Question

A milkman sold two of his buffaloes for $20000$ each on one he made a gain of $5\% $ and on other he made a loss of $10\% $. Find his overall gain or loss %?

Answer 1

Hint: Find the selling price of both the buffaloes, then with the help of the given condition i.e. one at loss% and other at gain% find the cost price of both the buffaloes, combine the cost price for overall cost price, Now if $cp > sp$,then the overall transaction is in loss, if $sp > cp$, then the overall transaction is in profit.

Complete step-by-step answer:
Before going to the solution, there are some formulas we need to understand
$
   \Rightarrow Gain = sp - cp \\
   \Rightarrow Loss = cp - sp \\
   \Rightarrow Gain\% = \dfrac{{Gain}}{{cp}} \times 100 \\
   \Rightarrow Loss\% = \dfrac{{Loss}}{{cp}} \times 100 \\
  Here,\,Loss\,and\,gain\,are\,calculated\,on\,cp \\
$
where $ sp = selling\,price\,\,and\,cp = \cos t\,price$
So, in the question they mentioned, a milkman sold two of his buffaloes for $20000$ each
Hence, Selling price(sp) of two buffaloes are
$
   \Rightarrow Total\,selling\,price = 2 \times Selling\,price\,of\,two\,buffaloes \\
   \Rightarrow Total\,selling\,price = 2 \times 20000 \\
   \Rightarrow Total\,selling\,price = 40000........(1) \\
 $
Now it comes to cost price, lets calculate the cost price of each buffaloes in two different cases,
Case (1):
On selling a buffalo at a gain of $5\% $, the cost price of the buffalo is
$
   \Rightarrow Gain\% = \dfrac{{sp - cp}}{{cp}} \times 100 \\
   \Rightarrow 5 = \dfrac{{20000 - c{p_1}}}{{c{p_1}}} \times 100 \\
   \Rightarrow \dfrac{5}{{100}} = \dfrac{{20000 - c{p_1}}}{{c{p_1}}} \\
   \Rightarrow \dfrac{1}{{20}} = \dfrac{{20000 - c{p_1}}}{{c{p_1}}} \\
   \Rightarrow c{p_1} = 400000 - 20c{p_1} \\
   \Rightarrow 21c{p_1} = 400000 \\
   \Rightarrow c{p_1} = \dfrac{{400000}}{{21}} \\
   \Rightarrow c{p_1} = 19047.619........\left( 2 \right) \\
 $
Case (2):
On selling a buffalo at a loss of $10\% $, the cost price of the buffalo is
$
   \Rightarrow Loss\% = \dfrac{{cp - sp}}{{cp}} \times 100 \\
   \Rightarrow 10 = \dfrac{{c{p_2} - 20000}}{{c{p_2}}} \times 100 \\
   \Rightarrow \dfrac{{10}}{{100}} = \dfrac{{c{p_2} - 20000}}{{c{p_2}}} \\
   \Rightarrow \dfrac{1}{{10}} = \dfrac{{c{p_2} - 20000}}{{c{p_2}}} \\
   \Rightarrow c{p_2} = 10c{p_2} - 200000 \\
   \Rightarrow 200000 = 9c{p_2} \\
   \Rightarrow c{p_2} = \dfrac{{200000}}{9} \\
   \Rightarrow c{p_2} = 22222.222.........\left( 3 \right) \\
 $
So, from (1) selling price of both the buffaloes are $40000$ and from (2)&(3) the combined cost price$c{p_1} + c{p_2} = 22222.222 + 19047.619 = 41269.841$
So, here total selling price is $40000$and total cost price is $41269.841$
Here it is clearly indicated that $cp > sp$ hence there is a loss in the transaction
$
   \Rightarrow Loss = cp - sp \\
   \Rightarrow Loss = 41269.841 - 40000 \\
   \Rightarrow Loss = 1269.841 \\
 $
So, the overall loss in the transaction is $1269.841$.

Note: If, sp >cp then it indicates that it is gain/profit, when it is cp > sp then it indicates that it is lost. Always gain/profit and loss is calculated on cost price and not on selling price.