GCSE Maths Number - How to Solve: Bounds and Error Intervals

How to Solve: Bounds and Error Intervals

Complete Guide (GCSE / A-Level Physics, Chemistry, Biology – Exam-Ready)

Introduction

"Mastering bounds and error intervals can add 4–6 marks to your exam—enough to boost your grade from a 5 to a 7 or a C to an A. Whether you’re measuring a reaction time in physics, a titration volume in chemistry, or a growth rate in biology, real-world data is never exact. This guide teaches you how to handle uncertainty like a pro."

WHAT YOU NEED TO KNOW FIRST

Before diving in, ensure you understand: 1. Rounding – How to round numbers to a given decimal place or significant figure. 2. Inequalities – How to read and write inequalities (e.g., 3.5 ≤ x < 3.6). 3. Measurement uncertainty – The idea that all measurements have a smallest possible unit (e.g., a ruler measures to the nearest mm).

KEY TERMS & FORMULAS

Key Terms

Formulas

1. Error Interval (Bounds)
2. If a number is rounded to n decimal places or s significant figures, its error interval is:
   Lower bound = Rounded value – 0.5 × (smallest unit)
   Upper bound = Rounded value + 0.5 × (smallest unit)

3. Example: A length of 5.2 cm (rounded to 1 d.p.) has a smallest unit of 0.1 cm.

   • Lower bound = 5.2 – 0.05 = 5.15 cm
   • Upper bound = 5.2 + 0.05 = 5.25 cm
   • Error interval: 5.15 ≤ x < 5.25

4. Absolute Error

5. Absolute error = 0.5 × (smallest unit)

6. Example: For 5.2 cm (1 d.p.), absolute error = 0.05 cm.

7. Percentage Error

8. Percentage error = (Absolute error / Rounded value) × 100%
9. Example: For 5.2 cm, percentage error = (0.05 / 5.2) × 100 ≈ 0.96%

MEMORISE THIS: - The smallest unit is the last digit’s place value (e.g., 0.1 for 1 d.p., 0.01 for 2 d.p.). - Upper bound is exclusive (e.g., x < 5.25, not x ≤ 5.25).

STEP-BY-STEP METHOD

How to Find Bounds and Error Intervals

Step 1: Identify the rounded value and its smallest unit. - Example: 3.7 kg (rounded to 1 d.p.) → smallest unit = 0.1 kg.

Step 2: Calculate the absolute error. - Absolute error = 0.5 × smallest unit. - Example: 0.5 × 0.1 = 0.05 kg.

Step 3: Find the lower bound. - Lower bound = Rounded value – absolute error. - Example: 3.7 – 0.05 = 3.65 kg.

Step 4: Find the upper bound. - Upper bound = Rounded value + absolute error. - Example: 3.7 + 0.05 = 3.75 kg.

Step 5: Write the error interval as an inequality. - Example: 3.65 ≤ x < 3.75 kg.

Step 6 (Optional): Calculate percentage error if needed. - Percentage error = (Absolute error / Rounded value) × 100%. - Example: (0.05 / 3.7) × 100 ≈ 1.35%.

Worked Example (Using Steps)

Question: A mass is recorded as 12.34 g (rounded to 2 d.p.). Find its error interval.

Solution: 1. Rounded value = 12.34 g, smallest unit = 0.01 g. 2. Absolute error = 0.5 × 0.01 = 0.005 g. 3. Lower bound = 12.34 – 0.005 = 12.335 g. 4. Upper bound = 12.34 + 0.005 = 12.345 g. 5. Error interval: 12.335 ≤ x < 12.345 g.

What we did and why: - We found the range of possible true values by accounting for rounding uncertainty. - The upper bound is exclusive because 12.345 g would round up to 12.35 g.

WORKED EXAMPLES

Example 1 – Basic (GCSE)

Question: A length is measured as 8.6 cm (rounded to 1 d.p.). Find its error interval.

Solution: 1. Rounded value = 8.6 cm, smallest unit = 0.1 cm. 2. Absolute error = 0.5 × 0.1 = 0.05 cm. 3. Lower bound = 8.6 – 0.05 = 8.55 cm. 4. Upper bound = 8.6 + 0.05 = 8.65 cm. 5. Error interval: 8.55 ≤ x < 8.65 cm.

What we did and why: - We applied the standard bounds formula to a simple measurement. - The interval shows all possible true lengths that would round to 8.6 cm.

Example 2 – Medium (A-Level)

Question: A reaction time is recorded as 0.23 seconds (rounded to 2 d.p.). Calculate: a) The error interval. b) The percentage error.

Solution: a) Error interval: 1. Rounded value = 0.23 s, smallest unit = 0.01 s. 2. Absolute error = 0.5 × 0.01 = 0.005 s. 3. Lower bound = 0.23 – 0.005 = 0.225 s. 4. Upper bound = 0.23 + 0.005 = 0.235 s. 5. Error interval: 0.225 ≤ x < 0.235 s.

b) Percentage error: - Percentage error = (0.005 / 0.23) × 100 ≈ 2.17%.

What we did and why: - We extended the basic method to include percentage error, which is common in A-Level questions. - The percentage error tells us how significant the uncertainty is relative to the measurement.

Example 3 – Exam-Style (Disguised)

Question (A-Level Physics): A student measures the speed of sound as 343 m/s (rounded to the nearest whole number). The true speed is known to be within ±0.5 m/s of the measured value. a) Write the error interval for the true speed. b) Another student claims the speed is 342.6 m/s. Is this consistent with the first measurement? Explain.

Solution: a) Error interval: 1. Rounded value = 343 m/s, smallest unit = 1 m/s. 2. Absolute error = 0.5 × 1 = 0.5 m/s. 3. Lower bound = 343 – 0.5 = 342.5 m/s. 4. Upper bound = 343 + 0.5 = 343.5 m/s. 5. Error interval: 342.5 ≤ x < 343.5 m/s.

b) Consistency check: - 342.6 m/s is within the interval (342.5 ≤ 342.6 < 343.5). - Conclusion: Yes, it is consistent.

What we did and why: - We applied bounds to a real-world physics scenario. - The second part tested understanding of whether a value fits within the error interval.

COMMON MISTAKES

EXAM TRAPS

1-MINUTE RECAP

"Here’s the night-before cheat sheet for bounds and error intervals: 1. Find the smallest unit – It’s the last digit’s place value (e.g., 0.1 for 1 d.p.). 2. Calculate absolute error – 0.5 × smallest unit. 3. Lower bound = Rounded value – absolute error. 4. Upper bound = Rounded value + absolute error (but not equal to it!). 5. Write the interval as lower ≤ x < upper. 6. Percentage error = (Absolute error / Rounded value) × 100%.

Examiners love testing: - Whether you exclude the upper bound. - If you mix up decimal places and significant figures. - Combined measurements (e.g., area = length × width).

Double-check your units, and you’ll nail every question!"