Question

Two cows were sold for Rs. $12,000$ each, one at a gain of $20\% $ and the other at a loss of $20\% $. Find the loss or gain in the entire transaction.

Answer 1

Hint: If the selling price of two items are the same, one sells at a profit of $x\% $ and another at a loss of $x\% $. Then in such cases, there is always a loss and this overall loss is equal to $\dfrac{{{x^2}}}{{100}}\% $.

Complete step-by-step answer:
Given, selling price (S.P.) of each cow= Rs. $12,000$
We know that if the selling price of two items are same, one sell at a profit of $x\% $ and another at a loss of $x\% $. Then in such cases, there is always a loss and this overall loss is equal to $\dfrac{{{x^2}}}{{100}}\% $.
Here, two cows were sold for Rs. $12,000$ each, one sell at a profit of $20\% $ and another at a loss of$20\% $.
 Therefore, overall loss %$ = \dfrac{{{{\left( {20} \right)}^2}}}{{100}}$$ = \dfrac{{400}}{{100}} = 4\% $
Given, two cows were sold for Rs. $12,000$ each.
Therefore, total selling price (S.P.) = $12,000 \times 2$= Rs. $24,000$
Total cost price = \[ = \left( {\dfrac{{100}}{{100 - Loss\% }}} \right) \times S.P.\]
\[ = \left( {\dfrac{{100}}{{100 - 4}}} \right) \times 24,000\]
\[ = \left( {\dfrac{{100}}{{96}}} \right) \times 24,000\]
\[ = \left( {\dfrac{{25}}{{24}}} \right) \times 24,000\]
\[ = 25 \times 1,000\]
 = Rs.\[25,000\]
Loss = Cost price $ - $ Selling price
=$25,000 - 24,000$
= Rs. $1,000$

Hence, loss in the entire transaction will be Rs. $1,000$.

Note: An another approach to solve this question is described below:
Given, selling price (S.P.) of each cow= Rs. $12,000$
One cow is sold at a gain of $20\% $, then cost price\[ = \left( {\dfrac{{100}}{{100 + Gain\% }}} \right) \times S.P.\]
\[ = \left( {\dfrac{{100}}{{100 + 20}}} \right) \times 12,000\]
\[ = \left( {\dfrac{{100}}{{120}}} \right) \times 12,000\]
\[ = 100 \times 100\]
 =Rs. \[10,000\]
Other cow is sold at a loss of $20\% $, then cost price\[ = \left( {\dfrac{{100}}{{100 - Loss\% }}} \right) \times S.P.\]
\[ = \left( {\dfrac{{100}}{{100 - 20}}} \right) \times 12,000\]
\[ = \left( {\dfrac{{100}}{{80}}} \right) \times 12,000\]
\[ = 100 \times 150\]
 = Rs. \[15,000\]
Therefore, total cost price = $10,000 + 15,000$= Rs. $25,000$
Given, two cows were sold for Rs. $12,000$ each.
Therefore, total selling price = $12,000 \times 2$= Rs. $24,000$
Here the selling price is less than the cost price, so there is a loss in the entire transaction.
Loss = Cost price $ - $ Selling price
=$25,000 - 24,000$
= Rs. $1,000$
Hence, loss in the entire transaction will be Rs. $1,000$.