Mathematics Standard • Year 11 • Module 2 • Lesson 13
Errors and Limits of Accuracy — Skill Drill
Build fluency with measurement uncertainty: absolute error from instrument precision, upper and lower bounds, percentage error, and bounds for areas/perimeters.
1. Quick recall
Answer each question in the space provided. 1 mark each
Q1.1 Complete the rules for measurement error.
Absolute error = ½ × ____________ . Upper bound = measurement + ____________ . Lower bound = measurement − ____________ .
Q1.2 Write the formula for percentage error of a measurement.
% error = ____________________________ × 100%
Q1.3 A ruler is marked in millimetres (smallest unit = 1 mm). What is the absolute error of any single measurement made with this ruler? ____________ mm
2. Worked example — bounds and percentage error
Follow each line of working. Every step has a reason on the right.
Problem. A length is measured as 34 cm using a ruler marked in centimetres (precision = 1 cm). Find (a) the absolute error, (b) the upper and lower bounds, and (c) the percentage error to 2 d.p.
Step 1 — Absolute error (½ × smallest unit).
AE = ½ × 1 cm = 0.5 cm
Reason: smallest graduation is 1 cm, so the largest rounding error is half of that.
Step 2 — Bounds (measurement ± AE).
Lower = 34 − 0.5 = 33.5 cm; Upper = 34 + 0.5 = 34.5 cm
Reason: true value lies in [33.5, 34.5] cm.
Step 3 — Percentage error.
% error = 0.5 ÷ 34 × 100 ≈ 1.47%
Reason: absolute error ÷ measurement × 100%.
Conclusion. AE = 0.5 cm; true value in [33.5, 34.5] cm; % error ≈ 1.47%.
3. Faded example — fill in the missing steps
A mass is recorded as 45 kg using scales with precision 0.5 kg. Find the absolute error, the bounds, and the percentage error to 2 d.p. Fill in each blank. 4 marks
Step 1 — Absolute error:
AE = ½ × ____________ = ____________ kg
Step 2 — Bounds:
Lower = 45 − ____________ = ____________ kg; Upper = 45 + ____________ = ____________ kg
Step 3 — Percentage error:
% error = ____________ ÷ 45 × 100 ≈ ____________ %
Conclusion. True mass lies in [____________ , ____________] kg; % error ≈ ____________ %.
4. Graduated practice — Error calculations
Show your working. State the AE with its unit, and round any percentages as requested.
Foundation — single-step bounds and AE (4 questions)
| Q | Problem | Answer |
|---|---|---|
| 4.1 1 | A length is 72 mm (precision 1 mm). State the absolute error. | |
| 4.2 1 | A jug measures 2.4 L (precision 0.1 L). State the absolute error. | |
| 4.3 1 | A car odometer reads 1 256.8 km (precision 0.1 km). State the upper bound of the true distance. | |
| 4.4 1 | A temperature is 37.2°C (precision 0.2°C). State the bounds. |
Standard — typical HSC difficulty (6 questions)
Show at least one line of substitution and clearly label your final answer with units.
4.5 A length of 85 cm is measured using a ruler with 1 mm precision. Find the percentage error to 3 significant figures. 2 marks
4.6 A measurement of 0.6 kg is recorded with precision 0.1 kg. Find the percentage error to 2 d.p. 2 marks
4.7 Student A records a length as 8 m (precision 0.5 m). Student B records the same length as 8.0 m (precision 0.1 m). Find both percentage errors and state which student is more precise. 2 marks
4.8 A balance reads 12 g (precision 0.5 g). Find the upper and lower bounds of the true mass. 2 marks
4.9 A 8.5 cm length is measured to the nearest mm. State the absolute error, the bounds, and the percentage error to 2 s.f. 2 marks
4.10 A square has side measured as 10 cm with absolute error 0.5 cm. Find the maximum possible area. 2 marks
Extension — compounding errors (2 questions)
4.11 A rectangle is measured as 12 cm × 9 cm with precision 1 mm. Find the maximum and minimum possible area (in cm², to 2 d.p.). 3 marks
4.12 Three lengths of rope each measure 2.5 m with precision 0.01 m. They are joined end to end. Find the maximum and minimum possible total length. 3 marks
5. Self-check the easy 3
Tick the first three once you've checked your method works.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1 — Error rules
AE = ½ × smallest unit. Upper bound = measurement + AE. Lower bound = measurement − AE.
Q1.2 — % error formula
% error = (absolute error ÷ measurement) × 100%.
Q1.3 — Ruler in mm
AE = 0.5 mm (half of 1 mm).
Q3 — Faded example (45 kg on 0.5 kg scales)
Step 1: AE = ½ × 0.5 = 0.25 kg.
Step 2: Lower = 45 − 0.25 = 44.75 kg; Upper = 45 + 0.25 = 45.25 kg.
Step 3: % error = 0.25 ÷ 45 × 100 ≈ 0.56%.
Conclusion: True mass in [44.75, 45.25] kg; % error ≈ 0.56%.
Q4.1 — 72 mm to nearest mm
AE = ½ × 1 = 0.5 mm.
Q4.2 — 2.4 L to nearest 0.1 L
AE = ½ × 0.1 = 0.05 L.
Q4.3 — Odometer 1 256.8 km upper bound
AE = 0.05 km. Upper = 1 256.8 + 0.05 = 1 256.85 km.
Q4.4 — 37.2°C to nearest 0.2°C
AE = 0.1°C. Bounds: [37.1, 37.3] °C.
Q4.5 — 85 cm with 1 mm precision
AE = 0.5 mm = 0.05 cm. % error = 0.05 ÷ 85 × 100 ≈ 0.0588% (to 3 s.f.).
Q4.6 — 0.6 kg with 0.1 kg precision
AE = 0.05 kg. % error = 0.05 ÷ 0.6 × 100 = 8.33%.
Q4.7 — Two students measuring the same length
Student A: AE = 0.25 m; % error = 0.25/8 × 100 = 3.125%. Student B: AE = 0.05 m; % error = 0.05/8 × 100 = 0.625%. Student B is more precise (smaller % error).
Q4.8 — Balance 12 g, precision 0.5 g
AE = 0.25 g. Bounds: [11.75, 12.25] g.
Q4.9 — 8.5 cm to nearest mm
AE = 0.05 cm. Bounds: [8.45, 8.55] cm. % error = 0.05 ÷ 8.5 × 100 ≈ 0.59% (to 2 s.f.).
Q4.10 — Square side 10 cm ± 0.5 cm, max area
Upper bound side = 10.5 cm. Max area = 10.5² = 110.25 cm².
Q4.11 — Rectangle 12 cm × 9 cm, precision 1 mm
AE = 0.5 mm = 0.05 cm. Bounds: length [11.95, 12.05]; width [8.95, 9.05]. Max area = 12.05 × 9.05 = 109.05 cm². Min area = 11.95 × 8.95 = 106.95 cm².
Q4.12 — Three 2.5 m ropes, precision 0.01 m
AE = 0.005 m. Each piece in [2.495, 2.505] m. Max total = 3 × 2.505 = 7.515 m. Min total = 3 × 2.495 = 7.485 m.