Question

Two radios were purchased at $Rs.99000$ each on one she gains $10\% $ and other her losses $10\% $ find the total ${\text{P % or L% }}{\text{.}}$

Answer 1

Hint- In this question we have to calculate cost price first. Then we will find the gain percentage and loss percentage using the formula.
${\text{Loss % = }}\dfrac{{{\text{total loss}} \times {\text{100}}}}{{{\text{cost price}}}}$.

Complete step-by-step solution -
Given that two radios purchased at $Rs{\text{ }}99000$
For the first gain $10\% $
As we know the formula for gain percentage
${\text{Gain% = }}\dfrac{{{\text{Gain}} \times {\text{100}}}}{{{\text{Cost price}}}}$
So, modifying the formula to find cost price
$
  {\text{Gain% = }}\dfrac{{{\text{Gain}} \times {\text{100}}}}{{{\text{Cost price}}}} \\
   \Rightarrow {\text{Gain% = }}\dfrac{{\left( {SP - CP} \right)}}{{{\text{CP}}}} \times {\text{100}} \\
   \Rightarrow {\text{Gain% = }}\left[ {\dfrac{{\left( {SP} \right)}}{{{\text{CP}}}} - \dfrac{{\left( {CP} \right)}}{{{\text{CP}}}}} \right] \times {\text{100 = }}\left[ {\dfrac{{\left( {SP} \right)}}{{{\text{CP}}}} - 1} \right] \times {\text{100}} \\
   \Rightarrow \dfrac{{\left( {SP} \right)}}{{{\text{CP}}}} - 1 = \dfrac{{gain\% }}{{100}} \\
   \Rightarrow \dfrac{{\left( {SP} \right)}}{{{\text{CP}}}} = \dfrac{{gain\% }}{{100}} + 1 = \dfrac{{100 + gain\% }}{{100}} \\
   \Rightarrow CP = \dfrac{{SP \times 100}}{{100 + gain\% }} \\
 $
Thus, the cost price is
$
  CP = \dfrac{{SP}}{{100 + gain\% }} \times 100 \\
   = \dfrac{{99000}}{{100 + 10}} \times 100 \\
   = 90000 \\
 $
For the second losses $10\% $
As we know the formula for loss percentage
\[{\text{Loss% = }}\dfrac{{{\text{Loss}} \times {\text{100}}}}{{{\text{Cost price}}}}\]
So, modifying the formula to find cost price
$
  {\text{Loss% = }}\dfrac{{{\text{Loss}} \times {\text{100}}}}{{{\text{Cost price}}}} \\
   \Rightarrow {\text{Loss% = }}\dfrac{{\left( {CP - SP} \right)}}{{{\text{CP}}}} \times {\text{100}} \\
   \Rightarrow {\text{Loss% = }}\left[ {\dfrac{{\left( {CP} \right)}}{{{\text{CP}}}} - \dfrac{{\left( {SP} \right)}}{{{\text{CP}}}}} \right] \times {\text{100 = }}\left[ {1 - \dfrac{{\left( {SP} \right)}}{{{\text{CP}}}}} \right] \times {\text{100}} \\
   \Rightarrow 1 - \dfrac{{\left( {SP} \right)}}{{{\text{CP}}}} = \dfrac{{loss\% }}{{100}} \\
   \Rightarrow \dfrac{{\left( {SP} \right)}}{{{\text{CP}}}} = 1 - \dfrac{{loss\% }}{{100}} = \dfrac{{100 - loss\% }}{{100}} \\
   \Rightarrow CP = \dfrac{{SP \times 100}}{{100 - loss\% }} \\
 $
Thus, the cost price is
$
  CP = \dfrac{{SP}}{{100 - loss\% }} \times 100 \\
   = \dfrac{{99000}}{{100 - 10}} \times 100 \\
   = 110000 \\
 $
The total SP is
$SP = 99000 \times 2 = 198000$
The total cost price is
$CP = 110000 + 90000 = 20,0000$
The total loss is given by
$
  {\text{Total loss = CP - SP}} \\
  {\text{ = 200000 - 198000}} \\
  {\text{ = 2000}} \\
 $
The loss percentage is
$
  {\text{Loss% = }}\dfrac{{{\text{total loss}} \times {\text{100}}}}{{{\text{cost price}}}} \\
   = \dfrac{{2000 \times 100}}{{20,0000}} \\
   = 1\% \\
 $
Hence, the total loss is $1\% $

Note- The price at which a particular item or article is purchased, is called cost price or in short CP. The price at which a particular item is sold, is called its selling price or in short SP. If the selling price is more than the cost price, then the vendor is said to have a gain or profit, if not then loss. These basic definitions of SP and CP must be remembered during solving these types of questions.