Question

# A shopkeeper fixes the MP of an item $35\%$ above the CP. The percentage of discount allowed to gain $8\%$ is:A. $43\%$B. $30\%$C. $20\%$D. $31\%$

Hint:In this problem they give the percentage of item MP a shopkeeper fixes above the CP. We have to find the percentage of discount allowed to gain $8\%$. We know that, if the cost price of an object is $a$. If the price increase $b\%$, the price of the object will become $a \times \dfrac{{100 + b}}{{100}}$.The selling price of an object is the sum of the cost price and the gain amount. Here first we consider the cost price of the item in some variable. Using that we calculate the market price and selling price with the gain of $8\%$. Subtracting the both to find the discount. Then we can convert it into percentage. This is our required result.

It is given that; a shopkeeper fixes the MP of an item $35\%$ above the CP. We have to find the percentage of discount that the shopkeeper allowed to gain $8\%$.
Let us consider, the cost price of the item is $Rs.{\rm{ }}x$.
The market price of the item is $35\%$ above the cost price.
That is marked price of the given item is found using the formula $\dfrac{{100 + b}}{{100}} \times {\rm{cost price}}$ where b is the percentage of the marked price.
By using the formula the marked price of the item will be$\dfrac{{100 + 35}}{{100}} \times x = \dfrac{{135x}}{{100}}$.
Now, it is given that the shopkeeper wants to gain $8\%$.
Hence the selling price of the item is given using the formula $\dfrac{{100 + b}}{{100}} \times {\rm{cost price}}$ where b is the percentage of the gain.
By using the above formula we get,
The selling price of the item will be$\dfrac{{100 + 8}}{{100}} \times x = \,\,\dfrac{{108x}}{{100}}$.
Now we have to find the amount of discount using the formula,
${\rm{marked price - selling price = discount}}$
Therefore the discount amount of the item is$\dfrac{{135x}}{{100}} - \dfrac{{108x}}{{100}} = \dfrac{{27x}}{{100}}$.
Now we have to find the percentage of discount for that we use the following formula,
${\rm{discount\% = }}\dfrac{{{\rm{discount}}}}{{{\rm{marked price}}}} \times 100$
Therefore the percentage of discount is $\dfrac{{\dfrac{{27x}}{{100}}}}{{\dfrac{{135x}}{{100}}}} \times 100\%$
On solving the percentage we get,
$\dfrac{{\dfrac{{27x}}{{100}}}}{{\dfrac{{135x}}{{100}}}} \times 100\% = \dfrac{{27x}}{{100}} \times \dfrac{{100}}{{135x}} \times 100 = \dfrac{{100}}{5} = 20\%$
The percentage of discount offered to gain $8\%$ profit is $20\% .$

So, the correct answer is “Option C”.

Note:The problem can be solved by another process.
Let us consider, the cost price of the item is $Rs.{\rm{ }}100$.The marked price of the item is $35\%$ above the cost price.
So, the marked price of the item will be $Rs.{\rm{ }}135$.
Now, the shopkeeper wants to gain $8\%$.
So, the selling price of the item will be $Rs.{\rm{ }}100 + 8 = Rs.{\rm{ }}108$
So, he has to give the discount of $Rs.\,135 - 108 = Rs.{\rm{ }}27$
Then, the percentage of discount is $\dfrac{{27}}{{135}} \times 100\%$
Simplifying we get, the percentage of discount is $20\% .$