cost price and selling price.