IGCSE Limits of Accuracy: Complete Guide for Cambridge IGCSE Mathematics

IGCSE limits of accuracy (also called bounds) are important topics in Cambridge IGCSE Mathematics that appear in both Paper 2 and Paper 4. Mastering upper and lower bounds and error calculations is essential for understanding measurement precision and solving problems involving rounded values.

This comprehensive IGCSE limits of accuracy guide covers everything you need to know, including finding upper and lower bounds, calculating bounds for sums, differences, products, and quotients, percentage error, step-by-step worked examples, common exam questions, and expert tips from Tutopiya’s IGCSE maths tutors. We’ll also show you how to avoid the most common mistakes that cost students valuable marks.

🎯 What you’ll learn: By the end of this guide, you’ll know how to find bounds for rounded numbers, calculate bounds for operations, find percentage error, and apply these skills to solve problems in IGCSE exams.

Already studying with Tutopiya? Practice these skills with our dedicated IGCSE Number practice deck featuring exam-style questions and instant feedback.

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Why IGCSE Limits of Accuracy Matter

IGCSE limits of accuracy are essential for understanding measurement and precision. Here’s why they’re so important:

• High frequency topic: Bounds questions appear regularly in IGCSE maths papers
• Real-world applications: Used in engineering, science, and measurement
• Exam weight: Typically worth 4-8 marks per paper
• Foundation for advanced topics: Essential for understanding error analysis
• Practical skills: Develops understanding of measurement precision

Key insight from examiners: Students often confuse which bound to use for maximum/minimum values, or forget that upper bounds are exclusive. This guide will help you master these systematically.

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Understanding Limits of Accuracy

When a number is rounded to a certain precision, there’s a range of possible values it could represent.

Example: If a length is given as 5.2 cm (to 1 decimal place), the actual length could be anywhere from 5.15 cm up to (but not including) 5.25 cm.

• Lower bound: 5.15 cm (the smallest possible value)
• Upper bound: 5.25 cm (exclusive - the actual value is less than this)

Notation: 5.15 ≤ length < 5.25

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Finding Upper and Lower Bounds

For Numbers Rounded to Decimal Places

Method:

1. Lower bound: Subtract half of the precision unit
2. Upper bound: Add half of the precision unit (exclusive)

Precision units:

• To 1 d.p.: precision unit = 0.1, half = 0.05
• To 2 d.p.: precision unit = 0.01, half = 0.005
• To 3 d.p.: precision unit = 0.001, half = 0.0005

Example 1: Find bounds for 7.3 (to 1 d.p.)

Solution:

• Precision unit: 0.1, half = 0.05
• Lower bound: 7.3 - 0.05 = 7.25
• Upper bound: 7.3 + 0.05 = 7.35 (exclusive)

Answer: 7.25 ≤ x < 7.35

Example 2: Find bounds for 12.47 (to 2 d.p.)

Solution:

• Precision unit: 0.01, half = 0.005
• Lower bound: 12.47 - 0.005 = 12.465
• Upper bound: 12.47 + 0.005 = 12.475 (exclusive)

Answer: 12.465 ≤ x < 12.475

For Numbers Rounded to Significant Figures

Method: Same principle - find the precision unit based on the position of the last significant figure.

Example: Find bounds for 340 (to 2 s.f.)

Solution:

• The last significant figure is in the tens place
• Precision unit: 10, half = 5
• Lower bound: 340 - 5 = 335
• Upper bound: 340 + 5 = 345 (exclusive)

Answer: 335 ≤ x < 345

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Bounds for Addition and Subtraction

Addition

To find maximum sum: Use upper bounds To find minimum sum: Use lower bounds

Formula:

• Maximum: (Upper bound of A) + (Upper bound of B)
• Minimum: (Lower bound of A) + (Lower bound of B)

Example: a = 5.2 (to 1 d.p.) and b = 3.7 (to 1 d.p.). Find bounds for a + b.

