Question

A person by selling an article for the Rs.\[450\], loses \[20\% \]. In order to make a profit of \[20\% \] what is the price at which he must sell?
A. Rs.\[500\]
B. Rs.\[475\]
C. Rs.\[575\]
D. Rs.\[675\]

Answer 1

Hint: Selling Price (S.P.) is the price at which article is sold and Cost Price (C.P.) is the price at which article is bought.
If ${\text{S}}{\text{.P}}{\text{. > C}}{\text{.P}}{\text{.}}$then the seller earn profit.
Such that \[{\text{profit = S}}{\text{.P}}{\text{. - C}}{\text{.P}}{\text{.}}\]
\[{\text{profit% = }}\dfrac{{{\text{profit}}}}{{{\text{C}}{\text{.P}}{\text{.}}}}{\text{x 100}}\]
If ${\text{S}}{\text{.P}}{\text{. < C}}{\text{.P}}{\text{.}}$then the seller incurred loss
Such that \[{\text{loss = C}}{\text{.P}}{\text{. - S}}{\text{.P}}{\text{.}}\]
\[{\text{loss% = }}\dfrac{{{\text{loss}}}}{{{\text{C}}{\text{.P}}{\text{.}}}}{\text{ times 100}}\]

Complete step by step solution:
Given that S.P. of the article is the Rs.\[450\] Incurred the loss of \[20\% \]
Need to find S.P of the article to earn profit of \[20\% \]
Step1 :
Finding C.P. of the article
Let C.P of the article be Rs.\[100\]
Since the person sold it at \[20\% \]loss.
We know that $loss\%\;=\;{\dfrac{loss}{C.P}}\times\;100$
$loss\;={\dfrac{loss\%\times{C.P}}{100}}$
Substituting the value
\[{\text{loss% = 20% }}\] & C.P.= Rs.\[100\]
$loss\;={\dfrac{20\times{100}}{100}}$
\[ \Rightarrow {\text{loss = Rs}}.20\]
And when there is loss
\[{\text{loss = C}}{\text{.P}}{\text{. - S}}{\text{.P}}{\text{.}}\]
\[ \Rightarrow {\text{S}}{\text{.P}}{\text{. = C}}{\text{.P}}{\text{. - }}{\kern 1pt} {\kern 1pt} {\text{loss}}\]
\[ \Rightarrow {\text{S}}{\text{.P}}. = 100 - 20 = 80\]
Using a unitary method we find the original C.P.
When S.P. = Rs.\[80\] at \[20\% \]loss , C.P.= Rs.\[100\]
And for S.P. = Rs.\[1\] at \[20\% \]loss, C.P. = Rs.\[\dfrac{{100}}{{80}}\]

Hence when S.P. = Rs.\[450\]at \[20\% \]loss , C.P = Rs\[\dfrac{{100}}{{80}} \times 450 = 562.5\].

Step 2:
Now finding amount of profit if article is sold at \[20\% \]profit
Since, $Profit\%\;=\;{\dfrac{Profit}{C.P}}\times\;100$
$loss\;={\dfrac{loss\%\times{C.P}}{100}}$
Substituting
\[{\text{profit% = 20% }}\]
\[\begin{gathered}
   \Rightarrow {\text{profit}} = \dfrac{{(20 \times 562.5)}}{{100}} \\
    \\
\end{gathered} \]
\[\begin{gathered}
   \Rightarrow {\text{profit}} = \dfrac{{(20 \times 562.5)}}{{100}} \\
    \\
\end{gathered} \]
\[{\text{profit = }}\dfrac{{562.5}}{5} = 112.5\]
\[{\text{profit}} = {\text{Rs}}.112.5\]
So the profit seller earn is Rs.\[112.5\]

Step 3:
Finding the required S.P. of the article,when earning profit of Rs.\[112.5\]
Since \[{\text{profit = S}}{\text{.P}}{\text{. - C}}{\text{.P}}{\text{.}}\]
\[ \Rightarrow {\text{S}}{\text{.P}}{\text{. = profit + C}}{\text{.P}}.\]
\[ \Rightarrow {\text{S}}{\text{.P}}{\text{. = 112}}{\text{.5 + 562}}{\text{.5}}\]
\[ \Rightarrow {\text{S}}{\text{.P}}{\text{. = 675}}\]

Hence the S.P. of the article is Rs.\[675\]. And the correct option is (D) Rs.\[675\].

Note: Discount is another factor used applied on marked price (marked in article to sell). This is a reduction in price offered on marked price.