Upper and Lower Bounds

In measurements and rounding, we often use approximations. The upper bound and lower bound define the range within which the true value lies. When a number is rounded to a certain degree of accuracy (like to the nearest 10, 0.1, or 2 significant figures), its true value could be slightly more or slightly less. Bounds define that range.

• Lower Bound: The smallest possible value that could round to the given number.
• Upper Bound: The largest possible value that could round to the given number.

These bounds are based on half of the degree of accuracy used in rounding.

For example: If a number is rounded to the nearest 10, the possible values could vary by ±5.

2. How to Find Bounds

To find bounds of a rounded value:

1. Determine the level of accuracy (nearest unit, 0.1, whole number, etc.)
2. Calculate \( \frac{\text{accuracy}}{2} \)
3. $\text{Lower Bound = Rounded Value − Half of Accuracy}$
4. $\text{Upper Bound = Rounded Value + Half of Accuracy}$

3. Decimal Place and Significant Figure Accuracy

• Nearest whole number: ±0.5
• 1 decimal place: ±0.05
• 2 decimal places: ±0.005
• 1 significant figure like 300: ±50

4. Applying Bounds in Calculations

When using measurements in formulas, use the maximum and minimum values (bounds) to estimate the maximum or minimum result.

• For multiplication: $\text{Max value = upper × upper, Min = lower × lower}$
• For division: $\text{Max = upper ÷ lower, Min = lower ÷ upper}$