Solution:

1. Bounds for a: 5.15 ≤ a < 5.25
2. Bounds for b: 3.65 ≤ b < 3.75
3. Maximum: 5.25 + 3.75 = 9.00 (but exclusive, so < 9.00)
4. Minimum: 5.15 + 3.65 = 8.80

Answer: 8.80 ≤ a + b < 9.00

Subtraction

To find maximum difference: Upper of first - Lower of second To find minimum difference: Lower of first - Upper of second

Formula:

• Maximum: (Upper bound of A) - (Lower bound of B)
• Minimum: (Lower bound of A) - (Upper bound of B)

Example: x = 12.4 (to 1 d.p.) and y = 5.8 (to 1 d.p.). Find bounds for x - y.

Solution:

1. Bounds for x: 12.35 ≤ x < 12.45
2. Bounds for y: 5.75 ≤ y < 5.85
3. Maximum: 12.45 - 5.75 = 6.70 (but exclusive)
4. Minimum: 12.35 - 5.85 = 6.50

Answer: 6.50 ≤ x - y < 6.70

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Bounds for Multiplication and Division

Multiplication

To find maximum product: Upper × Upper To find minimum product: Lower × Lower

Example: a = 4.2 (to 1 d.p.) and b = 3.6 (to 1 d.p.). Find bounds for a × b.

Solution:

1. Bounds for a: 4.15 ≤ a < 4.25
2. Bounds for b: 3.55 ≤ b < 3.65
3. Maximum: 4.25 × 3.65 = 15.5125 (exclusive)
4. Minimum: 4.15 × 3.55 = 14.7325

Answer: 14.73 ≤ a × b < 15.51 (to 2 d.p.)

Division

To find maximum quotient: Upper ÷ Lower To find minimum quotient: Lower ÷ Upper

Example: x = 15.0 (to 1 d.p.) and y = 3.0 (to 1 d.p.). Find bounds for x ÷ y.

Solution:

1. Bounds for x: 14.95 ≤ x < 15.05
2. Bounds for y: 2.95 ≤ y < 3.05
3. Maximum: 15.05 ÷ 2.95 = 5.1017... (exclusive)
4. Minimum: 14.95 ÷ 3.05 = 4.9016...

Answer: 4.90 ≤ x ÷ y < 5.10 (to 2 d.p.)

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Percentage Error

Percentage error shows how much a rounded value might differ from the actual value.

Formula: Percentage Error = (Maximum Error / Actual Value) × 100%

Maximum error is half the precision unit.

Example: Find the percentage error for 8.5 (to 1 d.p.)

Solution:

1. Precision unit: 0.1, half = 0.05
2. Maximum error: 0.05
3. Percentage error: (0.05 / 8.5) × 100% = 0.588%

Answer: 0.59% (to 2 s.f.)

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Step-by-Step Method for Bounds Problems

1. Identify the precision - Decimal places or significant figures?
2. Find the precision unit - What is the smallest unit?
3. Calculate half the precision unit - This is the maximum error
4. Find bounds - Lower = value - half, Upper = value + half (exclusive)
5. For operations: Use appropriate bounds (upper for max, lower for min)
6. Check your answer - Does it make sense?

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Worked Examples

Example 1: Finding Bounds

Find the upper and lower bounds for 6.8 (to 1 d.p.)

Solution:

• Precision: 0.1, half = 0.05
• Lower: 6.8 - 0.05 = 6.75
• Upper: 6.8 + 0.05 = 6.85 (exclusive)

Answer: 6.75 ≤ x < 6.85

Example 2: Bounds for Addition

a = 5.3 and b = 2.7 (both to 1 d.p.). Find bounds for a + b.

