Write the full range of values a rounded or truncated number could really have been, using inequalities. It is on Foundation and Higher, aimed at grade 3. Free questions with answers are on this page. Worked examples show each line before you try the questions.
When a measurement is rounded, the true value sits in a window around the printed number. Rounded to the nearest 10, a true value can be up to 5 either side; rounded to 1 decimal place, up to 0.05 either side. The error interval is written lower≤x<upper. The lower bound uses ≤ because that value would still round up to the printed number; the upper bound uses < because that exact upper value would round to the next number.
Truncating is chopping digits off, not rounding, so the true value is never smaller than what is written: value≤x<value+one unit of the last digit. 'Could it have been…?' questions compare two intervals and need a specific example that sits in both, or a reason why no such number exists.
You meet this as a 2-mark write-the-inequality, then as a 'could it have been…?' comparison. Halve the unit for rounding; do not halve for truncation. The upper bound is always a strict < so that exact value would round (or truncate) to the next number.
Objectives
If a length is 8 cm to the nearest cm, the unit is 1 cm.
The lower bound is 7.5 cm. The upper bound is 8.5 cm.
To the nearest 10, half the unit is 5.
Key points
Find half of the unit a number was rounded to (nearest 10 gives 5, nearest 0.1 gives 0.05).
Write an error interval as lower bound ≤x< upper bound.
Know the difference between rounding (goes either side) and truncating (only goes up).
Decide whether a value could or could not have been the original number.
Worked examples
Example 1
A length L is 10 cm to the nearest cm. Write the error interval for L.
Slip: The unit is 3 cm, so the half-unit is 0.5 cm. That changes a number from the question, so it answers a different question.
Slip: Lower bound: 13 − 0.5 = 9.5. Upper bound: 10 + 0.5 = 10.5. That changes a number in the working, so the step no longer follows from the one before.
Slip: The interval is 13.5 ≤ L < 10.5. That does not match the working above it.
Example 2
A number n is 40 to the nearest 10. Write the error interval for n.
Slip: 3 decimal place means a unit of 0.1, so the half-unit is 0.05. That changes a number from the question, so it answers a different question.
Slip: Lower bound: 5.6 − 0.05 = 2.55. Upper bound: 2.6 + 0.05 = 2.65. That changes a number in the working, so the step no longer follows from the one before.
Slip: The interval is 6.55 ≤ x < 2.65. That does not match the working above it.
Questions and answers
GCSE Maths error intervals questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
A length is 8 cm to the nearest cm. Give the lower bound, then the upper bound.
Show answer
Answer: 7.5, 8.5
Half the unit is 0.5.
The bounds are 7.5 and 8.5.
M1 — Half the unit is 0.5.
A1 — Answer 7.5, 8.5
Question 2 Easy2 marks
A length is 70 cm to the nearest 10 cm. Give the lower bound, then the upper bound.
Show answer
Answer: 65, 75
Half the unit is 5.
The bounds are 65 and 75.
M1 — Half the unit is 5.
A1 — Answer 65, 75
Question 3 Medium2 marks
A mass is 20 kg to the nearest kg. What is the lower bound?
Show answer
Answer: 19.5
The error is 0.5 kg.
20 − 0.5.
M1 — The error is 0.5 kg.
A1 — Answer 19.5
Question 4 Medium2 marks
30 to the nearest 10. What is the upper bound?
Show answer
Answer: 35
Nearest 10 uses an error of 5.
30 + 5.
M1 — Nearest 10 uses an error of 5.
A1 — Answer 35
Question 5 Hard2 marks
A square is 8 cm by 8 cm, both lengths to the nearest cm. What is the lower bound of the area?
Show answer
Answer: 56.25
Each lower bound is 7.5.
The area lower bound is 56.25.
M1 — Each lower bound is 7.5.
A1 — Answer 56.25
Question 6 Hard2 marks
A square is 10 cm by 10 cm, both to the nearest cm. What is the upper bound of the area?
Show answer
Answer: 110.25
Each upper bound is 10.5.
The area upper bound is 110.25.
M1 — Each upper bound is 10.5.
A1 — Answer 110.25
Question 7 Exam-style2 marks
A square is 9 cm by 9 cm, both lengths to the nearest cm. What is the lower bound of the area?
Show answer
Answer: 72.25
Each lower bound is 8.5.
The area lower bound is 72.25.
M1 — Each lower bound is 8.5.
A1 — Answer 72.25
Common mistakes
Most lost marks on this topic are one slip.
Using the whole unit instead of half.
The error is half the rounding unit.
Including the upper bound in the interval.
The upper bound rounds up to the next value, so the interval stops just before it.
Subtracting half a unit from a nearest-10 number as if the unit were 1.
Nearest 10 uses an error of 5.
Spot the slip
Half of 1 cm is
0.5 cm
1 cm
0.1 cm
Show the slip
0.5 cm Half the rounding unit is the error.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Write both bounds and both inequality symbols — a correct pair of numbers with the wrong symbols loses a mark.
Use the letter given in the question (if it says m, write m in your interval).
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.