Key Idea: Every real-world measurement is an approximation. This topic teaches you to round correctly (to decimal places or significant figures), measure how wrong an approximation is (percentage error), and find the range of values a rounded number could represent (upper and lower bounds).
Three things IB tests on this topic:
Round correctly
- To d.p.: count digits after decimal
- To s.f.: count from first non-zero digit
- e.g. 0.04863 to 3 s.f. = 0.0486
Percentage error
- ε = |vA − vE| ÷ vE × 100%
- Always divide by the EXACT value
- e.g. approx 240, exact 256 → 6.25%
Upper and lower bounds
- Rounded to nearest unit: ±0.5 unit
- Lower bound = value − 0.5 unit
- Upper bound = value + 0.5 unit
🎯 Rounding: d.p. vs s.f.
📐 Percentage error
- Upper bound
- The largest value the true measurement could be. If a length is 15 cm to the nearest cm, upper bound = 15.5 cm.
- Lower bound
- The smallest value the true measurement could be. Same measurement: lower bound = 14.5 cm. The true value satisfies: 14.5 ≤ length < 15.5
- Combining bounds
- Max sum = upper + upper. Min sum = lower + lower. Max product = upper × upper. Min quotient = lower ÷ upper. For differences: max difference = upper − lower.
Tip: Always divide by the exact (true) value vE — not the approximate one. Approx = 240, Exact = 256: ✓ ε = |240 − 256| ÷ 256 × 100 = 6.25% ✗ ε = |240 − 256| ÷ 240 × 100 = 6.67% ← wrong denominator
✏️ Worked examples
Round to significant figures
Round 0.048632 to 3 significant figures.
Step by step:
First significant figure: 4 (first non-zero digit)
2nd s.f.: 8, 3rd s.f.: 6
4th digit is 3 → do not round up
Answer: 0.0486
0.0486
Calculate percentage error
Estimated cost $240, actual cost $256. Find the percentage error.
Step by step:
vA = 240 (approximate), vE = 256 (exact)
ε = |240 − 256| ÷ 256 × 100
= 16 ÷ 256 × 100
= 6.25%
6.25%
Find bounds and max area
A box: length 15 cm, width 8 cm, each to the nearest cm. Find the maximum possible area.
Step by step:
Upper bound of length = 15.5 cm, upper bound of width = 8.5 cm
Maximum area = upper × upper = 15.5 × 8.5
= 131.75 cm²
Maximum area = 131.75 cm²
IB default: Give answers to 3 significant figures unless told otherwise. This applies after every GDC calculation. Percentage error: Identify the 'exact' or 'true' value — it goes in the denominator. The approximate value is what was estimated or measured. Bounds precision: The half-unit depends on the rounding used. Nearest cm → ±0.5 cm. Nearest 0.1 → ±0.05. Nearest 10 → ±5. Paper 1: Rounding questions are often 1–2 marks. Write clearly and show the digit you're rounding at.
IB-style question — model estimate and percentage error [5 marks]
A falling object's distance (in metres) after t seconds is modelled by d = 5t². (a) Use the model to estimate the distance fallen after 4 seconds. (b) The distance is actually measured as 76 m. Find the percentage error of the model's estimate, giving your answer to 3 significant figures.
Step by step:
(a) Substitute t = 4 into the model.
(b) Use the percentage error formula with the model value as the approximation and the measured value as the exact value.
Substitute the model estimate (80) and the measured value (76).
Evaluate and round to 3 significant figures.
(a) 80 m. (b) Percentage error ≈ 5.26%.