Question

By selling toffees at a rate of the $20$ for $\text{Rs 10,}$ a man loses 4%.To gain 20%, how many toffees must be sold for $\text{Rs 10}$?
A. $13$ toffees
B. $16$ toffees
C. $19$ toffees
D. $22$ toffees

Answer 1

Hint: In the problem they have mentioned the selling price of $20$ toffees as $\text{Rs 10}$, from this we will calculate the selling price of one toffee. Further they mentioned the value of loss percentage, so we will first calculate the loss by using Loss $=$Cost Price $-$ Selling Price, by assuming the value of cost price as $x$, from the value of loss we will calculate the loss percentage from Loss Percentage$=\dfrac{\text{Loss}}{\text{Cost Price}}\times 100$. Here we will equate the obtained Loss Percentage to the given percentage to get the value of Cost Price of one toffee.
Now we have to calculate the selling price of the one toffee to gain $20\%$, by using
Profit $=$ Selling Price $-$ Cost Price, Profit Percentage $=\dfrac{\text{Profit}}{\text{Cost Price}}\times 100$. But the required value is the number of toffees to be sold for $\text{Rs 10}$to get the profit of $20\%$. Hence, we will calculate the number of toffees by using the selling price of one toffee and the given selling price $\text{Rs 10}$.

Complete step by step answer:
Given that,
Number of toffees is $n=20$
Selling Price of $20$ toffees is $\text{S}\text{.}{{\text{P}}_{20}}=10$
Selling Price of one toffee is $\text{S}\text{.}{{\text{P}}_{1}}=\dfrac{10}{20}=\dfrac{1}{2}$
Let the cost price of one toffee is $\text{C}\text{.}{{\text{P}}_{1}}=x$
Now the loss is
$\begin{align}
  & \Rightarrow \text{L}=\text{C}\text{.}{{\text{P}}_{1}}-\text{S}\text{.}{{\text{P}}_{1}} \\
 &\Rightarrow \text{L} =x-\dfrac{1}{2}
\end{align}$
Then the loss percentage is
$\begin{align}
  & \text{L }\!\!\%\!\!\text{ }=\dfrac{\text{Loss}}{\text{C}\text{.}{{\text{P}}_{1}}}\times 100 \\
 &\Rightarrow \text{L }\!\!\%\!\!\text{ }=\dfrac{x-\dfrac{1}{2}}{x}\times 100
\end{align}$
But in the problem, they mentioned the loss percentage as $4\%$, then
$\begin{align}
  & \dfrac{x-\dfrac{1}{2}}{x}\times 100=4 \\
 &\Rightarrow 100x-50=4x \\
 &\Rightarrow 100x-4x=50 \\
 &\Rightarrow x=\dfrac{50}{96} \\
 &\Rightarrow x =\dfrac{25}{48}
\end{align}$
So, the cost price of one toffee is Rs.$\dfrac{25}{48}$
Let the selling price of one toffee to get $20\%$ profit is $\text{S}\text{.}{{\text{P}}_{2}}=x$
Now the profit is
$\begin{align}
  & \text{P}=\text{S}\text{.}{{\text{P}}_{2}}-\text{C}\text{.}{{\text{P}}_{1}} \\
 &\Rightarrow \text{P} =x-\dfrac{25}{48}
\end{align}$
And the profit percentage is
$\begin{align}
  & \text{P }\!\!\%\!\!\text{ }=\dfrac{\text{Profit}}{\text{Cost Price}}\times 100 \\
 &\Rightarrow \text{P }\!\!\%\!\!\text{ }=\dfrac{x-\dfrac{25}{48}}{\dfrac{25}{48}}\times 100
\end{align}$
But in the problem, they mentioned the value of profit percentage as $20\%$, then
$\begin{align}
  & \dfrac{x-\dfrac{25}{48}}{\dfrac{25}{48}}\times 100=20 \\
 &\Rightarrow 100x-\dfrac{25\times 100}{48}=\dfrac{25\times 20}{48} \\
 &\Rightarrow 100x=\dfrac{25\times 100}{48}+\dfrac{25\times 20}{48} \\
 &\Rightarrow 100x=\dfrac{25}{48}\left( 100+20 \right) \\
 &\Rightarrow x=\dfrac{25\times 120}{48\times 100} \\
 &\Rightarrow x =\dfrac{5}{8}
\end{align}$
Hence the selling price of one toffee to get $20\%$ profit is Rs.$\dfrac{5}{8}$.
Let the number of toffees to be sold at Rs.$10$ to get the profit of $20\%$ is ${{n}_{1}}$, then the selling price of ${{n}_{1}}$ toffees is ${{n}_{1}}\times \dfrac{5}{8}$. But the actual selling price is Rs.$10$, then
$\begin{align}
  & {{n}_{1}}\times \dfrac{5}{8}=10 \\
 &\Rightarrow {{n}_{1}}=10\times \dfrac{8}{5} \\
 &\Rightarrow {{n}_{1}} =16
\end{align}$
Hence the person has to be sold $16$ toffees.

So, the correct answer is “Option B”.

Note: When we are equating the calculated percentage to the given percentage, don’t write the given percentage in decimals or $\dfrac{\text{Profit Percentage}}{100}$, since we are multiplying $100$ while calculating the percentage, so there is no need to divide the given percentage with $100$.