GATE CS 2024 Set 2 Q7 — General Aptitude

We can solve this problem using two methods: a standard shortcut formula and a detailed step-by-step derivation.

Method 1: Using the Standard Formula

When two items are sold at the same selling price ($SP$), one at a profit of $x\%$ and the other at a loss of $x\%$, there is always a net loss. The net loss percentage is given by:

Given $x = 10$:

Method 2: Step-by-Step Derivation

Let the selling price ($SP$) of each item be $100$.

• For the first item (10% profit):
  $$CP_1 \times 1.10 = 100 \implies CP_1 = \frac{100}{1.1} = \frac{1000}{11} \approx 90.91$$
• For the second item (10% loss):
  $$CP_2 \times 0.90 = 100 \implies CP_2 = \frac{100}{0.9} = \frac{1000}{9} \approx 111.11$$

Now, calculate the total cost price ($CP$) and total selling price ($SP$):

• Total $SP = 100 + 100 = 200$
• Total $CP = \frac{1000}{11} + \frac{1000}{9} = 1000 \left( \frac{9 + 11}{99} \right) = \frac{20000}{99} \approx 202.02$

Since total $CP >$ total $SP$, there is a net loss:

Thus, the person made a total of 1% loss.