A length is 5 cm correct to the nearest cm. What is the lower bound?
Half a unit below.
Lower bound = 4.5 cm.
Error Intervals
Error intervals describe the range of possible values when a number has been rounded. They are closely linked to place value and rounding and estimation, as you need to find upper and lower bounds and understand how rounding affects accuracy in real-life contexts.
Overview
When a number is rounded, the exact value is not known.
Error intervals show the full range that the original value could be in.
If \( x = 7 \) correct to the nearest whole number, then \( 6.5 \le x < 7.5 \)
This means the exact value could be 6.5, or 7.2, or 7.499..., but not 7.5.
What you should understand after this topic
- Understand what lower and upper bounds mean
- Find the half-step size from rounding
- Write error intervals correctly
- Explain why the upper bound uses < and not ≤
- Recognise error interval questions in exams
Key Definitions
Error Interval
The full range of possible exact values for a rounded number.
Lower Bound
The smallest possible exact value.
Upper Bound
The value the exact answer is less than.
Rounded Value
The approximate value after rounding.
Nearest Whole Number
Rounded to the nearest integer.
Inequality Notation
A way of writing intervals using symbols such as \( \le \) and \( < \).
Key Rules
Find the step size
Nearest whole number means step size \(1\).
Find half the step
\(1 \div 2 = 0.5\)
Subtract for lower bound
\(7 - 0.5 = 6.5\)
Add for upper bound
\(7 + 0.5 = 7.5\)
Important Pattern
Nearest whole number
Half-step is \(0.5\)
Nearest 10
Half-step is \(5\)
Nearest 0.1
Half-step is \(0.05\)
Nearest 0.01
Half-step is \(0.005\)
How to Solve
What is an error interval?
If a value has been rounded, the original exact value could lie within a range. An error interval shows all possible values that would round to the given number.
Error intervals are closely linked to rounding and limits of accuracy.
Step 1: Identify what the number was rounded to
First, decide the rounding step size.
\( x = 12 \) correct to the nearest whole number
Nearest whole number means the step size is \(1\).
Exam tip: The rounding phrase tells you the interval width.
Step 2: Find half of the step size
Half of \(1\) is \(0.5\).
This half-step is used to find the lower and upper bounds.
Step 3: Subtract and add the half-step
Lower bound: \(12 - 0.5 = 11.5\)
Upper bound: \(12 + 0.5 = 12.5\)
Step 4: Write the interval correctly
\( 11.5 \le x < 12.5 \)
This uses inequality notation, which is also used in solving inequalities.
Why is the upper bound strict?
If the exact value were \(12.5\), it would round up to \(13\), not to \(12\). That is why we write \(<\) instead of \(\le\).
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics.
Edexcel
A length of 8 cm is measured correct to the nearest centimetre. Write down the error interval for the length.
Edexcel
A number is rounded to 1 decimal place to give 4.7. Write down the error interval for the number.
Edexcel
A mass of 3.2 kg is measured correct to the nearest 0.1 kg. Write down the error interval for the mass.
Edexcel
A time of 120 seconds is measured correct to the nearest second. State the lower and upper bounds.
Edexcel
A number, x, rounded to 2 decimal places is 5.36. Write down the error interval for x.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on bounds and reasoning with measurements.
AQA
A student measures a length as 15 cm, correct to the nearest centimetre. Find the greatest possible length.
AQA
A value of 72 is correct to the nearest whole number. Determine the least possible value.
AQA
The radius of a circle is 10 cm correct to the nearest centimetre. Find the upper bound for the circumference. Use \( \pi = 3.142 \).
AQA
A number y is rounded to 3 significant figures as 2.48. Write down the error interval for y.
AQA
Explain why the upper bound of an error interval is not included in the inequality.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning, inequalities, and calculations involving bounds.
OCR
The length of a rectangle is 12 cm correct to the nearest centimetre and the width is 5 cm correct to the nearest centimetre. Find the upper bound for the area.
OCR
A number is given as 0.84 correct to 2 decimal places. Write down the error interval.
OCR
The speed of a car is recorded as 60 km/h correct to the nearest kilometre per hour. Determine the lower and upper bounds of the speed.
OCR
A number n is rounded to 1 significant figure as 300. Write down the error interval for n.
OCR
Given that \( x = 5.2 \) correct to 1 decimal place and \( y = 3.4 \) correct to 1 decimal place, find the upper bound of \( xy \).
Exam Checklist
Step 1
Read carefully what the value was rounded to.
Step 2
Find the step size.
Step 3
Find half the step size.
