Number - Measures and accuracy: apply and interpret limits of accuracy (Foundation tier)

Overview

A measurement written after rounding is a useful estimate, not a promise that the object has exactly that size. This topic teaches you to turn a rounded value into the range of true values that could have produced it, then choose the end of the range that matters in a real situation. That could be checking whether a shelf is long enough, whether a quantity is safely below a limit, or whether a stated length could be bigger than it looks. It is a Number topic because careful use of decimals, inequalities and units protects the accuracy of every later calculation. In Foundation questions, candidates commonly state a simple error interval, read it from a number line, or explain a decision in context. N15 supports the rounding and inequality notation; N16 asks you to apply and interpret limits of accuracy. Read the degree of accuracy first, show the interval, and finish with a sentence that answers the actual question. The examples here are original illustrative questions, not Pearson questions or official mark allocations. 1 2

Key Concepts

Concept 1: A rounded value hides a range

If a shelf is recorded as 60 cm to the nearest cm, it might be 60 cm exactly, but it might also be 59.6 cm or 60.49 cm. All of these round to 60 cm. The rounding unit is 1 cm, so half a unit is 0.5 cm. The true length, (l), is therefore:

[59.5 \le l < 60.5\text{ cm}]

The lower limit is 59.5 cm because it rounds up to 60 cm. The upper limit is 60.5 cm, but it is excluded: 60.5 cm rounds to 61 cm, not 60 cm. This is why the final inequality is <, not ≤. There is no actual largest possible real measurement. Values can get arbitrarily close to 60.5 cm from below, such as 60.499 cm, but cannot equal it.

Why it works: rounding uses a halfway point. Values below the next halfway point remain nearer the stated rounded value; reaching that halfway point sends the value to the next rounded result. Treat the interval as a fence with a solid left gate and an open right gate.

Cover and recall: hide the interval. What is the half-unit? Which endpoint is included? Why is 60.5 cm excluded? Say the answers before looking again.

Concept 2: Find the half-unit and show it on a number line

First identify what the value was rounded to. For 60 cm to the nearest cm, the unit is 1 cm and the half-unit is 0.5 cm. For 340 cm to the nearest 10 cm, the unit is 10 cm and the half-unit is 5 cm, so (335 \le x < 345) cm. For 2.7 kg to the nearest 0.1 kg, the unit is 0.1 kg and the half-unit is 0.05 kg, so (2.65 \le m < 2.75) kg.

Number-line guidance: place the rounded value in the middle. Mark one half-unit either side. Put a filled circle at the lower limit because it is included, and an open circle at the upper limit because it is excluded. Shade or draw the segment between them. Label the units. Do not draw 59 cm and 61 cm for a value rounded to the nearest cm: that wrongly uses a whole unit each side.

This visual method is particularly useful when a question says write down the error interval or state the possible values. Translate the picture into an inequality only after checking the degree of accuracy. When reading an unfamiliar number line, check its scale rather than assuming each division represents one unit.

Concept 3: Choose the limit the context needs

An interval by itself may not answer the question. Ask, which possible value makes the situation hardest? If a box must fit on a shelf, consider a very large possible box length and a very small possible shelf length. If a minimum amount is required, consider the lower limit of the amount available. If a safety limit must not be exceeded, consider the upper limit of the measurement, remembering that the endpoint itself is not attained.

The following box-total reasoning is the requested bridge/enrichment. Foundation candidates should prioritise simple intervals and interpreting accuracy; formal compound-bound techniques are not the focus of this guide.

Suppose four box lengths are each given as 30 cm to the nearest cm. For one box, (29.5 \le b < 30.5) cm. Four such positive lengths have a total (118 \le T < 122) cm. The 122 cm figure is anupper limit, not an attainable maximum. Every box must be shorter than 30.5 cm, so their total must be shorter than 122 cm.

A shelf described as exactly 120 cm has no rounding interval. The boxes are not guaranteed to fit because an allowed set of boxes could be 30.4 cm each, totalling 121.6 cm. That does not mean they definitely will not fit: 29.9 cm each totals 119.6 cm, so fitting is also possible. These actual permitted measurements provide clear evidence without treating an excluded endpoint as a real measurement.

If the shelf is instead recorded as 120 cm to the nearest cm, write (119.5 \le S < 120.5) cm. For a guarantee question, test large boxes against a small shelf. For example, 121.6 cm of boxes will not fit on a permitted 119.6 cm shelf. A single valid counterexample disproves a guarantee.

Concept 4: Rounding, truncation and the supported N15 skill

N15 supplies the language behind the interval. Rounding selects the nearest stated unit. Truncationcuts digits off without rounding. For a positive value truncated to one decimal place as 4.2, the true value is (4.2 \le x < 4.3), because values from 4.20 up to 4.299... all truncate to 4.2. This is not the midpoint rule used for rounding.

