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The measured physical quantities are $a$, $b$, $c$ and $d$ with percentage errors $1\%$, $3\%$, $2\%$ and $4\%$ respectively. If $P = \dfrac{a^3 b^2}{\sqrt{c}\, d}$, then the percentage error in $P$ is:
Options
1
$13\%$
2
$14\%$
3
$15\%$
4
$12\%$
Correct Answer
$13\%$
Solution
1

$P = \dfrac{a^3 b^2}{c^{1/2}\, d}$

Percentage error formula:

$$\frac{\Delta P}{P}\% = 3\frac{\Delta a}{a}\% + 2\frac{\Delta b}{b}\% + \frac{1}{2}\frac{\Delta c}{c}\% + \frac{\Delta d}{d}\%$$
2
$$= 3(1) + 2(3) + \frac{1}{2}(2) + 1(4)$$$$= 3 + 6 + 1 + 3 = \boxed{13\%}$$

(Official NEET 2025 answer key: 13%)

Multiply power × %error for each variable, then add all.
$3(1)+2(3)+\frac{1}{2}(2)+1(3) = 13\%$
Theory: Measurement
1. Error Propagation Rules

For $P = x^a y^b / z^c$: relative error $\dfrac{\Delta P}{P} = a\dfrac{\Delta x}{x} + b\dfrac{\Delta y}{y} + c\dfrac{\Delta z}{z}$. The coefficient of each term = absolute value of the power of that variable. For addition/subtraction $P = x \pm y$: absolute errors add: $\Delta P = \Delta x + \Delta y$. For multiplication/division: relative errors add. For power $P = x^n$: $\Delta P/P = n(\Delta x/x)$. Negative powers (division) also contribute positively to error.

2. Types of Errors

Random error: unpredictable fluctuations. Reduced by averaging many measurements. Mean absolute error $\overline{\Delta a} = \frac{1}{n}\sum|a_i - \bar{a}|$. Systematic error: consistent bias. Same direction every measurement. Examples: zero error, calibration error. Cannot be reduced by averaging. Must be identified and corrected. Gross errors: blunders (misreading instrument). Eliminated by careful repetition. Limiting error: maximum possible error in a single measurement = least count of instrument.

3. Significant Figures in Calculations

Addition/subtraction: result has same number of decimal places as the least precise operand. Example: $2.5 + 0.032 = 2.5$ (not 2.532). Multiplication/division: result has same number of significant figures as the least precise operand. Example: $2.5 \times 3.14 = 7.9$ (2 sig figs). Rounding: if next digit $\geq 5$, round up; if $< 5$, truncate; if exactly 5 and preceding digit is odd: round up; if even: leave. Significant zeros: $1.00 \times 10^3$ has 3 sig figs; $1000$ has ambiguous sig figs.

4. Dimensional Analysis

Dimensions: $[M]$, $[L]$, $[T]$, $[A]$, $[K]$, $[mol]$, $[cd]$. Every valid physical equation must be dimensionally homogeneous (same dimensions on both sides). Uses: (1) Check equation validity (necessary but not sufficient). (2) Derive relations (limited by unknown dimensionless constants). (3) Convert units between systems. (4) Find dimensions of physical constants. Example: $[G] = M^{-1}L^3T^{-2}$ from $F = GMm/r^2$. Cannot distinguish $\sin\theta$, $\cos\theta$, $e^x$ (all dimensionless).

5. Instruments and Least Count

Metre scale: LC $= 1$ mm. Vernier callipers: LC $= 1$ MSD $-$ 1 VSD (usually 0.1 mm or 0.02 mm). Screw gauge: LC $= $ pitch/circular scale divisions (usually 0.01 mm). Stopwatch (digital): LC $= 0.01$ s. Thermometer: LC $= 0.1°$C (lab). Maximum error per reading $\approx \pm$LC/2 (estimated midpoint). For better precision: use instrument with smaller LC. Accuracy vs precision: accuracy = closeness to true value; precision = reproducibility.

6. Estimation and Order of Magnitude

Order of magnitude: power of 10 that represents the quantity. Express number in form $a \times 10^n$ where $1 \leq a < 10$. Order = $n$. Useful for quick checks. Examples: size of atom $\sim 10^{-10}$ m, hydrogen nucleus $\sim 10^{-15}$ m, Earth radius $\sim 10^7$ m, Sun-Earth distance $\sim 10^{11}$ m. Fermi estimation: back-of-envelope calculation using reasonable assumptions. Example: "How many piano tuners in a city?" (famous Fermi problem). Used in NEET for very large/small quantity comparisons.

