Units and estimating
Estimating means rounding every number to one significant figure and working with what is left. It is examined in its own right — "estimate the value of…" — and it is also how you check an answer from a calculator, including one that came out wrong because of a mistyped digit.
The method is simply to round, then calculate. 4.87 × 19.3 becomes 5 × 20 = 100, so an answer of 940 or 9.4 is wrong by a factor of ten and the calculation should be repeated.
When the question says "estimate", the rounded working must be shown — an exact answer scores nothing, because the exactness is not what is being marked.
Rounding and error intervals
Decimal places are counted after the point. Significant figures are counted from the first non-zero digit, wherever it sits.
So 0.004821 to 2 significant figures is 0.0048 — the leading zeros are placeholders and are not significant. But 0.004821 to 2 decimal places is 0.00, because there is nothing large enough to survive.
Round by looking at the next digit only: 5 or more rounds up, 4 or less rounds down. Do not round in stages — 4.4999 to 1 decimal place is 4.5, not 5.
An error interval states the range a rounded value could have come from, using inequality notation. If a length is 8 cm to the nearest centimetre, anything from 7.5 up to but not including 8.5 would have rounded to 8, so
7.5 ≤ length < 8.5
The lower bound uses ≤ because 7.5 does round to 8, and the upper bound uses < because 8.5 rounds to 9 instead. That asymmetry is the detail the mark scheme looks for.
Truncation is different from rounding: it cuts the digits off rather than rounding them. A value truncated to 8 could be anything from 8 up to just under 9, so the interval is 8 ≤ x < 9.
Common mistake
The upper bound is not 8.49 or 8.4999. It is 8.5, written with a strict inequality — there is no largest number below 8.5, which is exactly why the notation uses < rather than ≤.
- Significant figures
- digits counted from the first non-zero digit, wherever it sits; leading zeros are placeholders and are not significant
- Error interval
- the range of values a rounded number could have come from, written with the lower bound and upper bound
Upper and lower bounds in calculationsHigher tier
When rounded values are used in a calculation, the answer has bounds too — and working out which combination gives the largest and smallest result is the skill being tested.
The rules follow from thinking about what makes the answer big:
• Adding — the maximum uses both upper bounds, the minimum uses both lower bounds • Multiplying — the same, since making either factor larger makes the product larger • Subtracting — the maximum is upper minus lower; the minimum is lower minus upper • Dividing — the maximum is upper divided by lower; the minimum is lower divided by upper
Subtraction and division are the ones to think through rather than memorise: to make a difference as large as possible you want to take away as little as possible, and to make a quotient as large as possible you want to divide by as little as possible.
Worked example
A rectangle has length 12.4 cm and width 6.8 cm, each measured to 1 decimal place. Find the upper and lower bounds of its area, and give the area to a suitable degree of accuracy.
- Each value is to 1 decimal place, so the half-unit is 0.05.
- Length: 12.35 ≤ l < 12.45. Width: 6.75 ≤ w < 6.85.
- Area is a product, so the upper bound uses both upper bounds: 12.45 × 6.85 = 85.2825 cm²
- The lower bound uses both lower bounds: 12.35 × 6.75 = 83.3625 cm²
- Now decide the sensible accuracy by comparing the two bounds digit by digit.
- They agree on 8 and on 3 in the tens and units, then differ — 85.28… against 83.36… — so they do not even agree to 2 significant figures.
Answer: The area lies between 83.3625 cm² and 85.2825 cm². The bounds agree only on the leading 8, so the area can only be given as 80 cm² to 1 significant figure — quoting 84.32 cm² would claim an accuracy the measurements do not support.
That last step is what "apply and interpret limits of accuracy" means. A question asking for "a suitable degree of accuracy" wants you to compare the upper and lower bounds and quote only the digits on which they agree.
- Upper bound
- the largest value a rounded measurement could have been before rounding
- Lower bound
- the smallest value a rounded measurement could have been before rounding
Before you practise
- Area conversions square the length factor; volume conversions cube it.
- Speed = distance ÷ time, density = mass ÷ volume, pressure = force ÷ area.
- Time is not decimal: 2.5 hours is 2 hours 30 minutes.
- To estimate, round every number to 1 significant figure and show that working.
- Decimal places count after the point; significant figures count from the first non-zero digit.
- Round in one step, using the next digit only.
- An error interval for a value rounded to the nearest unit is lower ≤ x < upper, half a unit either side.
- Truncation cuts digits off, giving the interval x ≤ value < x + 1 unit.
- Adding and multiplying: both uppers for the maximum, both lowers for the minimum.
- Subtracting: upper minus lower for the maximum. Dividing: upper divided by lower for the maximum.
- For a suitable degree of accuracy, quote only the digits on which the two bounds agree.