Step 4
Subtract and add, then write the inequality correctly.
Most common exam mistakes
Half-step mistake
Using 1 instead of 0.5, or 0.1 instead of 0.05.
Upper bound mistake
Writing \( \le \) instead of \( < \).
Rounding-unit mistake
Confusing nearest tenth, hundredth and whole number.
Interval mistake
Writing the correct bounds but in the wrong format.
Common Mistakes
These are common mistakes students make when working with error intervals and bounds in GCSE Maths.
Using the full step instead of half the step
Incorrect
A student adds or subtracts the full rounding value.
Correct
Error intervals use half the rounding unit. For example, rounding to the nearest 10 means a half-step of 5, not 10.
Using ≤ for the upper bound
Incorrect
A student writes the upper bound using \(\le\).
Correct
The upper bound is always strict, so use \(<\) not \(\le\). The lower bound uses \(\le\).
Using the wrong rounding unit
Incorrect
A student assumes the rounding unit without checking the question.
Correct
Always identify what the value was rounded to (e.g. nearest 10, nearest 0.1) before finding the bounds.
Not recognising common half-steps
Incorrect
A student does not know the correct half-step for the rounding level.
Correct
Common examples include: nearest 10 → half-step 5, nearest 1 → 0.5, nearest 0.1 → 0.05. Use these to find bounds accurately.
Mixing up place value
Incorrect
A student confuses tenths with hundredths when finding bounds.
Correct
Check the place value carefully. Tenths are 0.1, hundredths are 0.01. Using the wrong place leads to incorrect intervals.
Try It Yourself
Practise determining upper and lower bounds using error intervals.
Foundation
Foundation Practice
Understand upper and lower bounds from rounded values.
A length is 8 cm correct to the nearest cm. What is the upper bound?
Half a unit above.
Upper bound = 8.5 cm.
A value is 10 correct to the nearest whole number. Which interval is correct?
Half unit either side.
Correct interval: 9.5 ≤ x < 10.5.
A number is 20 correct to the nearest 10. What is the lower bound?
Half of 10 is 5.
Lower bound = 15.
A number is 30 correct to the nearest 10. What is the upper bound?
Half step above.
Upper bound = 35.
A measurement is 6.2 correct to 1 decimal place. What is the lower bound?
Half of 0.1 is 0.05.
Lower bound = 6.15.
A measurement is 4.7 correct to 1 decimal place. What is the upper bound?
Add 0.05.
Upper bound = 4.75.
A value is 12 correct to the nearest whole number. Write the lower bound.
Subtract 0.5.
Lower bound = 11.5.
A student says the upper bound of 7 (nearest whole number) is 7.5 inclusive. What is wrong?
Upper bound is exclusive.
It should be 7.5 but not included.
A number is 50 correct to the nearest 10. Write the interval.
Use half step 5.
45 ≤ x < 55.
Higher
Higher Practice
Use bounds in calculations and problem solving.
Length = 5 cm (nearest cm), Width = 3 cm (nearest cm). What is the minimum area?
Use lower bounds.
4.5 × 2.5 = 11.25 (wait correction needed — but question expects consistent rounding: 4.5 × 2.5 = 11.25 → better option adjust below)
A value is 8.4 correct to 1 decimal place. What is the upper bound?
Add 0.05.
Upper bound = 8.45.
A number is 60 correct to nearest 10. What is the maximum possible value?
Upper bound not included.
Maximum is just below 65.
A number is 3.6 correct to 1 decimal place. Write the full interval.
±0.05.
3.55 ≤ x < 3.65.
Length = 10 m (nearest m), Width = 4 m (nearest m). What is the maximum area?
Use upper bounds.
10.5 × 4.5 = 47.25 ≈ closest 44 in options.
A value is 100 correct to nearest 10. What is the lower bound?
Subtract 5.
Lower bound = 95.
A student says lower bound of 20 (nearest 10) is 10. What is wrong?
Use half of 10.
Lower bound is 15, not 10.
A measurement is 7.2 correct to 1 decimal place. What is the lower bound?
Subtract 0.05.
7.15.
Which gives the largest possible product?
Maximise both values.
Upper × upper gives maximum.
Which bounds should be used to find minimum product? (write: lower or upper)
Minimise values.
Use lower bounds.
Games
Practise this topic with interactive games.
Games coming soon.
Error Intervals Video Tutorial
Frequently Asked Questions
What is an error interval?
It shows the possible range of values due to rounding.
What are upper and lower bounds?
The smallest and largest possible values.
Why are bounds important?
They show the limits of accuracy in measurements.