Lower & Upper Bound Calculator
Enter a rounded value and the unit it was rounded to (nearest 1, 0.1, 10, etc.).
What are lower and upper bounds?
When a number is rounded, the original value is not known exactly. Instead, it lies in an interval. The lower bound is the smallest possible original value, and the upper bound is the largest possible original value (usually written as a value the original is less than).
Example: If a length is given as 12 cm to the nearest 1 cm, the true length is at least 11.5 cm and less than 12.5 cm. So:
- Lower bound = 11.5
- Upper bound = 12.5
- Interval form: 11.5 ≤ x < 12.5
Formula used by this calculator
If a value R is rounded to the nearest unit U, then:
- Lower bound = R − U/2
- Upper bound = R + U/2
This works for whole numbers, decimal places, and larger place values.
Examples
1) Rounded to nearest whole number
Rounded value = 84, unit = 1. Half-unit = 0.5. Bounds are 83.5 and 84.5, so 83.5 ≤ x < 84.5.
2) Rounded to nearest tenth
Rounded value = 7.2, unit = 0.1. Half-unit = 0.05. Bounds are 7.15 and 7.25, so 7.15 ≤ x < 7.25.
3) Rounded to nearest ten
Rounded value = 350, unit = 10. Half-unit = 5. Bounds are 345 and 355, so 345 ≤ x < 355.
Common mistakes to avoid
- Using the full rounding unit instead of half the unit.
- Forgetting that the upper end is usually written as “less than” (<), not “less than or equal to.”
- Mixing decimal place rounding and significant-figure rounding rules.
- Not matching units (for example, centimeters vs meters).
Where bounds are useful
Lower and upper bounds are used in many practical contexts:
- Measurement uncertainty in science labs
- Error intervals in engineering and manufacturing
- Estimation questions in school math exams
- Data reporting where numbers are rounded before publishing
Quick study tip
If you remember only one thing, remember this: take half of the rounding step and go down for lower, up for upper.