Question

Peter sells two watches for ${\text{Rs}}{\text{.198}}$ each; gaining $20\% $ on one and losing $20\% $ on other. Find his gain % or loss % on the whole.

Answer 1

Hint: Initially, calculate the actual price of the watch then add the profit. Sale price is the sum of the cost and the profit and cost price is the difference of profit from the sale. Use the formula of gain or loss percentage.

Complete step-by-step solution:
We know that the profit and loss is studied in any institution, business, industry, etc. While the sale of a good, one can either gain or loss, which is generally calculated in terms of percentage. The profit is equal to the difference of selling price and cost price while the loss is equal to the difference of cost price and selling price.
In the above question, It is given that $20\% $ gain on watch$\left( {{w_1}} \right)$, $20\% $ loss suffered on watch $\left( {{w_2}} \right)$, and The selling price of each watch is ${\text{Rs}}{\text{.198}}$
We can write the Gain percentage formula as,
${\text{Gain\% }} = \dfrac{{{\text{Gain}}}}{{{\text{CP}}}} \times 100$
Here, the cost price of an object is ${\text{CP}}$
Similarly, we can write the loss percentage formula as,
${\text{loss\% }} = \dfrac{{{\text{loss}}}}{{{\text{CP}}}} \times 100$
Let us consider the cost price of watch 1 is ${\left( {{\text{CP}}} \right)_{w1}}$, the cost price of watch 2 is ${\left( {{\text{CP}}} \right)_{w2}}$, and the selling price of each the watches is ${\text{SP}}$.
Now, calculate the cost of first watch as,
 \[
  {\left( {{\text{CP}}} \right)_{w1}} = \left( {\dfrac{{{\text{SP}}}}{{100 + {\text{gain\% }}}}} \right) \times 100 \\
   = \left( {\dfrac{{198}}{{100 + 20}}} \right) \times 100 \\
   = \dfrac{{198}}{{120}} \times 100 \\
   = {\text{Rs}}{\text{.168}}
 \]
Similarly, we can calculate the cost price of the second watch as,
$
  {\left( {{\text{CP}}} \right)_{w2}} = \left( {\dfrac{{{\text{SP}}}}{{100 - {\text{loss\% }}}}} \right) \times 100 \\
   = \left( {\dfrac{{198}}{{100 - 20}}} \right) \times 100 \\
   = \dfrac{{198}}{{80}} \times 100 \\
   = {\text{Rs}}{\text{. }}247.5
 $
Now, calculate the total sale price of two watches as,
$
  {\text{selling price}} = 198 \times 2 \\
   = {\text{Rs}}{\text{. }}396
 $
Similarly, calculate the total cost price of two watches as,
\[
  {\text{cost price}} = {\left( {{\text{CP}}} \right)_{w1}} + {\left( {{\text{CP}}} \right)_{w2}} \\
   = 165 + 247.5 \\
   = {\text{Rs}}{\text{. }}412.5
 \]
Now, compare the total selling price and the total cost price, the total selling price is less than the cost price. So, Peter suffered loss.
Now, calculate the loss suffered by Peter as,
$
  {\text{loss}} = {\text{cost price}} - {\text{selling price}} \\
   = {\text{412}}{\text{.5}} - {\text{396}} \\
   = {\text{Rs}}{\text{.16}}{\text{.5}}
 $
Finally, we calculate the loss percentage as,
$
  {\text{loss}}\% = \left( {\dfrac{{{\text{loss}}}}{{{\text{cost price}}}}} \right) \times 100 \\
   = \dfrac{{16.5}}{{412.5}} \times 100\% \\
   = 4\%
 $

Therefore, Peter’s loss percentage is $4\% $.

Note: In the profit or loss problem, please keep in mind that, if the cost price is greater than the selling price than loss will occur or if the selling price is greater than the cost price than the profit will occur.