Solution:

1. a: 5.25 ≤ a < 5.35
2. b: 2.65 ≤ b < 2.75
3. Maximum: 5.35 + 2.75 = 8.10 (exclusive)
4. Minimum: 5.25 + 2.65 = 7.90

Answer: 7.90 ≤ a + b < 8.10

Example 3: Bounds for Multiplication

A rectangle has length 8.4 cm and width 5.2 cm (both to 1 d.p.). Find bounds for the area.

Solution:

1. Length: 8.35 ≤ L < 8.45
2. Width: 5.15 ≤ W < 5.25
3. Maximum area: 8.45 × 5.25 = 44.3625 (exclusive)
4. Minimum area: 8.35 × 5.15 = 43.0025

Answer: 43.00 cm² ≤ Area < 44.36 cm² (to 2 d.p.)

Example 4: Percentage Error

Find the percentage error for 250 (to 2 s.f.)

Solution:

1. Precision: tens place, so precision unit = 10, half = 5
2. Maximum error: 5
3. Percentage error: (5 / 250) × 100% = 2%

Answer: 2%

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Common Examiner Traps (and How to Dodge Them)

• Upper bound is exclusive - Use < not ≤ for upper bounds
• Wrong bounds for operations - Remember: max sum = upper + upper, min sum = lower + lower
• Subtraction bounds - Max difference = upper - lower (of different numbers!)
• Division bounds - Max quotient = upper ÷ lower, min = lower ÷ upper
• Precision unit errors - Count carefully for significant figures
• Forgetting to state bounds - Always give both lower and upper bounds

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IGCSE Limits of Accuracy Practice Questions

Question 1: Finding Bounds

Find bounds for 12.8 (to 1 d.p.)

Solution:

• Precision: 0.1, half = 0.05
• Lower: 12.75, Upper: 12.85 (exclusive)

Answer: 12.75 ≤ x < 12.85

Question 2: Addition Bounds

p = 7.3 and q = 4.6 (both to 1 d.p.). Find bounds for p + q.

Solution:

• p: 7.25 ≤ p < 7.35
• q: 4.55 ≤ q < 4.65
• Min: 7.25 + 4.55 = 11.80
• Max: 7.35 + 4.65 = 12.00 (exclusive)

Answer: 11.80 ≤ p + q < 12.00

Question 3: Multiplication Bounds

Length 6.0 m and width 4.0 m (both to 1 d.p.). Find bounds for area.

Solution:

• Length: 5.95 ≤ L < 6.05
• Width: 3.95 ≤ W < 4.05
• Min: 5.95 × 3.95 = 23.5025
• Max: 6.05 × 4.05 = 24.5025 (exclusive)

Answer: 23.50 m² ≤ Area < 24.50 m² (to 2 d.p.)

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Frequently Asked Questions About IGCSE Limits of Accuracy

What are upper and lower bounds?

When a number is rounded, bounds show the range of possible values:

• Lower bound: Smallest possible value (inclusive)
• Upper bound: Largest possible value (exclusive)

Why is the upper bound exclusive?

Because if the actual value equaled the upper bound, it would round up to the next value, not the given value.

How do I find bounds for addition?

• Maximum sum: upper + upper
• Minimum sum: lower + lower

How do I find bounds for subtraction?

• Maximum difference: upper (first) - lower (second)
• Minimum difference: lower (first) - upper (second)

How do I find bounds for multiplication?

• Maximum product: upper × upper
• Minimum product: lower × lower

What is percentage error?

Percentage error = (Maximum Error / Value) × 100%, where maximum error is half the precision unit.

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Related IGCSE Maths Resources

Strengthen your IGCSE Mathematics preparation with these comprehensive guides:

• IGCSE Estimation and Rounding: Complete Guide - Master rounding and significant figures
• IGCSE Standard Form: Complete Guide - Master scientific notation
• IGCSE Maths Revision Notes, Syllabus and Preparation Tips - Complete syllabus overview, topic breakdown, and revision strategies
• IGCSE Past Papers Guide - Access free IGCSE past papers and exam resources

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