Mathematics 0580 · syllabus C1.10, E1.10
Cambridge IGCSE Mathematics 0580 — Upper and lower bounds topical past papers
The upper and lower bounds of measurements rounded to a given accuracy, and on the Extended route the bounds of areas, speeds and other calculated values.
- Questions
- 154
- Years
- 2009–2026
- With mark scheme
- 89%
- Questions 2022–2026
- 49
Anyone can browse these pages. Opening the questions and their mark schemes takes a free Momentum account.
What Upper and lower bounds covers
A measurement rounded to a stated accuracy could have been anywhere in an interval around the value given. A length of 34 cm to the nearest centimetre lies between 33.5 cm and 34.5 cm: the lower bound is 33.5 cm and the upper bound 34.5 cm, half a unit of rounding either side. The interval is written 33.5 ≤ l < 34.5, because 34.5 itself would round up to 35. Core candidates find the bounds of a single rounded value, such as a mass to the nearest 10 g or a time to the nearest tenth of a second.
Extended candidates also find the bounds of results calculated from rounded values. For a sum or a product, the upper bounds combine to give the upper bound; for a difference or a quotient, the largest result comes from the upper bound of the first value with the lower bound of the second. So the upper bound of a speed is the upper bound of the distance divided by the lower bound of the time. Questions ask for bounds of areas, perimeters, speeds and densities, or for the greatest number of items that could fit.
Bounds of a single rounded value are Core content, so they can be set on all four papers; the bounds of calculated results are Extended, examined only on Papers 2 and 4. The topic grows out of rounding, and it appears inside mensuration and rates questions, so the formulas for area, perimeter, speed and density need to be secure first.
In the syllabus
The headings this topic has in the Cambridge International 2025–2027 syllabus, as the board prints them.
- C1.10 Limits of accuracy
- E1.10 Limits of accuracy
What you should be able to do
- Write down the bounds of a measurement given to the nearest whole unit, tenth or other accuracy
- Express the interval as an inequality, such as 33.5 ≤ l < 34.5
- Choose the correct combination of bounds for sums, differences, products and quotients (Extended)
- Find the bounds of areas, perimeters, speeds and densities calculated from rounded values (Extended)
- Interpret a bound in context, such as the greatest whole number of items that must fit
Key ideas
- The bounds lie half a unit of rounding either side: a value of 70 to the nearest 10 lies in 65 ≤ x < 75
- Upper bound of a + b = upper a + upper b; upper bound of a − b = upper a − lower b
- Upper bound of a × b = upper a × upper b; upper bound of a ÷ b = upper a ÷ lower b, for positive values
- Lower bounds follow the same pattern with every bound swapped
How Upper and lower bounds has been examined
From 2022 to 2026, Upper and lower bounds was on Paper 4 · Calculator (Extended) in 11 of 14 exam series, worth about 4% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 1 · Non-calculator (Core) | 16 | 2% | 10 of 14 |
| Paper 2 · Non-calculator (Extended) | 7 | 1% | 6 of 14 |
| Paper 3 · Calculator (Core) | 6 | 1% | 6 of 14 |
| Paper 4 · Calculator (Extended) | 20 | 4% | 11 of 14 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 3.
Filed in Momentum under: Upper and Lower Bound.
A worked example
A car travels 240 km, correct to the nearest 10 km, in 3.2 hours, correct to the nearest 0.1 hour. Find the upper bound of its average speed.
- Distance to the nearest 10 km: half of 10 is 5, so 235 ≤ d < 245.
- Time to the nearest 0.1 hour: half of 0.1 is 0.05, so 3.15 ≤ t < 3.25.
- Speed = distance ÷ time, so the largest speed comes from the largest distance divided by the smallest time.
- Upper bound = 245 ÷ 3.15 = 77.777…, which is 77.8 km/h to 3 significant figures.
Answer: 77.8 km/h (3 s.f.)
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Using ±0.5 whatever the accuracy: 2.4 kg to the nearest 0.1 kg lies between 2.35 kg and 2.45 kg, not between 1.9 kg and 2.9 kg.
- Using the upper bound of the time to find the upper bound of a speed: dividing by a larger number gives a smaller answer, so the upper bound of the speed needs the lower bound of the time.
- Writing the upper bound of 34 cm, to the nearest centimetre, as 34.49: the upper bound is 34.5, even though 34.5 itself would round up.
- Subtracting lower bound from lower bound for the upper bound of a difference: the largest difference is the upper bound of the first value minus the lower bound of the second.
- Rounding a calculated bound before the end: keep full values, and round only if the question asks.
Exam technique
- Write the bounds of every rounded value in a short list before doing any calculation.
- For each value in a formula, ask whether making it bigger makes the result bigger or smaller, then choose the upper or lower bound to match.
- Check that the lower bound is less than the upper bound and that the original value lies between them.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.