How to Solve: Bounds and Error Intervals

How to Solve: Bounds and Error Intervals

(GCSE / A-Level Maths – Exam-Ready!)

Introduction

Mastering bounds and error intervals lets you answer real-world questions like: "A bridge is built to hold 500 kg, measured to the nearest 10 kg. What’s the smallest weight that could actually collapse it?" On exams, this topic appears in 1–2 questions per paper (5–10 marks). Lose these marks, and you drop a grade. Nail them, and you secure top marks.

What You Need To Know First

Before starting, you must understand: 1. Rounding – How to round numbers to significant figures, decimal places, or nearest units. 2. Inequalities – How to write and interpret expressions like a ≤ x < b. 3. Upper and Lower Bounds – The idea that a rounded number has a range of possible true values.

Key Vocabulary

Formulas To Know

1. Upper and Lower Bounds for Rounded Numbers

Formula: - Lower Bound (LB) = Rounded number – (0.5 × precision) - Upper Bound (UB) = Rounded number + (0.5 × precision)

Variables: - Precision = The smallest unit the number is rounded to (e.g., 0.1 for 1 d.p., 10 for nearest 10). - Rounded number = The number given after rounding.

MEMORISE THIS – You’ll use this for every bounds question.

2. Error Interval

Formula: LB ≤ x < UB (Note: The upper bound is not included because the true value could be just below it.)

Example: If a length is 12 cm to the nearest cm, the error interval is: 11.5 cm ≤ x < 12.5 cm

MEMORISE THIS – Examiners always test this notation.

3. Absolute Error

Formula: Absolute Error = 0.5 × precision

Example: A mass is 6.2 kg (1 d.p.). Absolute error = 0.5 × 0.1 = 0.05 kg.

Given on exam sheet (but you should still understand it).

4. Combining Bounds (Multiplication/Division)

When calculating with bounds, use these rules to find the maximum/minimum possible result:

MEMORISE THIS – Examiners love testing combined bounds.

Step-by-Step Method

Step 1: Identify the Precision

• Look at the last digit of the rounded number.
• The precision is 1 unit of that digit’s place value.
• Example: 3.7 (1 d.p.) → Precision = 0.1
• Example: 500 (nearest 10) → Precision = 10

Step 2: Calculate the Lower Bound (LB)

• LB = Rounded number – (0.5 × precision)
• Example: 3.7 (1 d.p.) → LB = 3.7 – (0.5 × 0.1) = 3.65

Step 3: Calculate the Upper Bound (UB)

• UB = Rounded number + (0.5 × precision)
• Example: 3.7 (1 d.p.) → UB = 3.7 + (0.5 × 0.1) = 3.75

Step 4: Write the Error Interval

• LB ≤ x < UB
• Example: 3.65 ≤ x < 3.75

Step 5: Apply to Calculations (If Needed)

• If the question asks for maximum/minimum possible values of a calculation (e.g., area, speed), use the combining bounds rules from the formulas section.

Worked Examples

Example 1 – Basic: Finding Bounds

Question: A length is measured as 12.3 cm to 1 decimal place. Find the error interval.

Solution: 1. Precision = 0.1 (since it’s rounded to 1 d.p.) 2. Lower Bound (LB) = 12.3 – (0.5 × 0.1) = 12.25 cm 3. Upper Bound (UB) = 12.3 + (0.5 × 0.1) = 12.35 cm 4. Error Interval = 12.25 cm ≤ x < 12.35 cm

What we did and why: - We found the smallest and largest possible true values of 12.3 cm. - The upper bound is not included because 12.35 cm would round up to 12.4 cm.

Example 2 – Medium: Combining Bounds (Addition)

Question: Two masses are measured as 5.1 kg and 3.2 kg (both to 1 d.p.). What is the maximum possible total mass?

Solution: 1. Find bounds for each mass:
- 5.1 kg → LB = 5.05 kg, UB = 5.15 kg
- 3.2 kg → LB = 3.15 kg, UB = 3.25 kg 2. Maximum total mass = UB₁ + UB₂ (from combining bounds rules)
= 5.15 + 3.25 = 8.4 kg

What we did and why: - To find the maximum possible total, we added the largest possible values of each mass. - If we added the lower bounds, we’d get the minimum possible total.

Example 3 – Exam-Style: Division with Bounds

Question: A car travels 120 km (nearest 10 km) in 2.5 hours (1 d.p.). What is the minimum possible average speed?

Solution: 1. Find bounds for distance and time:
- Distance: 120 km (nearest 10 km) → LB = 115 km, UB = 125 km
- Time: 2.5 hours (1 d.p.) → LB = 2.45 hours, UB = 2.55 hours 2. Minimum speed = LB (distance) ÷ UB (time) (from combining bounds rules)
= 115 ÷ 2.55 ≈ 45.1 km/h (1 d.p.)

What we did and why: - Speed = Distance ÷ Time. - To find the minimum speed, we used the smallest distance and the largest time (since dividing by a larger number gives a smaller result).

Common Mistakes

Exam Traps

1-Minute Recap (Night Before the Exam)

"Okay, listen up—this is your 60-second crash course on bounds and error intervals. First, precision is key. If a number is rounded to 1 decimal place, the precision is 0.1. For the nearest 10, it’s 10. Got it? Good. Now, lower bound = number – (0.5 × precision), upper bound = number + (0.5 × precision). Write the error interval as LB ≤ x < UB—never include the upper bound! If you’re combining bounds, remember: maximum = UB + UB, minimum = LB + LB for addition. For division, minimum = LB ÷ UB, maximum = UB ÷ LB. Watch out for truncation—no rounding, just chopping digits. And always check units! Now go smash that exam!