17. The volume of a liquid is measured to be 100 ml with an uncertainty of ± 3 ml, and its
mass is measured to be 100 g with an uncertainty of ± 4 g. By calculating mass over
volume, the density is reported to be 1 g/ml. Assuming the errors are uncorrelated, what
is the uncertainty in the reported density, measured in g/ml?
a.1/100
b.5/100
c.7/100
d.10/100
Introduction
This problem determines the uncertainty in a reported density of 1 g/ml using the error
propagation formula for uncorrelated measurements, where relative uncertainties combine in quadrature.
It is a classic CSIR NET Life Sciences quantitative aptitude example.
Step-by-Step Solution
Given: Mass (m) = 100 g ± 4 g; Volume (V) = 100 ml ± 3 ml.
1. Density formula:
ρ = m / V = 100 / 100 = 1 g/ml
2. Relative uncertainties:
Δm/m = 4 / 100 = 0.04 (4%)
ΔV/V = 3 / 100 = 0.03 (3%)
3. Propagation of relative errors for quotient:
(Δρ / ρ) = √((Δm/m)² + (ΔV/V)²)
= √(0.04² + 0.03²)
= √(0.0016 + 0.0009)
= √(0.0025)
= 0.05 (5%)
4. Absolute uncertainty:
Δρ = 0.05 × 1 = 0.05 g/ml = 5/100 g/ml
Final Answer: The uncertainty in the reported density is 5/100 g/ml.
Option Analysis
- (a) 1/100 — Too low; ignores the dominant 4% mass error and incorrectly uses linear addition.
- (b) 5/100 — Correct; matches quadrature rule: √(4² + 3²) = 5.
- (c) 7/100 — Incorrect; linear sum (0.04 + 0.03 = 0.07) violates propagation rule.
- (d) 10/100 — Wrong; represents maximum error estimation, not statistical propagation.
Concept Highlight
In measurements involving division or multiplication, the total relative uncertainty adds in quadrature:
(Δρ / ρ)² = (Δm / m)² + (ΔV / V)²
This 3-4-5 triangle error pattern (3%, 4%, 5%) is frequently tested.
Recognizing it allows quick identification of the 5% result.
