Question

By selling $90$ ball pens for Rs.$160$, a person loses $20\%$. How many ball pens should be sold for Rs.$96$, so as to have a profit of $20\%$.
A. $20$
B. $30$
C. $36$
D. $26$

Answer 1

Hint: The problem can be solved by dividing it into two parts in order to get a better clarity. The first part is to find the Cost price of each ball pen and from that value we will assume the Cost Price of the required number of pens. In part two we will find the profit percentage from the value of Cost price of the required number of pens which is obtained in part one, the value of Selling Price given in the problem. Here we will get an equation for the profit and percentage of profit. In the problem they mentioned the percentage of profit, so we can equate them to get the number of required ball pens.

Complete step by step answer:
Given that the Selling Price of $90$ ball pens $=$ Rs.$160$,
Loss percentage $=20\%$
Let the Cost Price of $90$ ball pens are $x$, then loss is
$\begin{align}
  & \text{Loss}=\text{C}\text{.P}-\text{S}\text{.P } \\
 & =x-160
\end{align}$
Now the loss percentage is
$\begin{align}
  & \text{Loss Percentage}=\dfrac{\text{Loss}}{\text{Cost Price}}\times 100 \\
 & =\dfrac{x-160}{x}\times 100
\end{align}$
But we have loss percentage as $20\%$, then
$\begin{align}
  &\Rightarrow \dfrac{x-160}{x}\times 100=20 \\
 & \Rightarrow \left( x-160 \right)100=20x \\
 & \Rightarrow 100x-20x=160\times 100 \\
 &\Rightarrow 80x=160\times 100 \\
 &\Rightarrow x=\dfrac{160\times 100}{80} \\
 &\Rightarrow x =200
\end{align}$
Hence the cost price of $90$ ball pens is Rs.$200$, so the Cost Price of one ball pen is $\dfrac{200}{90}=\dfrac{20}{9}$.
Now we will assume that the number pens to be sold for a selling price of Rs.$96$ to get a profit of $20\%$ as $x$. Now the Cost price of $x$ ball pens is
$\text{C}\text{.}{{\text{P}}_{x}}=\dfrac{20}{9}x$
Given that Selling Price is $\text{S}\text{.}{{\text{P}}_{x}}=96$ and Profit is
$\begin{align}
  & {{\text{P}}_{x}}=\text{S}\text{.P}-\text{C}\text{.P} \\
 & \text{=}96-\dfrac{20}{9}x
\end{align}$
And the Profit Percentage is
$\begin{align}
  & {{\text{P}}_{x}}\%=\dfrac{\text{Profit}}{\text{Cost Price}}\times 100 \\
 & =\dfrac{96-\dfrac{20}{9}x}{\dfrac{20}{9}x}\times 100
\end{align}$
But we have profit percentage as $20\%$, then
$\begin{align}
  &\Rightarrow \dfrac{96-\dfrac{20}{9}x}{\dfrac{20}{9}x}\times 100=20 \\
 & \Rightarrow \left( 96-\dfrac{20}{9}x \right)100=20\times \dfrac{20}{9}x \\
 &\Rightarrow 96-\dfrac{20}{9}x=\dfrac{4x}{9} \\
 &\Rightarrow 96=\dfrac{20}{9}x+\dfrac{4}{9}x \\
 &\Rightarrow \dfrac{24}{9}x=96 \\
 &\Rightarrow x=\dfrac{96\times 9}{24} \\
 & \Rightarrow x=36
\end{align}$

So the person has to sell $36$ ball pens with selling price Rs.$96$ to get $20\%$ profit.

Note: For this kind of Problems we need to know the following data.
Cost Price (CP): The cost price is the amount of money spent on making the product. For example, if Leela is making cotton skirts, the amount of money spent on getting the cotton fabric, buttons, and other materials are referred to as the cost price.
Selling Price (SP): After investing in making a product, a businessman always aims to earn more than what he spent. So, in this case, Leela is selling her skirts at a price above her cost price. The cost at which the product is sold is called Selling Price. It is the price the customer (buyer) pays for the product.
Profit (P): In math, profit is the difference between the selling price and the cost price. But only if the SP > CP, then there is a profit.
Loss (L): The loss is the difference between the selling price and the cost price. But only if the CP > SP, then there is a loss. Loss is bad for business as the businessman loses a part of the money he spent on making the product. Loss can also be defined as a negative profit.
Profit percentage: We can write profit percentage as
Profit Percentage $=\dfrac{\text{Profit}}{\text{Cost Price}}\times 100$
Loss Percentage: We can use below formula to calculate the loss percentage
Loss Percentage $=\dfrac{\text{Loss}}{\text{Cost Price}}\times 100$