Question

Two articles are sold for the same price, losing 10% on one and gaining 20% on the other. Find the cost price of each article, if Rs. 40 is gained on the whole transaction.  
A. Rs. 654 and Rs. 510
B. Rs. 725 and Rs. 546
C. Rs. 800 and Rs. 600
D. None of these

Answer 1

Hint: In these questions, it is important to first know all the terms that are included which are as follows
The price at which an article is sold is called the selling price or the S.P.
The price at which an article is bought is called the cost price or the C.P.
The important formulae that would be used in this question are as follows
\[profit\%=\dfrac{profit}{\cos t\ price}\times 100\]
\[loss\%=\dfrac{loss}{\cos t\ price}\times 100\]

Complete step-by-step answer:
As mentioned in the question, we have to find the cost price of each article.
Now, let x be the selling price for both the articles.
Now, one article makes 10% loss for x being the selling price, so we can write as follows
\[\begin{align}
  & 10=\dfrac{C.P{{}_{1}}-x}{C.P{{}_{1}}}\times 100 \\
 & \dfrac{10}{100}=\dfrac{C.P{{}_{1}}-x}{C.P{{}_{1}}} \\
 & 9C.P{{}_{1}}=10x \\
 & C.P{{}_{1}}=\dfrac{10}{9}x\ \ \ \ \ \ ...(a) \\
\end{align}\]
Now, the other article makes 20% gain for x being the selling price, so we can write as follows
\[\begin{align}
  & 20=\dfrac{x-C.P{{}_{2}}}{C.P{{}_{2}}}\times 100 \\
 & \dfrac{1}{5}=\dfrac{x-C.P{{}_{2}}}{C.P{{}_{2}}} \\
 & 6C.P{{}_{2}}=5x \\
 & C.P{{}_{2}}=\dfrac{5}{6}x\ \ \ \ \ \ ...(b) \\
\end{align}\]
Now, for finding the cost prices of each article, we can write as follows
\[\begin{align}
  & 40=gain \\
 & 40=2x-\left( C.P{{}_{1}}+C.P{{}_{2}} \right) \\
\end{align}\]
Now, using the equations (a) and (b), we can write as follows
\[\begin{align}
  & 40=2x-\left( \dfrac{10}{9}x+\dfrac{5}{6}x \right) \\
 & 40=2x-x\dfrac{105}{54}=\dfrac{3x}{54}=\dfrac{x}{18} \\
 & x=720 \\
\end{align}\]
Now, using the value of x, we can get the cost prices of the two articles using the equations (a) and (b) as follows
\[C.P{{}_{1}}=\dfrac{10}{9}\times 720=Rs.800\]
\[C.P{{}_{2}}=\dfrac{5}{6}\times 720=Rs.600\]

NOTE: -
The students can make an error if they don’t know about the formulae that are given in the hint as follows
\[profit\%=\dfrac{profit}{\cos t\ price}\times 100\]
\[loss\%=\dfrac{loss}{\cos t\ price}\times 100\]
Without knowing these formulae one cannot get to the right answer.