Round to decimal places and significant figures, estimate by rounding to one significant figure, and find and use upper and lower bounds, including in calculations with compound measures.

What this dot point is asking

This CCEA Number topic is about accuracy: how to round numbers sensibly, how to estimate a calculation quickly, and how to capture the range a rounded measurement could really lie in using upper and lower bounds. You must round to decimal places and significant figures, estimate by rounding to one significant figure, and find and use bounds, including in compound-measure calculations at Higher tier. Bounds questions are reliable marks because the method is the same every time.

Rounding

To round to a given number of decimal places, look at the first digit you are removing. If it is 5 or more, round up; if it is less than 5, round down. So $3.247$ to 2 decimal places is $3.25$.

Significant figures count from the first non-zero digit. In $0.004073$ the first significant figure is the $4$, so to 2 significant figures this is $0.0041$. To 1 significant figure, $38.7$ is $40$ and $0.0529$ is $0.05$. Be careful to keep the place value: rounding $4872$ to 1 significant figure gives $5000$, not $5$.

Estimating

To estimate the value of a calculation, round every number to one significant figure and work out the simpler sum. This is a fast check that a calculator answer is reasonable. For $\tfrac{61.3 \times 4.8}{2.1}$, estimate $\tfrac{60 \times 5}{2} = 150$. The estimate need not be exact; it just needs to be close enough to catch a mistake of a factor of ten.

Upper and lower bounds

A rounded measurement does not give the exact value, only a range. If a length is $7$ cm to the nearest centimetre, the true length is anything from $6.5$ cm up to (but not including) $7.5$ cm. The trick is to find the half-unit of the rounding and add or subtract it.

Bounds in calculations

When you combine bounded measurements, you must choose the right bound for each to make the answer as large or as small as possible.

Why this matters

Approximation underpins sensible answers throughout the exam: a quick estimate catches calculator slips, and rounding correctly to the requested accuracy is the difference between full and partial marks. Bounds questions show that measurement is never perfectly exact, an idea that links to compound measures, area and volume, and they are predictable Higher-tier marks once the half-unit method is automatic.