Rounding also needs an appropriate degree of accuracy. An answer measured only to the nearest centimetre should not be reported as 60.000 cm. Extra decimal places suggest certainty the measurement does not have. In a multi-step question, avoid premature rounding of calculations, but state the final conclusion at a sensible accuracy and with units.

For significant figures, identify the place value of the final retained digit. For example, 340 to two significant figures is rounded to the nearest 10, so its interval is (335 \le x < 345). The number of significant figures is not itself the rounding unit.

Mathematical/Scientific Relationships

The following number relationships must be memorised. The Pearson 2026 Foundation exam aid does not supply an error-interval or limits-of-accuracy relationship. 3

Must memorise, rounding to the nearest unit (u): if a positive measurement (x) is recorded as (r), then:

[r-\frac{u}{2}\le x<r+\frac{u}{2}]

Here (r) is the rounded reading, (u) is the rounding unit, and (x) is the true value. Subtract half the rounding unit for the lower limit and add it for the upper limit.

Must memorise, positive truncation to unit (u):

[r\le x<r+u]

The displayed value is the lower limit, not the centre. This version is stated for positive quantities; do not apply it blindly to negative truncation examples.

Must memorise, repeated lengths as bridge/enrichment: for (n) positive objects each recorded as (r) to nearest (u), reason from:

[n(r-u/2)\le T<n(r+u/2)]

Here (n) is an exact count and (T) is the total true length. This is repeated addition of intervals. Formal bounds for products of two uncertain measurements, quotients and compound measures are Higher-tier extensions and are omitted.

State the units clearly. An inequality should make three ideas visible: lower limit included, actual values between the endpoints, upper limit excluded. Do not write (59.5<l<60.5), because 59.5 cm really does round to 60 cm. Do not write (59.5\le l\le60.5), because 60.5 cm does not.

Unit conversions and accuracy

Convert before comparing lengths. Must memorise: (1\text{ m}=100\text{ cm}), (1\text{ cm}=10\text{ mm}), and (1\text{ kg}=1000\text{ g}). A half-unit must be in the same unit as the reading.

For example, 1.20 m to the nearest 0.01 m gives (1.195\le L<1.205) m. In centimetres this becomes (119.5\le L<120.5) cm. Convert the whole interval, not merely one endpoint. The half-unit 0.005 m becomes 0.5 cm, not 0.005 cm.

Practical Applications

Measurements are rounded because rulers, scales and manufacturing machines have limited resolution. A carpenter allows a margin so parts fit; a delivery company checks whether a plausible package will fit; a laboratory reports a balance reading to its resolution rather than inventing digits. In each case the question decides the direction of the check.

For a rule such as “must be under 5 kg”, a displayed value of 5.0 kg to the nearest 0.1 kg represents (4.95\le m<5.05) kg. It does not guarantee compliance: a permitted true mass could exceed 5 kg. A useful conclusion identifies an allowed counterexample rather than simply repeating the rounded reading.

Required practicals

None apply to this GCSE Mathematics topic. The practical skill is clear mathematical communication: state the interval, identify the relevant extreme, calculate with consistent units, and interpret it in words.

Exam Focus, Method and Common Mistakes

Use the HALF routine: Highlight accuracy, Amount halved, Limits written, Finish in context. Circle the accuracy phrase, identify the rounding step, halve that step, and write the interval before attempting the contextual decision. For positive truncation, replace the midpoint approach with the full-step interval above the displayed value.

A frequent error is treating 120 cm as exact when the question says “correct to the nearest cm”. Another is calling 122 cm the largest total of four rounded 30 cm boxes. It is only the excluded upper limit. A third error is saying the boxes “will not fit”. The evidence proves only that fitting is not guaranteed; a fitting case and a non-fitting case can both exist.

Pearson mark schemes use M for method, P for process, A for accuracy following a method or process, B for unconditional accuracy, and C for communication. The allocation depends on the individual question. 2 Candidates should show a short logical chain, but should not assume that every numerical question has the same marking pattern. The questions and exact mark allocations in this guide are original illustrative material.

Tier Boundary and Formula-Sheet Check

For the Foundation core, confidently use simple error intervals, inequality notation and contextual interpretation. The formal phrase upper and lower bounds is explicitly added in Higher N16. 1 The box-total examples are included as requested bridge/enrichment, not as a promise that such a formal compound-bound question is routine Foundation assessment.

The 2026 Foundation exam aid provides no bounds or error-interval formula to copy. Learn the half-unit rule and the open upper endpoint rather than expecting the formula sheet to supply them. 3

Ten-minute Tutor Podcast

Listen once, then answer the recall questions without looking at the text. The episode begins with an introduction at 00:00, core concepts at 01:00, exam tips at 06:00, a quick-fire quiz at 08:00 and a summary at 09:00.

References