Upper and lower bounds, explained with examples
When a measurement is rounded, the true value could be a little more or a little less. Upper and lower bounds describe that range, and bounds calculations are a favourite Higher tier GCSE question.
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What bounds are
If a length is 8 cm to the nearest centimetre, the true length could be anything that rounds to 8. The smallest value that rounds to 8 is 7.5 cm; this is the lower bound. The upper bound is 8.5 cm, even though 8.5 itself would round up to 9. It is the boundary of the interval, so we use it in calculations.
The quick rule: halve the degree of accuracy, then subtract it for the lower bound and add it for the upper bound.
- To the nearest 1: ±0.5
- To the nearest 10: ±5
- To 1 decimal place (nearest 0.1): ±0.05
- To 2 significant figures: halve the place value of the second significant figure.
Error intervals
An error interval writes the bounds as an inequality. For x = 8 to the nearest whole number:
7.5 ≤ x < 8.5
The lower bound uses ≤ because 7.5 rounds up to 8. The upper bound uses < because 8.5 would round to 9, so it is not included. Getting these two symbols right is often a mark on its own.
Example 1: decimal places
A mass is 3.6 kg to 1 decimal place. Write its error interval.
- Accuracy 0.1, so half is 0.05.
- 3.55 ≤ m < 3.65.
Example 2: significant figures
A town has 24,000 people to 2 significant figures. Write the error interval for the population p.
- The second significant figure is in the thousands, so half is 500.
- 23,500 ≤ p < 24,500.
Truncation
Truncating means cutting off digits without rounding. If a number is truncated to 7.3, the true value could be anywhere from 7.3 up to, but not including, 7.4. So the error interval is 7.3 ≤ x < 7.4. Read questions carefully: “truncated” and “rounded” give different intervals.
Calculating with bounds (Higher)
To find the largest or smallest possible result of a calculation, choose the bounds that push the answer furthest in the direction you want.
- Addition: max = UB + UB; min = LB + LB.
- Subtraction: max = UB − LB; min = LB − UB.
- Multiplication: max = UB × UB; min = LB × LB (for positive values).
- Division: max = UB ÷ LB; min = LB ÷ UB.
The subtraction and division rules surprise many students. To make a difference as large as possible, start big and take away as little as possible. To make a quotient as large as possible, divide a big number by a small one.
Example 3: perimeter
A rectangle measures 12 cm by 7 cm, each to the nearest cm. Find the upper bound of the perimeter.
- Upper bounds: 12.5 and 7.5.
- Perimeter = 2(12.5 + 7.5) = 40 cm.
Example 4: area
Find the lower bound of the area of the same rectangle.
- Lower bounds: 11.5 and 6.5.
- Area = 11.5 × 6.5 = 74.75 cm².
Example 5: speed
A car travels 150 km, to the nearest 10 km, in 2.4 hours, to 1 decimal place. Find the upper bound of its average speed.
- Distance bounds: 145 to 155. Time bounds: 2.35 to 2.45.
- Max speed = max distance ÷ min time = 155/2.35 ≈ 65.96 km/h.
Example 6: subtraction
a = 9.4 and b = 3.2, both to 1 d.p. Find the lower bound of a − b.
- Min = LB of a − UB of b = 9.35 − 3.25 = 6.1.
Example 7: a suitable degree of accuracy
A square has side 8.3 cm to 1 decimal place. Give its area to a suitable degree of accuracy.
- Bounds of the side: 8.25 and 8.35. Bounds of the area: 8.25² = 68.0625 and 8.35² = 69.7225.
- To 2 significant figures these are 68 and 70, which differ. To 1 significant figure both are 70.
- So the area is 70 cm² to 1 significant figure, because that is the most accurate value both bounds agree on.
Example 8: a formula with several steps
Density = mass ÷ volume. A cube has side 4 cm to the nearest cm and mass 450 g to the nearest 10 g. Find the lower bound of its density.
- For the smallest density, use the smallest mass and the largest volume.
- Mass LB = 445 g. Side UB = 4.5 cm, so volume UB = 4.5³ = 91.125 cm³.
- Density LB = 445 ÷ 91.125 ≈ 4.88 g/cm³. Decide whether each quantity should be large or small before substituting.
Choosing a suitable degree of accuracy
Some questions ask you to give an answer “to a suitable degree of accuracy” and justify it. Work out both bounds of the answer, then round each to fewer and fewer significant figures until they agree. That shared value is the most accurate answer you can justify. For example, if the bounds are 4.372 and 4.358, both round to 4.4 to 2 significant figures, but 4.37 and 4.36 to 3 s.f. differ, so the answer is 4.4.
Common mistakes
- Using ≤ at the upper bound of an error interval.
- Halving the wrong amount, such as using ±0.5 for a value given to 1 decimal place.
- Using UB − UB for the maximum of a difference.
- Using UB ÷ UB for the maximum of a quotient.
- Mixing up truncation and rounding.
Practice questions
Try each one before opening the answer.
Q1. x = 50 to the nearest 10. Write the error interval.
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Q2. y = 0.37 to 2 d.p. Write the error interval.
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Q3. A square has side 6 cm to the nearest cm. Find the upper bound of its area.
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Q4. p = 20 and q = 4, both to the nearest whole number. Find the upper bound of p ÷ q.
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Q5. A number is truncated to 2 decimal places as 5.18. Write its error interval.
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Frequently asked questions
Are bounds on the Foundation paper?
Error intervals for rounding and truncation can appear on Foundation papers. Calculations with bounds, especially for subtraction and division, are Higher tier.
Why is the upper bound 8.5 when 8.5 rounds to 9?
The upper bound is the edge of the interval. Values like 8.4999 round to 8 and get as close to 8.5 as you like, so 8.5 is the boundary, written with < to show it is not included.
Key takeaways
- Halve the degree of accuracy to find the bounds.
- Error intervals use ≤ at the lower end and < at the upper end.
- Truncation intervals start at the given value.
- Max of a difference: UB − LB. Max of a quotient: UB ÷ LB.
- For a suitable accuracy, round both bounds until they agree.