Limits of Accuracy
O Level and IGCSE Mathematics Cheat Sheet – 1.10 Limits of Accuracy
Key Idea
- Measurements are never exact — they are rounded to a certain accuracy.
- Upper bound = highest possible actual value.
- Lower bound = lowest possible actual value.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Finding Bounds for a Single Measurement
- Identify rounding unit (nearest 1 m, 1 cm, 0.1 s, etc.).
- Lower bound = rounded value − half of rounding unit.
- Upper bound = rounded value + half of rounding unit.
Examples:
- Length = 12 m (nearest metre)
Lower bound = 11.5 m, Upper bound = 12.5 m - Time = 7.3 s (nearest 0.1 s)
Lower bound = 7.25 s, Upper bound = 7.35 s
Bounds in Calculations
- Addition / Perimeter / Area →
- Upper bound = use all upper bounds
- Lower bound = use all lower bounds
- Subtraction →
- Upper bound = first number’s upper bound − second number’s lower bound
- Lower bound = first number’s lower bound − second number’s upper bound
- Multiplication / Division / Speed, Density, etc. →
- Upper bound = (Upper ÷ Lower) or (Upper × Upper) depending on formula
- Lower bound = (Lower ÷ Upper) or (Lower × Lower)
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Worked Example 1 – Perimeter
Rectangle: Length = 8.4 m (nearest 0.1 m), Width = 5.2 m (nearest 0.1 m)
- Length bounds: 8.35 m ≤ L < 8.45 m
- Width bounds: 5.15 m ≤ W < 5.25 m
- Upper bound perimeter = 2 × (8.45 + 5.25) = 27.4 m
- Lower bound perimeter = 2 × (8.35 + 5.15) = 27.0 m
Worked Example 2 – Speed
Distance = 120 m (nearest metre), Time = 9.6 s (nearest 0.1 s)
- Distance bounds: 119.5 ≤ d < 120.5
- Time bounds: 9.55 ≤ t < 9.65
- Upper bound speed = 120.5 ÷ 9.55 ≈ 12.62 m/s
- Lower bound speed = 119.5 ÷ 9.65 ≈ 12.38 m/s
Exam Special Tips
- Always write bounds as inequalities in working to show understanding.
- In perimeter/area problems, use upper bounds for maximum possible value.
- In subtraction, watch order — swapping bounds gives wrong answer.
- For speed, density, etc., always use formula with correct bound substitution.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Common Mistakes & Confusions
- Forgetting to halve the rounding unit.
- Using wrong bound in division (e.g., upper ÷ upper instead of upper ÷ lower).
- Writing rounded bounds instead of leaving them exact.
- Mixing up < and ≤ in inequalities.