Question

Find the cost price when $SP = 2431$ and loss $ = 6\dfrac{1}{2}\% $

Answer 1

Hint : First, we shall analyze the given information so that we are able to solve the given problem. We are given $SP = 2431$ and loss $ = 6\dfrac{1}{2}\% $ .
We are asked to find the cost price.
We know the formula to calculate the loss percent. Since we are given the selling price and the loss percent, we need to change the formula of loss percent for our convenience to obtain the desired answer.

Formula used:
a) The formula to calculate the loss percent is as follows.
$loss\% = \dfrac{{CP - SP}}{{CP}} \times 100$
b) The formula to calculate the cost price is as follows.
$CP = \dfrac{{100}}{{100 - loss\% }} \times SP$

Complete step-by-step solution:
It is given that the selling price is $2431$ and loss $ = 6\dfrac{1}{2}\% $
We are asked to find the cost price.
It is given that loss $ = 6\dfrac{1}{2}\% $
We shall convert the mixed fraction of the above into an improper fraction.
If we are given $a\dfrac{b}{c}$ , then we can write it as $\dfrac{{ac + b}}{c}$ .
Hence,
$6\dfrac{1}{2} = \dfrac{{12 + 1}}{2}$
         $ = \dfrac{{13}}{2}$
Hence, we get
$loss\% = \dfrac{{13}}{2}$
We know the formula, $loss\% = \dfrac{{CP - SP}}{{CP}} \times 100$
We shall change the above formula as follows.
$loss\% = \dfrac{{CP - SP}}{{CP}} \times 100$
$ \Rightarrow \dfrac{{loss\% }}{{100}} = \dfrac{{CP}}{{CP}} - \dfrac{{SP}}{{CP}}$
$ \Rightarrow \dfrac{{loss\% }}{{100}} = 1 - \dfrac{{SP}}{{CP}}$
$ \Rightarrow \dfrac{{loss\% }}{{100}} - 1 \times \dfrac{1}{{SP}} = - \dfrac{1}{{CP}}$
$ \Rightarrow \dfrac{{loss\% - 100}}{{100}} \times \dfrac{1}{{SP}} = - \dfrac{1}{{CP}}$
$ \Rightarrow \dfrac{{loss\% - 100}}{{100}} \times \dfrac{1}{{SP}} = - CP$
$ \Rightarrow \dfrac{{100}}{{100 - loss\% }} \times SP = CP$
$CP = \dfrac{{100}}{{100 - loss\% }} \times SP$
This is the required formula to calculate the given problem.
Now, we shall apply the given values in the above formula.
$CP = \dfrac{{100}}{{100 - \dfrac{{13}}{2}}} \times 2431$
$ \Rightarrow CP = \dfrac{{100}}{{\dfrac{{200 - 13}}{2}}} \times 2431$
$ \Rightarrow CP = \dfrac{{200}}{{187}} \times 2431$
$ \Rightarrow CP = 200 \times 13$
$ \Rightarrow CP = 2600$
Hence the required cost price is $Rs.2600$

Note:Here, we are asked to calculate the cost price when the selling price and the loss percent are given.
We need to change the formula of the loss percent for our convenience. If we are asked to calculate the cost price when the selling price and the profit percent are given, we need to change the formula of profit percent for our convenience in the same way we did in this problem.