Question

What percentage of mark price should be raised then after getting $2.5%$ discount will give $22.5%$ profit?
A. $21%$
B. $23%$
C. $25%$
D. $17%$

Answer 1

Hint: We find the relation between the hiked price and the discounted price. Using the given data, we find out those equations and form a linear equality. Then we solve the linear equation to find the hike percentage.

Complete step-by-step answer:
So, the shopkeeper who is selling the thing will hike the actual cost price to get the marked price so that even after giving $2.5%$ discount on the marked price, he will profit $22.5%$.
Let’s take two variables x and a where x is the cost price and a is the hike percentage on the cost price.
If P is the real principal value, M is the change in percentage of the value then the new value will be $P\left( 1+\dfrac{x}{100} \right)$. If the change is in a negative sense like loss or discount then we use a negative sign for M.
So, the seller hiked $a%$ on the value of the thing which is x.
So, the marked price will be $x\left( 1+\dfrac{a}{100} \right)$.
Now the seller is giving $2.5%$ discount on the hiked price.
So, the discounted price will be $x\left( 1+\dfrac{a}{100} \right)\left( 1-\dfrac{2.5}{100} \right)$ …..(i)
Even after giving a $2.5\%$ discount the seller profited $22.5%$.
The profit always is on the marked price which is x.
The selling price is $22.5%$ profited value on x.
This value is $x\left( 1+\dfrac{22.5}{100} \right)$ ……(ii)
We get that the selling price and the discounted price are the same. So, we can equate them.
Equating (i) and (ii) we get $x\left( 1+\dfrac{a}{100} \right)\left( 1-\dfrac{2.5}{100} \right)=x\left( 1+\dfrac{22.5}{100} \right)$.
We have x on both sides, $x\ne 0$. We eliminate x to get $\left( 1+\dfrac{a}{100} \right)\left( 1-\dfrac{2.5}{100} \right)=\left( 1+\dfrac{22.5}{100} \right)$.
Now we solve the linear equation to find the value of a.
\[\begin{align}
  & \left( 1+\dfrac{a}{100} \right)\left( 1-\dfrac{2.5}{100} \right)=\left( 1+\dfrac{22.5}{100} \right) \\
 & \Rightarrow \left( 1+\dfrac{a}{100} \right)\left( \dfrac{97.5}{100} \right)=\left( \dfrac{122.5}{100} \right) \\
 & \Rightarrow 1+\dfrac{a}{100}=\dfrac{122.5}{100}\times \dfrac{100}{97.5} \\
 & \Rightarrow 1+\dfrac{a}{100}=\dfrac{49}{39} \\
 & \Rightarrow \dfrac{a}{100}=\dfrac{49}{39}-1=\dfrac{10}{39} \\
 & \Rightarrow a=\dfrac{10\times 100}{39}=\dfrac{1000}{39}=25\dfrac{25}{39} \\
\end{align}\]
‘a’ is the hiked percentage on the cost price.
Now we need to find the hike percentage on the marked price.
The profit on the thing was $\left( \dfrac{22.5x}{100} \right)$. The marked price is $x\left( 1+\dfrac{a}{100} \right)$.
So, hike percentage on the marked price will be $\dfrac{\left( \dfrac{22.5x}{100} \right)}{x\left( 1+\dfrac{a}{100} \right)}\times 100$.
Solving the equation, we get $\dfrac{\left( \dfrac{22.5x}{100} \right)}{x\left( 1+\dfrac{a}{100} \right)}\times 100=\dfrac{22.5}{1+\dfrac{a}{100}}$
We put value of a in the equation to get \[\dfrac{22.5}{1+\dfrac{a}{100}}=\dfrac{22.5}{1+\dfrac{1000}{39}\times \dfrac{1}{100}}=\dfrac{22.5\times 39}{39+10}=17.9%\]
So, the hike percentage is $17$ (approx). The correct option is (D).

So, the correct answer is “Option D”.

Note: We need to remember we can use 100 as the value of x. the given data are all in percentage. So, taking 100 as principal eliminates that. Also, the principal value in any case is the least important thing as that gets eliminated in the solving of the equation.