A retailer sells two laptop models at an identical selling price. On the first laptop, she makes a profit of 15%, while on the second, she incurs a loss of 15%. Determine her overall percentage gain or loss on the entire transaction.

The correct answer is 2.25% loss.

When two items are sold at the same identical selling price, one at a profit of x% and the other at a loss of x%, the overall transaction always results in a net loss. This can be solved using both a standard shortcut formula and a detailed step-by-step cost price derivation.

Method 1: Using the Shortcut Formula

For any transaction where two items have the same selling price and equal percentage gain and loss (x%), the overall percentage loss is given by:

$\text{Overall Percentage Loss}=\frac{x^2}{100}\%$

Given that
$x=15$
:

$\text{Overall Percentage Loss}=\frac{{15}^2}{100}=\frac{225}{100}=2.25\%$

Method 2: Step-by-Step Derivation

Let the identical selling price (SP) of each laptop be
$S$
.

Step 1: Calculate the Cost Price of the first laptop
The first laptop is sold at a 15% profit.

$\text{SP}={\text{CP}}_1\times \left(1+\frac{15}{100}\right)=1.15\times {\text{CP}}_1$

${\text{CP}}_1=\frac{S}{1.15}=\frac{20S}{23}$

Step 2: Calculate the Cost Price of the second laptop
The second laptop is sold at a 15% loss.

$\text{SP}={\text{CP}}_2\times \left(1-\frac{15}{100}\right)=0.85\times {\text{CP}}_2$

${\text{CP}}_2=\frac{S}{0.85}=\frac{20S}{17}$

Step 3: Find Total Selling Price and Total Cost Price

$\text{Total SP}=S+S=2S$

$\text{Total CP}={\text{CP}}_1+{\text{CP}}_2=\frac{20S}{23}+\frac{20S}{17}=20S\times \left(\frac{1}{23}+\frac{1}{17}\right)=20S\times \frac{40}{391}=\frac{800S}{391}$

Step 4: Compute the Net Percentage Loss
Since Total CP is greater than Total SP, there is a loss.

$\text{Net Loss}=\text{Total CP}-\text{Total SP}=\frac{800S}{391}-2S=\frac{18S}{391}$

$\text{Percentage Loss}=\left(\frac{\text{Net Loss}}{\text{Total CP}}\right)\times 100=\left(\frac{\frac{18S}{391}}{\frac{800S}{391}}\right)\times 100=\frac{18}{800}\times 100=2.25\%$

Therefore, the retailer incurs an overall percentage loss of 2.25% on the entire transaction.