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Smallest unit and limits of accuracy

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A parcel mass is 640 g when rounded to the nearest 10 g. Halfway values round up. Which interval describes the possible mass m?

Year 10, Chapter 1, 4 difficulty levels, No calculatorBuild a worksheet

Bounds from a rounded measurement

These are rounding-only bounds for positive measurements with halfway values rounded up. Use the stated rounding unit; trailing zeros in an unexplained whole-number reading can be ambiguous

\[\begin{gathered}\text{lower}=x-\frac{u}{2}\\\text{upper}=x+\frac{u}{2}\end{gathered}\]
  1. State the rounding unit Use the stated precision. A reading such as 640 g alone does not tell you whether it was rounded to 1 g or 10 g.
  2. Allow half a rounding unit For rounding to the nearest unit u, the rounding uncertainty extends u/2 on each side. This is a bound, not the actual error of every reading \(\displaystyle \text{lower}=x-\frac{u}{2}\)
  3. Keep the upper boundary open For positive measurements with halfway values rounded up, the lower boundary is included and the upper boundary is excluded \(\displaystyle x-\frac{u}{2}\le t<x+\frac{u}{2}\)
  4. Separate other errors These bounds model rounding only. A badly calibrated instrument or an inconsistent measuring method can introduce additional error

Worked example

A length is 8.2 cm rounded to the nearest 0.1 cm. Find the possible interval

\[x=8.2\text{ cm},\quad u=0.1\text{ cm}\]
  1. \(\displaystyle u/2=0.05\text{ cm}\)Use half of the rounding unit
  2. \(\displaystyle 8.2-0.05=8.15\text{ cm}\)The lower bound is included
  3. \(\displaystyle 8.2+0.05=8.25\text{ cm}\)The upper bound is excluded
  4. \(\displaystyle 8.15\le t<8.25\)Here t is measured in centimetres. This interval does not cover additional calibration or method error

Common mistakes

Using a full unit
✕\(\displaystyle 5.6\le t<5.8\)✓\(\displaystyle 5.65\le t<5.75\)
A mass of 5.7 kg rounded to 0.1 kg has half-unit uncertainty 0.05 kg, not 0.1 kg
Treating an upper bound as attainable
✕\(\displaystyle P_{\max}=130.2\)✓\(\displaystyle P<130.2\)
For four equal sides each less than 32.55 cm, the perimeter can approach 130.2 cm but cannot equal it

Difficulty levels

introIdentify the rounding unit from an explicitly stated recording convention
standardCalculate lower and upper bounds of one- and two-decimal measurements
challengeRecover recorded values or combine bounds for repeated lengths and perimeters
examRecognise the correct pair of bounds and reject common half-unit errors

Questions about this topic

Does 640 g always mean nearest 10 g?+

No. The rounding precision must be stated. Whole-number trailing zeros alone do not establish the resolution

Is half the rounding unit the actual measurement error?+

No. It bounds rounding uncertainty under the stated convention. The actual error depends on the unknown true value, and other measurement errors may exist

Why is there no greatest value below the upper bound?+

Any value below the boundary can be increased a little while remaining below it. The boundary is a strict upper bound, not a value the measurement can attain

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 10 Mathematics, chapter 1: Measurement

In this chapter

Convert measurement unitsPerimeter of standard shapesFind an unknown side from a perimeterCircumference of a circlePerimeter of composite shapesArea of standard shapesArea of a circleArea of a sectorArea of composite shapes1F Prism surface area: Surface area of solidsSurface area of a triangular prismExposed surface area of joined cuboids1G Cylinder surface area: Surface area of solidsSurface area of a cylinder portionSurface area of a cylinder joined to a cuboidVolume of prisms and cylindersCapacity of a containerVolume of a cylinder portionJoined and hollow prism-cylinder volumesModel rainwater collection and storageSmallest unit and limits of accuracyMeasurement bounds in calculationsVolume of cones, pyramids and spheres (optional extension)1J Level 10A extension: pyramids, cones and spheres: Surface area of solids (optional extension)Surface areas of advanced solids and composites (optional extension)