Q.A physical quantity $X$ is related to four measurable quantities $a$, $b$, $c$ and $d$ as follows: $X = a^2 b^3 c^{5/2} d^{-2}$. The percentage error in the measurement of $a$, $b$, $c$ and $d$ are 1%, 2%, 3% and 4%, respectively. What is the percentage error in quantity $X$? If the value of $X$ calculated on the basis of the above relation is 2.763, to what value should you round off the result.

Absolute uncertainty on $X=2.763$: $\Delta X = 0.235 \times 2.763 \approx 0.65$ — comparable to the ones digit, so only the whole-number part of $X$ …

Percentage error in a power-law combination

For $X = a^m b^n c^p d^q$, errors combine as:

Here $X = a^2b^3c^{5/2}d^{-2}$, so $m=2$, $n=3$, $p=5/2$, $|q|=2$, and the given percentage errors are $1\%$, $2\%$, $3\%$, $4\%$ respectively.

Computing the total

Deciding how to round X = 2.763

The percentage error tells us the absolute uncertainty on the given value $X = 2.763$:

$\Delta X = \frac{23.5}{100} \times 2.763 \approx 0.649$ …

Method: Propagation of Errors (for products and powers)

This method uses the rule that for a quantity expressed as a product of powers, the relative error adds up with the powers as coefficients.

Steps

1. Write the general formula for relative error

If

$X = a^m \, b^n \, c^p \, d^q$

then the relative error in $X$ is:

Here, the signs of exponents don’t matter for error addition — we take absolute values of the powers.

2. Identify the powers and given percentage errors

From $X = a^2 \, b^3 \, c^{5/2} \, d^{-2}$:

• Power of $a$: $m = 2$
• Power of $b$: $n = 3$
• Power of $c$: $p = 5/2 = 2.5$
• Power of $d$: $q = -2$ → use $|q| = 2$

Given percentage errors:

• $\frac{\Delta a}{a} \times 100 = 1\%$
• $\frac{\Delta b}{b} \times 100 = 2\%$
• $\frac{\Delta c}{c} \times 100 = 3\%$
• $\frac{\Delta d}{d} \times 100 = 4\%$

3. Compute percentage error in $X$ …

🔍 Mistake 1: Forgetting the power rule for errors

The mistake:

Students often write the percentage error in $X$ as just the sum of the percentage errors in $a$, $b$, $c$, and $d$:

Why it’s wrong:

The formula for error propagation in multiplication/division with powers is:

Percentage error in $X$ = sum of (power × percentage error in each quantity)

So for $X = a^2 b^3 c^{5/2} d^{-2}$, the correct expression is:

How to avoid:

Always multiply each percentage error by its exponent (absolute value of the power) before adding. The sign of the exponent doesn’t matter — errors always add.

🔍 Mistake 2: Using the wrong sign for $d^{-2}$

The mistake:

Some students treat $d^{-2}$ as subtracting the error:

Why it’s wrong:

Errors always add in quadrature (or directly in this simplified method). The negative exponent means $d$ is in the denominator, but the uncertainty still adds to the total error.

How to avoid:

Remember: for $X = a^m b^n$,

Take absolute values of the powers.

🔍 Mistake 3: Arithmetic slip with $\frac{5}{2} \times 3\%$

The mistake:

Students often compute $\frac{5}{2} \times 3\%$ incorrectly — either as $7.5\%$ (correct) or mistakenly as $\frac{15}{2}\% = 7.5\%$ (same), but sometimes they write $\frac{5}{2} \times 3 = \frac{15}{2} = 7.5$ and forget the % sign.

How to avoid:

Write each term with the % sign clearly:

Then sum carefully:

So percentage error in $X$ = 23.5%.

🔍 Mistake 4: Rounding off the final value incorrectly

The mistake:

Given $X = 2.763$ and error $= 23.5\%$, students often round to 2 or 3 decimal places without considering the error magnitude.

Why it’s wrong:

The error determines the number of significant figures in the result.

Here, $23.5\%$ of $2.763$ is: …