What are bounds?
Any measurement that has been rounded (see the Rounding and Estimation page) could actually be any value within a range around the rounded figure. The lower bound is the smallest value that would round to the given figure; the upper bound is the smallest value that would round up to the next figure.
The general rule: find half of the rounding interval, then subtract it for the lower bound and add it for the upper bound.
Finding bounds from a rounded value
Worked Example 1
A length is measured as 8cm, correct to the nearest cm. Find the lower and upper bounds.
1
The rounding interval is 1cm (nearest whole cm), so half of it is 0.5cm
2
Lower bound: 8 − 0.5 = 7.5. Upper bound: 8 + 0.5 = 8.5
Answer7.5cm ≤ length < 8.5cm
The true value could be anywhere from 7.5 up to (but not including) 8.5 — both would round to 8.
Worked Example 2
A mass is measured as 12.4kg, correct to the nearest 0.1kg. Find the bounds.
1
The rounding interval is 0.1kg, so half of it is 0.05kg
2
Lower bound: 12.4 − 0.05 = 12.35. Upper bound: 12.4 + 0.05 = 12.45
Answer12.35kg ≤ mass < 12.45kg
Worked Example 3
A number is given as 250, correct to 2 significant figures. Find the bounds.
1
2 significant figures of 250 means rounded to the nearest 10, so the rounding interval is 10, and half of it is 5
2
Lower bound: 250 − 5 = 245. Upper bound: 250 + 5 = 255
Answer245 ≤ number < 255
Calculating with bounds
When a calculation uses several rounded values, work out which combination of bounds gives the largest or smallest possible result — this isn't always simply "use all upper bounds" or "use all lower bounds".
| To find the... | Use these bounds |
|---|---|
| Maximum of a sum or product | upper bound of every value |
| Minimum of a sum or product | lower bound of every value |
| Maximum of a subtraction (a − b) | upper bound of a, lower bound of b |
| Minimum of a subtraction (a − b) | lower bound of a, upper bound of b |
| Maximum of a division (a ÷ b) | upper bound of a, lower bound of b |
| Minimum of a division (a ÷ b) | lower bound of a, upper bound of b |
Worked Example 4
A rectangle has length 8cm and width 5cm, both correct to the nearest cm. Find the upper bound for the area.
1
Length bounds: 7.5 to 8.5. Width bounds: 4.5 to 5.5
2
For the maximum area (a product), use the upper bound of both: 8.5 × 5.5
Answer46.75 cm²
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1A length is 15cm, correct to the nearest cm. Find the lower and upper bounds.Foundation
Q2A mass is 60kg, correct to the nearest 10kg. Find the bounds.Foundation
Q3A number is 3.6, correct to 1 decimal place. Find the bounds.Foundation
Q4A distance is 200m, correct to the nearest 10m. Find the bounds.Foundation
Q5A time is 45 minutes, correct to the nearest 5 minutes. Find the bounds.Foundation
Higher (Grade 5–7)
Q6A rectangle has length 7.2cm and width 4.5cm, both correct to the nearest 0.1cm. Find the upper bound of the perimeter.Higher
Q7Find the lower bound of the area of a rectangle measuring 9.4cm by 6.2cm, both correct to the nearest 0.1cm.Higher
Q8A number is 400, correct to 1 significant figure. Find the bounds.Higher
Q9Two numbers, 58 and 22, are both correct to the nearest whole number. Find the upper bound of their sum.Higher
Q10Find the lower bound of 84 ÷ 12, where both numbers are correct to the nearest whole number, to 2 decimal places.Higher
Higher — Hard (Grade 8–9)
Q11A cube has a side length of 5cm, correct to the nearest cm. Find the upper bound for the volume.Grade 8–9
Q12Using v = u + at, find the maximum possible value of v, given u = 12 (nearest whole number), a = 3 (nearest whole number) and t = 4.5 (nearest 0.1).Grade 8–9
Q13A length of 25cm is measured to the nearest cm. Find the maximum possible percentage error.Grade 8–9
Q14Two numbers, 8 and 5, are both correct to the nearest whole number. Find the lower bound of 8 ÷ 5, to 4 decimal places.Grade 8–9
Q15A rectangle has an area of 42cm² and a width of 6cm, both correct to the nearest whole number. Find the upper bound for the length, to 3 significant figures.Grade 8–9
Answers
Foundation (Q1–Q5)
Q114.5cm ≤ length < 15.5cm
Q255kg ≤ mass < 65kg
Q33.55 ≤ number < 3.65
Q4195m ≤ distance < 205m
Q542.5 min ≤ time < 47.5 min
Higher (Q6–Q10)
Q623.6cm(2 × (7.25 + 4.55))
Q757.5025 cm²(9.35 × 6.15)
Q8350 ≤ number < 450
Q981(58.5 + 22.5)
Q106.68(83.5 ÷ 12.5 — lower bound of numerator, upper bound of denominator)
Higher — Hard (Q11–Q15)
Q11166.375 cm³(5.5³)
Q1228.425(12.5 + 3.5 × 4.55)
Q132%(0.5 ÷ 25 × 100)
Q141.3636(7.5 ÷ 5.5 — lower bound of numerator, upper bound of denominator)
Q157.73 cm(42.5 ÷ 5.5 — upper bound of area, lower bound of width)
Common mistakes
Common Mistake 1
Using the wrong half-interval
For a number rounded to the nearest 10, the half-interval is 5, not 0.5 — always match the half-interval to the actual rounding accuracy, not just halve the last digit shown.
Common Mistake 2
Assuming the upper bound is included
The upper bound is the value that would round up to the next figure, so it's technically excluded (use <, not ≤) — though in most GCSE calculations this distinction doesn't affect the numerical working.
Common Mistake 3
Using all upper bounds (or all lower bounds) for every calculation
This only works for addition and multiplication. For subtraction or division, the maximum result comes from mixing an upper bound with a lower bound — always check the reference table for which combination applies.
Common Mistake 4
Forgetting that 2 significant figures doesn't always mean "1 decimal place"
For 250 to 2 significant figures, the rounding interval is 10 (not 0.1) — always work out what accuracy the number was actually rounded to before finding the half-interval.
Exam tips
💡 Exam Tip 1
Always identify the rounding interval first
Before finding any bound, write down exactly what accuracy the value was rounded to (nearest cm, nearest 0.1kg, nearest 10) — this is the single most important step.
💡 Exam Tip 2
Use the reference table for combined calculations
For any question combining several rounded values, write out which bound (upper or lower) you need for each value before calculating — guessing leads to the wrong combination more often than you'd expect.
💡 Exam Tip 3
Check your answer's direction makes sense
A "maximum" answer should always be bigger than the value calculated using the rounded figure directly, and a "minimum" answer should always be smaller — use this as an instant sanity check.
💡 Exam Tip 4
State bounds using inequality notation when asked
If a question says "write down the error interval" or asks for bounds explicitly, use inequality notation (e.g. 7.5 ≤ x < 8.5) rather than just listing the two numbers — see the Error Intervals page for more on this notation.
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