Lower and Upper Bounds

🔍 What Are Bounds?

When a value is rounded or truncated, it is no longer exact. The lower bound and upper bound show the range within which the actual value lies.

🧮 Key Concepts

1. Rounding to the Nearest Whole Number / Decimal / Significant Figure

• When a value is rounded, it could be slightly less or more than the rounded value.

• The range of possible values is from:

  • Lower Bound = rounded value − half the degree of accuracy

  • Upper Bound = rounded value + half the degree of accuracy

✅ How to Find Bounds

Rounding To	Step Size	Lower Bound	Upper Bound
Nearest 10	10	value − 5	value + 5
1 dp	0.1	value − 0.05	value + 0.05
2 sf	Depends	Half the unit place	Half the unit place

📝 Example:
If a length is 12.4 cm (rounded to 1 decimal place):

• Step size = 0.1

• Lower bound = 12.4 − 0.05 = 12.35 cm

• Upper bound = 12.4 + 0.05 = 12.45 cm

➕ Calculations with Bounds

⚠️ Important Rule:

• Addition/Subtraction: Use extreme combinations to find min and max possible results.

• Multiplication/Division: Use extreme combinations that increase or decrease the result as much as possible.

➕ 1. Addition

Use lower + lower for the smallest result
Use upper + upper for the largest result

📝 Example:
Length A = 5.3 m (±0.05) ⇒ LB = 5.25, UB = 5.35
Length B = 2.6 m (±0.05) ⇒ LB = 2.55, UB = 2.65

• Min total = 5.25 + 2.55 = 7.80 m

• Max total = 5.35 + 2.65 = 8.00 m

➖ 2. Subtraction

Use lower − upper for the smallest result
Use upper − lower for the largest result

📝 Example:
A = 8.1 cm (±0.05) ⇒ LB = 8.05, UB = 8.15
B = 3.4 cm (±0.05) ⇒ LB = 3.35, UB = 3.45

• Min difference = 8.05 − 3.45 = 4.60 cm
• Max difference = 8.15 − 3.35 = 4.80 cm

✖️ 3. Multiplication

Use lower × lower for the smallest result
Use upper × upper for the largest result

📝 Example:
Length = 4.2 m (±0.05) ⇒ LB = 4.15, UB = 4.25
Width = 1.8 m (±0.05) ⇒ LB = 1.75, UB = 1.85

• Min area = 4.15 × 1.75 = 7.26 m²

• Max area = 4.25 × 1.85 = 7.86 m²

➗ 4. Division

Use lower ÷ upper for the smallest result
Use upper ÷ lower for the largest result

📝 Example:
Distance = 120 km (±0.5) ⇒ LB = 119.5, UB = 120.5
Time = 2 h (±0.05) ⇒ LB = 1.95, UB = 2.05

• Min speed = 119.5 ÷ 2.05 = ≈ 58.29 km/h

• Max speed = 120.5 ÷ 1.95 = ≈ 61.79 km/h

✨ Summary Table

Operation	Min Value (Lower Bound)	Max Value (Upper Bound)
Addition	LB + LB	UB + UB
Subtraction	LB − UB	UB − LB
Multiplication	LB × LB	UB × UB
Division	LB ÷ UB	UB ÷ LB

🧠 Quick Tip:

Always identify the degree of accuracy first (nearest 0.1, 1, 10, etc.) and calculate the bounds before doing any operations.