Limits of Accuracy (Copy)
IGCSE CORE Mathematics Cheat Sheet: Upper and Lower Bounds
1. What Are Bounds?
When a value is rounded to a specified degree of accuracy, the actual value could be slightly more or less than the rounded value.
- The Lower Bound is the smallest possible value.
- The Upper Bound is the largest possible value.
These define the range within which the true value lies.
2. Key Rule for Bounds
If a value is rounded to the nearest unit (e.g., 10, 1, 0.1, etc.):
- Lower Bound = Rounded Value − Half of the Unit
- Upper Bound = Rounded Value + Half of the Unit
3. Examples by Type of Rounding
a) Nearest Whole Number
- A length is 7 m, measured to the nearest metre.
- Lower Bound = 7 − 0.5 = 6.5 m
- Upper Bound = 7 + 0.5 = 7.5 m
b) Nearest 10, 100, etc.
- A number is 350, correct to the nearest 10.
- Lower Bound = 350 − 5 = 345
- Upper Bound = 350 + 5 = 355
c) Decimal Places
- A weight is 4.2 kg, correct to 1 decimal place.
- The unit is 0.1.
- Lower Bound = 4.2 − 0.05 = 4.15 kg
- Upper Bound = 4.2 + 0.05 = 4.25 kg
d) Significant Figures
- A value is 500, correct to 1 significant figure.
- The unit is 100.
- Half of 100 = 50
- Lower Bound = 500 − 50 = 450
- Upper Bound = 500 + 50 = 550
4. Quick Reference Table
| Rounding Method | Example | Lower Bound | Upper Bound |
|---|---|---|---|
| Nearest 1 | 7 | 6.5 | 7.5 |
| Nearest 10 | 120 | 115 | 125 |
| 1 d.p. | 3.6 | 3.55 | 3.65 |
| 2 s.f. (value = 0.034) | 0.034 | 0.0335 | 0.0345 |
5. How to Find Bounds
- Identify the accuracy (nearest 10, 1, 0.1, etc.).
- Divide that unit by 2.
- Subtract for the Lower Bound, add for the Upper Bound.
6. Example Questions
- Write down the upper bound of 250 cm measured to the nearest 10 cm:
Unit = 10 → Half = 5
Upper Bound = 250 + 5 = 255 cm - A number is given as 12.8 correct to 1 decimal place. Find the bounds:
Unit = 0.1 → Half = 0.05
Lower Bound = 12.8 − 0.05 = 12.75
Upper Bound = 12.8 + 0.05 = 12.85
7. Tips for Exams
- Always divide the unit by 2 to get bounds.
- Watch whether you’re dealing with:
- Decimal places
- Significant figures
- Nearest 10, 100, etc.
- Remember:
- Lower Bound = Value − Half Unit
- Upper Bound = Value + Half Unit
- If asked only for the upper bound, give just that.