Question

By selling $100$ pencils a shopkeeper gains the S.P of $20$ pencils. His gain percent is
A. $25\%$
B. $20\%$
C. $15\%$
D.$12\%$

Answer 1

Hint: To solve this problem, we will first assume the Selling price and Cost Price of on pencil. Then we will calculate the total Selling Price and Cost price of $100$ pencils. From the Selling Price and Cost Price of $100$ pencils we will find the profit for the shopkeeper. From the given data they mentioned the gain as Selling Price of $20$pencils. Here we will equate both the values to get an equation which involves the relation between the selling price and cost price of the pencil. From this relation we will calculate the Profit Percentage.

Complete step by step answer:
Let the Selling Price of the one pencil is $\text{S}\text{.P}=x$
     Cost Price of the one pencil is $\text{C}\text{.P}=y$
Now the Selling Price of $100$ pencils are $100\left( \text{S}\text{.P} \right)=100x$
       Cost Price of $100$ pencils are $100\left( \text{C}\text{.P} \right)=100y$
Given that, the shopkeeper will gain profit by selling $100$ pencils. We know that the profit gained is equal to the difference between the Selling Price and Cost Price. So, the profit gained by the shopkeeper by selling $100$ pencils are
$\begin{align}
  & \text{P}=\text{S}\text{.P}-\text{C}\text{.P} \\
 & \text{=100}x-100y \\
 & =100\left( x-y \right)
\end{align}$
But in the problem, they mentioned that the profit gained by the shopkeeper is equal to S.P of $20$ pencils. So, the Selling price of $20$ pencils are $\text{S}\text{.}{{\text{P}}_{20}}=20x$, hence
$\begin{align}
  & \text{P}=\text{S}\text{.}{{\text{P}}_{20}} \\
 & 100\left( x-y \right)=20x \\
 & 5x-5y-x=0 \\
 & 4x=5y.....\left( \text{i} \right)
\end{align}$
Calculating the percentage of profit by using the formula Percentage of Profit $=\dfrac{\text{Profit}}{\text{Cost Price}}\times 100$
$\begin{align}
  & \text{P }\!\!\%\!\!\text{ }=\dfrac{100\left( x-y \right)}{100y}\times 100 \\
 & =\dfrac{x-y}{y}\times 100 \\
 & =100\left( \dfrac{x}{y}-1 \right)
\end{align}$
From equation $\left( \text{i} \right)$ substituting the value of $\dfrac{x}{y}$ as $\dfrac{5}{4}=1.25$, then
$\begin{align}
  & \text{P }\!\!\%\!\!\text{ }=100\left( \dfrac{x}{y}-1 \right) \\
 & =100\left( 1.25-1 \right)\% \\
 & =100\left( 0.25 \right)\% \\
 & =25\%
\end{align}$

So, the correct answer is “Option A”.

Note: We can also simply solve the above problem by using the following method.
Given that, the profit gained by the shopkeeper by selling $100$ pencils are equal to the Selling Price of $20$ Pencils, then
Selling Price of $100$ pencils $-$ Cost Price of $100$ pencils $=$ Selling Price of $20$ Pencils
Selling Price of $100$ pencils $-$ Selling Price of $20$ Pencils $=$ Cost Price of $100$ pencils
Selling Price of $80$Pencils $=$ Cost Price of $100$ pencils
If the Cost Price of pencil is Rs.$1$
Then S.P of $80$Pencils $=100$
C.P of $80$Pencils $=80$, then
$\begin{align}
  & \text{Profit}\%=\dfrac{\text{S}\text{.P of 80 Pencils }-\text{ C}\text{.P of 80 Pencils}}{\text{C}\text{.P of 80 Pencils}}\times 100 \\
 & =\dfrac{100-80}{80}\times 100 \\
 & =\dfrac{20}{80}\times 100 \\
 & =25\%
\end{align}$