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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.