What Are Bounds?

A range where the true value lies. The Lower Bound (LB) is the smallest possible value; the Upper Bound (UB) is the first value that rounds into the next category.

Formula: LB ≤ x < UB

Example: 5.3 cm (to 1 d.p.) LB = 5.25, UB = 5.35

Finding LB and UB

Add and subtract HALF the degree of accuracy.

Error Interval Notation

Always uses a ≤ for the lower bound and a < for the upper bound.

Formula: LB ≤ x < UB

Example: 120 rounded to nearest 10: 115 ≤ x < 125

Truncation

Truncating is different from rounding. Truncating to 1 d.p. means simply "cutting off" the extra digits.

Example: 5.39 truncated to 1 d.p. is 5.3 (LB=5.3, UB=5.4)

Half-Unit Rule

The range is always ± half of the rounding unit.

Example: To nearest 0.1 → ± 0.05 To nearest 100 → ± 50

Bounds in Area

Max Area = UB_length × UB_width. Min Area = LB_length × LB_width.

Example: l=5.3, w=3.1 (both 1 d.p.) Max = 5.35 × 3.15 = 16.8525

Bounds in Fractions

To get the MAX result: Max ÷ Min. To get the MIN result: Min ÷ Max.

Formula: Upper = UB_top / LB_bottom Lower = LB_top / UB_bottom

Significant Figure Bounds

If a number is 400 to 1 s.f., the degree of accuracy is 100. Half is 50.

Example: LB = 350, UB = 450

Complex Calculation Bounds

Calculate each part separately using its specific bound, then combine using the rules of arithmetic.

Bounds in Subtraction

To get the MAX difference: UB_first − LB_second. To get the MIN difference: LB_first − UB_second.