Bounds & Rounding
Rounding, significant figures, error intervals and (Higher) upper and lower bounds in calculations.
Watch the lesson
Every point below, worked through in one video, and each point links straight to its own part.
Learn
Bounds & Rounding, explained point by point
Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.
Round numbers to decimal places and significant figures
Decimal places count digits after the point; significant figures count from the first non-zero digit. So 0.030462 is 0.030 to 3 d.p. but 0.0305 to 3 s.f. Look one digit beyond where you are rounding: 5 or more rounds up.
Estimate calculations by rounding every number to 1 significant figure
Round each number to 1 s.f. and then calculate: 4.87 × 21.2 ≈ 5 × 20 = 100. Estimates check whether a calculator answer is sensible, and "estimate" questions expect exactly this method - not the accurate answer.
Write the error interval of a rounded or truncated value using inequality notation
A length of 5.6 cm rounded to 1 d.p. could really be anything from 5.55 up to (but not including) 5.65: written 5.55 ≤ x < 5.65. Truncated values chop rather than round, so 5.6 truncated gives 5.6 ≤ x < 5.7. The strict < always sits on the upper end.
Find upper and lower bounds of a rounded measurement
The bounds sit half a unit of rounding either side. A mass of 340 g to the nearest 10 g has lower bound 335 g and upper bound 345 g. Even though 345 itself would round up, we still call it the upper bound for calculations.
Calculate with bounds to find the maximum or minimum possible answerHigher only
Choose the bounds that push the answer the way you want. A maximum area uses both upper bounds; a maximum for a ÷ b uses the biggest a with the SMALLEST b. Write down which bound you used for each quantity - that reasoning earns the marks.
Use bounds to give an answer to an appropriate degree of accuracyHigher only
Work out the upper and lower bounds of the answer, then round both until they agree: if the bounds are 8.336 and 8.341, both round to 8.3, so the answer is 8.3 to 2 s.f. Quote both bounds and the agreement - that is the expected justification.
Common mistakes in Bounds & Rounding
The slips that cost marks year after year, and what to do instead.
Using the upper bound of both numbers to find the maximum of a division or a subtraction.Higher only
A maximum for a ÷ b or a − b needs the largest a and the smallest b. With a = 12 and b = 4 to the nearest whole number, the maximum of a ÷ b is 12.5 ÷ 3.5 ≈ 3.57.
Writing both ends of an error interval with ≤, as in 5.55 ≤ x ≤ 5.65.
The upper end is not included, because 5.65 would round up to 5.7. Write 5.55 ≤ x < 5.65.
Putting the bounds a whole unit either side, so 340 g to the nearest 10 g becomes 330 g to 350 g.
The bounds sit half a unit either side: 340 g to the nearest 10 g has a lower bound of 335 g and an upper bound of 345 g.
Estimating with the exact numbers, or rounding to the nearest whole number, instead of rounding to 1 significant figure.
Round every number to 1 significant figure first: 4.87 × 21.2 ≈ 5 × 20 = 100. The rounded numbers show the method, and the method earns the marks.
Rounding to 3 decimal places when the question asks for 3 significant figures, so 0.030462 becomes 0.030.
Significant figures start at the first non-zero digit, here the 3. So 0.030462 is 0.0305 to 3 s.f., but 0.030 to 3 d.p.
Practice
Try a Bounds & Rounding question
A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.
x = 4.6 correct to 1 decimal place. Write down the error interval for x.
- 4.55 ≤ x < 4.65
- 4.55 < x ≤ 4.65
- 4.5 ≤ x < 4.7
- 4.6 ≤ x < 4.7
Show the answer and the working
Answer: 4.55 ≤ x < 4.65
Values from 4.55 up to (but not including) 4.65 all round to 4.6 to 1 decimal place.
- Rounding to 1 decimal place uses a half-unit of 0.05.
- Lower bound: 4.6 − 0.05 = 4.55, which is included.
- Upper bound: 4.6 + 0.05 = 4.65, which is not included.
- So the error interval is 4.55 ≤ x < 4.65.