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A man sold two cycles for Rs 3000 each, gaining 20 percent on one and losing 20 percent on the other. Find his gain or loss percentage on the whole transaction?

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Hint- Here, we will be proceeding by using the formulas SP$ = {\text{CP}}\left[ {1 + \dfrac{{{\text{Profit percent}}}}{{100}}} \right]$ and SP$ = {\text{CP}}\left[ {1 - \dfrac{{{\text{Loss percent}}}}{{100}}} \right]$ in order to determine the cost price of each cycle. Then, we will compare the total CP and the total SP of both the cycles to find whether profit is occurring or loss is occurring.

Given, the selling price of two cycles are equal to Rs 3000
i.e., S.P of first cycle = S.P of second cycle = Rs 3000
Let the cost price of the first cycle and second cycle be x and y respectively.
i.e., CP of first cycle = x, CP of second cycle = y
Also, given by selling first cycle, there occurs a gain of 20 percent (i.e., Profit percent = 20 percent)
As we know that relation between selling price and cost price of any item when there is profit or gain occurs is given by
SP$ = {\text{CP}}\left[ {1 + \dfrac{{{\text{Profit percent}}}}{{100}}} \right]$ where SP is the selling price of the item and CP is the cost price of the item.
Using the above formula for the selling of first cycle, we get
So, SP of first cycle= CP of first cycle$\left[ {1 + \dfrac{{{\text{Profit percent}}}}{{100}}} \right]$
$
   \Rightarrow 3000 = x\left[ {1 + \dfrac{{20}}{{100}}} \right] \\
   \Rightarrow 3000 = x\left[ {\dfrac{{100 + 20}}{{100}}} \right] \\
   \Rightarrow 3000 = x\left[ {\dfrac{{120}}{{100}}} \right] \\
   \Rightarrow x = 3000\left[ {\dfrac{{100}}{{120}}} \right] = 2500 \\
 $
Therefore, the cost price of the first cycle is Rs 2500.
Also, given by selling second cycle, there occurs a loss of 20 percent (i.e., Loss percent = 20 percent)
As we know that relation between selling price and cost price of any item when loss occurs is given by
SP$ = {\text{CP}}\left[ {1 - \dfrac{{{\text{Loss percent}}}}{{100}}} \right]$ where SP is the selling price of the item and CP is the cost price of the item.
Using the above formula for the selling of first cycle, we get
So, SP of first cycle= CP of first cycle$\left[ {1 - \dfrac{{{\text{Loss percent}}}}{{100}}} \right]$
$
   \Rightarrow 3000 = y\left[ {1 - \dfrac{{20}}{{100}}} \right] \\
   \Rightarrow 3000 = y\left[ {\dfrac{{100 - 20}}{{100}}} \right] \\
   \Rightarrow 3000 = y\left[ {\dfrac{{80}}{{100}}} \right] \\
   \Rightarrow y = 3000\left[ {\dfrac{{100}}{{80}}} \right] = 3750 \\
 $
Therefore, the cost price of the first cycle is Rs 3750.
Now, the total cost price of both the cycles is obtained by adding the individual cost price of the two cycles.
i.e., Total CP of both the cycles = CP of first cycle + CP of second cycle = 2500+3750 = Rs 6250
Also, the total selling price of both the cycles is obtained by adding the individual selling price of the two cycles.
i.e., Total SP of both the cycles = SP of first cycle + SP of second cycle = 3000+3000 = Rs 6000
Clearly, the total CP is greater than the total SP. Hence, overall loss will occur.
Since, Loss percentage$ = \left[ {\dfrac{{{\text{CP}} - {\text{SP}}}}{{{\text{CP}}}}} \right] \times 100$ percent
Here, Loss percentage on the whole transaction$ = \left[ {\dfrac{{{\text{Total CP of both cycles}} - {\text{Total SP of both cycles}}}}{{{\text{Total CP of both cycles}}}}} \right] \times 100$
$ \Rightarrow $ Loss percentage on the whole transaction$ = \left[ {\dfrac{{{\text{6250}} - {\text{6000}}}}{{{\text{6250}}}}} \right] \times 100 = \left[ {\dfrac{{{\text{250}}}}{{{\text{6250}}}}} \right] \times 100 = 4$ percent
Hence, the loss percentage on the whole transaction is 4 percent.

Note- In this particular problem, the total cost price of both the cycles is greater than the total selling price of both the cycles that’s why loss is occurring. If we had a case with total selling price of both the cycles greater than the total cost price of both the cycles which means overall profit is occurring then the formula used would be Profit percentage$ = \left[ {\dfrac{{{\text{SP}} - {\text{CP}}}}{{{\text{CP}}}}} \right] \times 100$ percent.
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