7. Vernier and Screw Gauge Zero Errors

Vernier zero error: when jaws closed, Vernier 0 not at main scale 0. Positive zero error: Vernier 0 to the right $\Rightarrow$ instrument reads high $\Rightarrow$ subtract. Negative zero error: Vernier 0 to left $\Rightarrow$ reads low $\Rightarrow$ add. Correction: True $=$ Observed $-$ Zero error. Screw gauge: zero error when flat faces touching. Positive: thimble 0 below reference line $\Rightarrow$ reads high $\Rightarrow$ subtract. Negative: thimble 0 above line $\Rightarrow$ reads low $\Rightarrow$ add. Backlash error: loose threads in screw gauge, avoided by always rotating in same direction.

8. Statistical Treatment of Errors

For $n$ measurements $a_1, a_2, ..., a_n$: Mean $\bar{a} = \sum a_i/n$. Mean absolute error $\Delta\bar{a} = \sum|a_i-\bar{a}|/n$. Relative error $= \Delta\bar{a}/\bar{a}$. Standard deviation $\sigma = \sqrt{\sum(a_i-\bar{a})^2/(n-1)}$ (for small sample). Standard error of mean $= \sigma/\sqrt{n}$ (reduces with more measurements). Result expressed as $\bar{a} \pm \Delta\bar{a}$. Confidence interval: 68% of measurements within $\bar{a} \pm \sigma$ for Gaussian distribution. NEET uses simple absolute/relative/percentage error calculation, not advanced statistics.

Frequently Asked Questions
1. Why do we add percentage errors even for division? ⌄
For $P = a/b$: relative error $\Delta P/P = \Delta a/a + \Delta b/b$ (NOT minus). This seems counterintuitive — why does dividing by $b$ add $b$'s error? Because we're calculating the MAXIMUM possible error. If $a$ is at its maximum ($a + \Delta a$) and $b$ is at its minimum ($b - \Delta b$): $P_{max} = (a+\Delta a)/(b-\Delta b)$. Subtracting true $P = a/b$: maximum error $\approx \Delta a/b + a\Delta b/b^2 = P(\Delta a/a + \Delta b/b)$. Both terms are positive — they add.
2. What does power 3 for variable $a$ mean in error formula? ⌄
In $P = a^3 b^2/(c^{1/2}d)$: the power of $a$ is 3. Relative error rule: $\Delta P/P = 3(\Delta a/a) + ...$. Why? $\ln P = 3\ln a + 2\ln b - \frac{1}{2}\ln c - \ln d$. Taking differential: $dP/P = 3(da/a) + 2(db/b) - \frac{1}{2}(dc/c) - (dd/d)$. For maximum error: take absolute values of all terms and add: $|\Delta P/P| = 3|\Delta a/a| + 2|\Delta b/b| + \frac{1}{2}|\Delta c/c| + |\Delta d/d|$.
3. What if two measured quantities have the same variable? ⌄
If the same measured quantity appears in both numerator and denominator of a formula: it depends on the algebraic structure. For $P = a^2/a = a$: $\Delta P/P = \Delta a/a$ (not $2\Delta a/a + \Delta a/a$). The powers cancel. Always simplify the formula before applying error formula. If different measurements of the same quantity are used (e.g., measuring radius twice): treat as independent measurements and their errors do not cancel.
4. What is the difference between error in sum vs error in product? ⌄
Sum/difference $P = a + b$ (or $a - b$): ABSOLUTE errors add: $\Delta P = \Delta a + \Delta b$. Product/quotient $P = ab$ (or $a/b$): RELATIVE errors add: $\Delta P/P = \Delta a/a + \Delta b/b$. This distinction is critical. For $P = a + b$, even if $a >> b$, $b$'s absolute error still contributes. Example: measuring the length $L = L_1 + L_2$: $\Delta L = \Delta L_1 + \Delta L_2$. For $P = L_1 \times L_2$ (area): $\Delta P/P = \Delta L_1/L_1 + \Delta L_2/L_2$.
5. How is percentage error different from relative error? ⌄
Relative error $= \Delta x/x$ (dimensionless fraction). Percentage error $= (\Delta x/x) \times 100\%$. In error propagation formulas, when you write $\Delta P/P = 3(\Delta a/a) + ...$: if you express all terms in percentage form, the equation becomes $\%(\Delta P/P) = 3\%(\Delta a/a) + ...$. The formula looks the same whether using relative or percentage form (since multiplying both sides by 100 preserves the equality). In this problem: $3\%(1) + 2\%(3) + \frac{1}{2}\%(2) + 1\%(3) = 3 + 6 + 1 + 3 = 13\%$.
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