The Logic Behind the Math: Relative Change

The logic of ROI is rooted in the concept of Relative Error or Fractional Change $(\Delta x / x)$. To find the ROI, we take the change in value, divide it by the starting point, and multiply by 100 to get a percentage.Mathematically: $ROI = \frac{\Delta V}{V_{initial}} \times 100$.

Step-by-Step Solved Example

Problem: A student spends ₹5,000 on a specialized JEE crash course. After the course, they secure a scholarship worth ₹12,500. What is the ROI of the course?

• Step 1: Identify Initial Investment ($P$). $P = 5,000$.
• Step 2: Identify Final Value ($G$). $G = 12,500$.
• Step 3: Calculate Net Gain. $Gain = G - P = 12,500 - 5,000 = 7,500$.
• Step 4: Form the Ratio. $Ratio = 7,500 / 5,000$.
• Step 5: Simplify. $75/50 = 3/2 = 1.5$.
• Step 6: Convert to Percentage. $1.5 \times 100 = 150\%$.

Alternative Methods: The "Multiplier" Approach

Instead of calculating the gain first, divide the Final Value by the Initial Value.$12,500 / 5,000 = 2.5$.Subtract 1 from the result ($2.5 - 1 = 1.5$) and multiply by 100. This "Multiplier Method" is much faster for mental math during exams.

Exam Trap Alert: The "Total Cost" Oversight

In many complex problems, students forget to include hidden costs in the initial investment.

Example: If you buy a machine for ₹10,000 but spend ₹2,000 on transport and ₹1,000 on installation, your $V_{initial}$ is ₹13,000, not ₹10,000. In JEE Physics, this is like forgetting to include the internal resistance of a battery when calculating total circuit resistance.

Practice Problem (Competitive Level)

Question: An investor buys two stocks, A and B. Stock A rises by 20% and Stock B falls by 10%. If the investor put twice as much money into Stock A as Stock B, what is the total ROI of the portfolio?Hint: Use weighted averages. $ROI_{total} = \frac{w_1r_1 + w_2r_2}{w_1 + w_2}$.