Question

One trader calculates the percentage of on the C.P and other calculates on S.P when their selling prices are the same, then the difference in their actual profit is 85 rupees, and both claim to have \[20\%\] profit. What is the selling price of each?
(a) Rs. 3525
(b) Rs. 3250
(c) Rs. 2550
(d) Rs. 2200
(e) None of these

Answer 1

Hint: For solving this problem we use the formula for calculating the percentage of profit on C.P as \[\text{Profit percentage}=\dfrac{S.P-C.P}{C.P}\times 100\]. Similarly, the percentage of profit on S.P is \[\text{profit percentage}=\dfrac{S.P-C.P}{S.P}\times 100\]. We are given that difference in C.P of both traders and also both profit percentages. So we have three equations and three unknowns. We need to solve those equations to get a selling price.

Complete step-by-step solution
Let us consider the first trader and assume
Cost price of the first trader as \[{{\left( C.P \right)}_{1}}\], selling price as \[\left( S.P \right)\].
We are given that the first trader calculated profit percentage on C.P and is equal to \[20\%\].
We know that formula for profit percentage on C.P is
\[\text{Profit percentage}=\dfrac{S.P-C.P}{C.P}\times 100\]
By using the formula we get
\[\begin{align}
  & \Rightarrow 20=\dfrac{S.P-{{\left( C.P \right)}_{1}}}{{{\left( C.P \right)}_{1}}}\times 100 \\
 & \Rightarrow \dfrac{S.P-{{\left( C.P \right)}_{1}}}{{{\left( C.P \right)}_{1}}}=\dfrac{1}{5} \\
 & \Rightarrow 5\left( S.P \right)-5{{\left( C.P \right)}_{1}}={{\left( C.P \right)}_{1}} \\
 & \Rightarrow {{\left( C.P \right)}_{1}}=\dfrac{5}{6}\left( S.P \right) \\
\end{align}\]
Let us consider the second trader and assume
Cost price of second trader as \[{{\left( C.P \right)}_{2}}\], selling price as \[\left( S.P \right)\].
We are given that the second trader calculated the profit percentage on S.P and is equal to \[20{\scriptstyle{}^{0}/{}_{0}}\].
We know that formula for profit percentage on S.P is
\[\text{Profit percentage}=\dfrac{S.P-C.P}{S.P}\times 100\]
By using the formula we get
\[\begin{align}
  & \Rightarrow 20=\dfrac{S.P-{{\left( C.P \right)}_{2}}}{S.P}\times 100 \\
 & \Rightarrow \dfrac{S.P-{{\left( C.P \right)}_{2}}}{S.P}=\dfrac{1}{5} \\
 & \Rightarrow 5\left( S.P \right)-5{{\left( C.P \right)}_{2}}=\left( S.P \right) \\
 & \Rightarrow {{\left( C.P \right)}_{2}}=\dfrac{4}{5}\left( S.P \right) \\
\end{align}\]
We know that profit can be calculated by using the formula
\[P=S.P-C.P\]
Let us assume that profit of first and second traders as \[{{P}_{1}},{{P}_{2}}\] respectively then we can write
\[\begin{align}
  & {{P}_{1}}=S.P-{{\left( C.P \right)}_{1}} \\
 & {{P}_{2}}=S.P-{{\left( C.P \right)}_{2}} \\
\end{align}\]
Now, we are given that the difference in actual profit of two traders is Rs. 85. So, we can take
\[\Rightarrow \left| {{P}_{1}}-{{P}_{2}} \right|=85\]
By substituting the values of \[{{P}_{1}},{{P}_{2}}\] in above equation we get
\[\begin{align}
  & \Rightarrow \left| \left[ S.P-{{\left( C.P \right)}_{1}} \right]-\left[ S.P-{{\left( C.P \right)}_{2}} \right] \right|=85 \\
 & \Rightarrow \left| {{\left( C.P \right)}_{2}}-{{\left( C.P \right)}_{1}} \right|=85 \\
\end{align}\]
By substituting the values of \[{{\left( C.P \right)}_{1}}\] and \[{{\left( C.P \right)}_{2}}\] in above equation we get
\[\begin{align}
  & \Rightarrow \left| \dfrac{4}{5}S.P-\dfrac{5}{6}S.P \right|=85 \\
 & \Rightarrow S.P\left( \dfrac{1}{30} \right)=85 \\
 & \Rightarrow S.P=2550 \\
\end{align}\]
Therefore the selling price of both the traders is equal to Rs. 2,550. So, option (c) is the correct answer.

Note: Students may make mistakes in taking the percentage of profit on S.P this is similar to the percentage of profit on C.P. that means they apply the same formula of profit to both the traders which will be wrong. Also, they may not consider the modulus of the difference in actual profit. At \[\left| {{P}_{1}}-{{P}_{2}} \right|\]. The modulus is important because we don’t know which value is bigger. If we do not consider the modulus we may get a negative answer which is impossible for the amount.