The Logic Behind the Math: Square-Law Sensitivity

The BMI formula is $BMI = Mass / Height^2$.Because the height is squared, the result is highly sensitive to height changes.If you increase height by $10\%$, the BMI doesn't drop by $10\%$; it drops by approximately $19\%$ (since $1.1^2 = 1.21$, and $1/1.21 \approx 0.82$). In JEE Physics, this sensitivity is called Relative Error Propagation.

Step-by-Step Solved Example (Metric)

Problem: A student weighs 70 kg and is 175 cm tall. Calculate their BMI.

• Step 1: Identify Variables. $W = 70$, $H = 175$.
• Step 2: Convert Units. Height must be in meters. $175 / 100 = 1.75m$.
• Step 3: Square the Height. $1.75 \times 1.75 = 3.0625$.
• Step 4: Divide Mass by $H^2$. $70 / 3.0625$.
• Step 5: Solve. $70 / 3.0625 \approx 22.85$.
• Step 6: Interpret. 22.85 falls in the "Normal" range.

Alternative Methods: The Imperial Shortcut

If you are given weight in pounds ($lbs$) and height in inches ($in$):$$BMI = \frac{Weight(lbs)}{Height(in)^2} \times 703$$The constant $703$ is a conversion factor that accounts for $(lb \to kg)$ and $(in \to m)^2$. In exams, remembering this constant is faster than doing four separate unit conversions.

Exam Trap Alert: The Unit Conversion Slip

The most common error in NEET/JEE for this type of ratio is forgetting to convert cm to m before squaring.

Trap: Squaring 175 ($30,625$) instead of 1.75 ($3.06$). This leads to a BMI of $0.002$, which is physically impossible for a human. Always perform a Sanity Check on your final number.

Practice Problem (NEET Physiology/Physics)

Question: If a person's height is measured with a $2\%$ error and their weight with a $1\%$ error, what is the maximum percentage error in their calculated BMI?Hint: Use the error formula $\Delta Z/Z = \Delta A/A + 2(\Delta B/B)$.