JEE PYQ: Units & Measurements - Question ID eeab58ef8206 (JEE Main 2025)

Core Formula & Concept:

In the chapter Units & Measurements, one of the key concepts is significant figures and how to report the result of a calculation when the input values have a certain number of significant digits.

When multiplying or dividing measured quantities, the result must be reported with the same number of significant figures as the input value that has the fewest significant figures. This rule ensures that the precision of the result is not overstated.

In the given expression: $y = \frac{32.3 \times 1125}{27.4}$ each number has the following significant figures:

• $32.3$ → 3 significant figures
• $1125$ → 4 significant figures
• $27.4$ → 3 significant figures

Since the least number of significant figures among the inputs is 3, the final result must be reported with 3 significant figures.

Step-by-Step Derivation:

Let’s compute the value of $y$ step-by-step while keeping track of significant figures.

1. Numerator Calculation: $32.3 \times 1125$ First, multiply $32.3$ and $1125$: $32.3 \times 1125 = 32.3 \times (1000 + 100 + 20 + 5)$ Breaking it down: $$32.3 \times 1000 = 32300 \\ 32.3 \times 100 = 3230 \\ 32.3 \times 20 = 646 \\ 32.3 \times 5 = 161.5$$ Adding them together: $$32300 + 3230 = 35530 \\ 35530 + 646 = 36176 \\ 36176 + 161.5 = 36337.5$$ So, the numerator is $36337.5$.
2. Division Step: Now, divide the numerator by the denominator $27.4$: $y = \frac{36337.5}{27.4}$ Performing the division: $$27.4 \times 1300 = 35620 \\ 36337.5 - 35620 = 717.5 \\ 27.4 \times 20 = 548 \\ 717.5 - 548 = 169.5 \\ 27.4 \times 6 = 164.4 \\ 169.5 - 164.4 = 5.1 \\$$ So, the result is approximately: $y \approx 1300 + 20 + 6 = 1326$ More precisely, using a calculator: $y = 1326.1861313868613 \quad (\text{exact computation})$
3. Applying Significant Figures Rule: The input with the fewest significant figures is $32.3$ and $27.4$, each having 3 significant figures. Thus, the final result must be rounded to 3 significant figures.
   The computed value is $1326.186...$. Rounding this to 3 significant figures:
   • The first three significant digits are $1$, $3$, and $2$.
   • The fourth digit is $6$, which is $\geq 5$, so we round up the third digit ($2$) to $3$.
   • The remaining digits are replaced with zeros to maintain the magnitude.
   Thus: $y = 1330$

Common Traps & Exam Tip:

Students often make the following mistakes in such questions:

• Ignoring Significant Figures: Many compute the exact value ($1326.186$) and select option A or C without rounding to the correct number of significant figures.
• Incorrect Rounding: Some may round to 4 significant figures (e.g., $1326$) or incorrectly round to 3 significant figures as $1320$ (which is wrong because $1326$ rounds to $1330$).
• Overlooking the Least Precise Input: Students may count the significant figures in $1125$ (4) and report the answer with 4 significant figures, ignoring that $32.3$ and $27.4$ have only 3.

Exam Tip: Always identify the input with the fewest significant figures before performing the calculation. The result must match this precision, even if the intermediate computation yields more digits.