Numerical/problem-solving

1. TITLE

How to solve Maximum Percentage Error in powers in NEET — Step-by-step

2. PROBLEM TYPE INTRO

This numerical belongs to the Errors in Measurement (Uncertainty Combination) category. To solve it, we need the combination rule for products and quotients, which states that the error of a product or division is found by summing their relative errors. The most typical mistake students make is subtracting errors when dealing with division or forgetting to multiply the relative error by the power of the quantity.

3. MCQ

Problem:
A physical quantity $X$ is calculated from the relation $X = A^2 B^3$. If the percentage errors in the measurement of quantities $A$ and $B$ are $1\%$ and $2\%$ respectively, what is the maximum percentage error in the calculation of $X$?

Options:
(a) $8\%$
(b) $-4\%$
(c) $12\%$
(d) $3\%$

4. STEP-BY-STEP SOLUTION

SOLUTION

Given:
* Percentage error in $A$ $\left(\frac{\Delta A}{A} \times 100\%\right)$ = $1\%$
* Percentage error in $B$ $\left(\frac{\Delta B}{B} \times 100\%\right)$ = $2\%$

To Find:
* Maximum percentage error in $X$ $\left(\frac{\Delta X}{X} \times 100\%\right)$

Formula:
* Combination Rule for Products: The error of a product of experimental values is found by summing their relative errors.
* Explanation: Since powers are just repeated multiplication ($X = A \times A \times B \times B \times B$), we add the relative error of $A$ two times and the relative error of $B$ three times.

Step 1: Express the relation in terms of relative errors
Since errors are always combined by summing them to find the maximum possible error, the relative error for $X = A^2 B^3$ is written as:

$$\frac{\Delta X}{X} = 2\left(\frac{\Delta A}{A}\right) + 3\left(\frac{\Delta B}{B}\right)$$

 

Step 2: Convert to percentage errors
Multiply the entire equation by $100\%$ to work directly with the given percentage values:

$$\text{\% Error in } X = 2(\text{\% Error in } A) + 3(\text{\% Error in } B)$$

 

Step 3: Substitute the given values and calculate

$$\text{\% Error in } X = 2(1\%) + 3(2\%)$$

 

$$\text{\% Error in } X = 2\% + 6\%$$

 

$$\text{\% Error in } X = 8\%$$

 

Answer:
$8\%$ (Option a)

5. WHERE STUDENTS GO WRONG

  • ❌ Mistake 1: Subtracting errors instead of adding them. In both multiplication and division, relative errors are combined by summing them. Students often mistakenly calculate $2(1\%) - 3(2\%) = -4\%$.
  • ❌ Mistake 2: Multiplying the percentage errors. Instead of adding the relative errors of the components, students mistakenly multiply them together, resulting in $12\%$ or other incorrect magnitudes.
  • ❌ Mistake 3: Forgetting the powers. Students sometimes ignore the exponents entirely and just add the base percentage errors ($1\% + 2\% = 3\%$).

6. QUICK FORMULA REFERENCE

Formulas Used:
* Relative/Percentage Error — Use when converting an absolute uncertainty into a percentage: $\text{Relative Error} = \left(\frac{\text{Absolute Uncertainty}}{\text{Measured Value}}\right) \times 100\%$.
* Combination Rule (Product/Quotient/Power) — Use when calculating total error for a formula with multiplied or divided terms: Sum the relative errors of all variables.

7. SIMILAR PROBLEM HINT

"If NEET changes the values, the method stays the same. Practice with: $X = \frac{A^3}{B^2}$ (Remember, even if $B$ is in the denominator, its relative error is still added!)"

8. NCERT REFERENCE

Based on: NCERT Class 11, Chapter 1 (Units and Measurement).

9. CTA

More solved numericals → Physics and Measurement Complete Guide
Practice problems → [Mock exams]