Question

Raman buys $5$ pens and $30$ pencils for Rs. $1000$ . He sells the pens at a profit of $15\%$ and pencils at a profit of $10\%$ and makes a total profit of Rs. $120$ . Find the cost of a pen and a pencil.
A. Rs. 70, Rs. 30
B. Rs. 85, Rs. 15
C. Rs.80, Rs. 20
D. Data inadequate

Answer 1

Hint: We will start by assuming the cost price of pens and pencils to be x and y respectively and then we will put them in the conditions given in the question. After that we will find the profit of the pens and pencils in terms of the variables we have assumed and this can be done by using the formula for selling price and then finally we will get two equations in two variables, we will solve the equations to get the cost price of pens and pencils.

Complete step by step answer:
We will start by assuming the cost price of $5$ pens to be $x$ and the cost price of $30$ pencils to be $y$. Now it is given that the total cost price of $5$ pens and $30$ pencils is Rs. $1000$ .
Therefore, $x+y=1000\text{ }.......\text{Equation 1}\text{.}$
We know that the formula for selling price is as follows: $S.P.=\left\{ \dfrac{\left( 100+\text{Profit }\!\!\%\!\!\text{ } \right)}{100} \right\}\times C.P.$
Now next it is given that Raman earns a profit of $15\%$ by selling the pens, now we have already assumed that the cost price of the pens is $x$ . Putting it in the above equation to calculate the selling price:
$\begin{align}
  & S.P.=\left\{ \dfrac{\left( 100+\text{Profit }\!\!\%\!\!\text{ } \right)}{100} \right\}\times C.P.=\left\{ \dfrac{\left( 100+\text{15} \right)}{100} \right\}\times x \\
 & S.P.=\left( \dfrac{115}{100} \right)\times x\Rightarrow S.P.=1.15x \\
\end{align}$
We will now find the profit made by selling the pens, we know that the formula for finding the profit is: $\text{Profit}=S.P.-C.P.$ ,
Therefore, $1.15x-x=0.15x$ . Hence, profit made by selling the pens is $0.15x$ Rs.
Similarly, it is given that Raman earns a profit of $10\%$ by selling the pencils, now we have already assumed that the cost price of the pencils is $y$ . Putting it in the formula to calculate the selling price:
$\begin{align}
  & S.P.=\left\{ \dfrac{\left( 100+\text{Profit }\!\!\%\!\!\text{ } \right)}{100} \right\}\times C.P.=\left\{ \dfrac{\left( 100+\text{10} \right)}{100} \right\}\times y \\
 & S.P.=\left( \dfrac{110}{100} \right)\times y\Rightarrow S.P.=1.1y \\
\end{align}$
Now the profit made by selling the pencils would be: $S.P.-C.P.\Rightarrow 1.1y-y=0.1y$ . Hence, profit made by selling the pens is $0.1y$ Rs.
It is given in the question that total profit made by Raman is $120$. Therefore, $0.15x+0.1y=120\text{ }..........\text{Equation 2}\text{.}$
We have with us two equations that are equation 1 and equation 2 in two variables. We will now find the value of $x$ and $y$ :
$\begin{align}
  & x+y=1000\text{ }............\left( 1 \right) \\
 & 0.15x+0.1y=120\text{ }............\left( 2 \right) \\
\end{align}$
From $\left( 1 \right)$ , $x=1000-y\text{ }.......\text{(3)}$ , we will put this in $\left( 2 \right)$ :
$\begin{align}
  & 0.15x+0.1y=120 \\
 & \Rightarrow 0.15\left( 1000-y \right)+0.1y=120\Rightarrow 150-.15y+0.1y=120 \\
 & \Rightarrow 30=0.05y\Rightarrow y=\dfrac{30}{0.05} \\
 & \Rightarrow y=600 \\
\end{align}$
Putting this value of $y$ in $\left( 3 \right)$ , therefore:
$\begin{align}
  & x=1000-y\Rightarrow 1000-600=400 \\
 & x=400 \\
\end{align}$
We now have values of $x$ and $y$ .
We have assumed $x$ as the cost price of 5 pens and $y$ as the cost price of 30 pencils.
Therefore, cost of a single pen is : $\Rightarrow \dfrac{x}{5}=\dfrac{400}{5}=80$
And the cost of a single pencil is : $\Rightarrow \dfrac{y}{30}=\dfrac{600}{30}=20$

So, the correct answer is “Option C”.

Note: Here, we have used the selling price formula to find out the profit you can also directly go with the percent values given in the question such as for pens: $15\%\text{ of }x=0.15x$ , similarly for pencils we will get: $10\%\text{ of }y=0.1y$. And then follow the same procedure as shown above.