AQA GCSE Mathematics 8300: Measures and accuracy – Revision Notes

These notes condense section 3.1.3, Measures and accuracy (N13 to N16), of the AQA GCSE Mathematics (8300) specification, for teaching from September 2015 with exams from May/June 2017 onwards (version 1.0). For full explanations and worked examples, use the Measures and accuracy study guide. Units, estimation, rounding, error intervals and limits of accuracy are on both tiers. Calculating with upper and lower bounds is Higher tier only and is marked (Higher). Any of the three papers can test this; Paper 1 is non-calculator, Papers 2 and 3 allow a calculator.

Links: Measures and accuracy practice questions, Number revision notes for the rest of Topic 1, the AQA GCSE Mathematics hub, the printable checklist and the free 10-minute diagnostics.

Spec at a glance

Ref	Skill	Tier
N13	Standard units, compound measures, metric conversions (length, area, volume, capacity)	Both
N14	Estimate; check answers, including calculator answers; evaluate results	Both
N15	Round to d.p., s.f. or a sensible accuracy in context; no rounding mid-calculation	Both
N15	Error intervals from rounding or truncation, using inequalities	Both
N16	Apply and interpret limits of accuracy	Both
N16	Upper and lower bounds	Higher tier only

N13: Conversion factors

Measure	Facts to know
Length	10 mm = 1 cm; 100 cm = 1 m; 1000 m = 1 km
Mass	1000 g = 1 kg; 1000 kg = 1 tonne
Capacity	1000 ml = 1 litre; 1 cm³ = 1 ml; 1000 cm³ = 1 litre; 1 m³ = 1000 litres
Area	1 cm² = 100 mm²; 1 m² = 10 000 cm²; 1 km² = 1 000 000 m²
Volume	1 cm³ = 1000 mm³; 1 m³ = 1 000 000 cm³
Time	60 s = 1 min; 60 min = 1 h; 24 h = 1 day

Imperial conversions (miles, pounds, pints and so on) are given in the question.

Method: converting a unit

1. Write the length conversion (e.g. 1 m = 100 cm).
2. Square it for area, cube it for volume.
3. Big unit -> small unit: multiply.  Small unit -> big unit: divide.
4. Check: the number of small units should be larger.

Worked reminders. 0.8 m³ = 0.8 × 1 000 000 = 800 000 cm³. 6500 m² = 6500 ÷ 1 000 000 = 0.0065 km².

Time. Minutes ÷ 60 gives decimal hours: 2 h 48 min = 2 + 48/60 = 2.8 hours. Decimal hours × 60 gives minutes: 0.75 h = 45 min.

Compound measures (speed, density, pressure, rates of pay, unit price) follow the same unit rules. See the Ratio, proportion and rates of change revision notes for the formulas.

N14: Estimation

Method: estimate a calculation

1. Round every number to 1 significant figure.
2. Write the rounded calculation down (this earns the method mark).
3. Work it out mentally.  Dividing by 0.5 doubles; dividing by 0.2 multiplies by 5.
4. If asked, say whether the estimate is too big or too small, and why.

Over or under?

You rounded…	Effect on the estimate
a number being multiplied or added up	too big
a number being multiplied or added down	too small
a divisor up	too small
a divisor down	too big
in both directions	cannot tell without more working

Use your estimate to check a calculator answer. If the estimate is 10 and the display shows 103.7, a key or decimal point is wrong.

Worked reminder. 29 notebooks at £1.95 each: 30 × £2 = £60. Both numbers were rounded up and multiplied, so £60 is an overestimate (the true cost is £56.55).

N15: Rounding

• Decimal places: count after the point. Keep trailing zeros: 3.996 to 2 d.p. is 4.00.
• Significant figures: start at the first non-zero digit. 0.07049 to 2 s.f. is 0.070 (the final zero is significant and should be shown).
• Keep place value: 349 500 to 2 s.f. is 350 000, not 35.
• Context: round up when everyone or everything must be included (buses, tins of paint); round down for “how many whole ones can you make or afford”.
• Money: pounds to 2 d.p.
• Never round mid-calculation. Store full values; round once at the end.

N15: Error intervals

Method: rounding

1. Find the unit of rounding (nearest 0.1, nearest 100, 2 s.f. -> what place?).
2. Halve it.
3. Lower limit = value − half unit  (≤)
   Upper limit = value + half unit  (<)

Method: truncation

Lower limit = the value itself     (≤)
Upper limit = value + one whole unit (<)

Given	Error interval
p = 14.7 to 1 d.p.	14.65 ≤ p < 14.75
q = 5000 to the nearest 100	4950 ≤ q < 5050
r = 9.2 truncated to 1 d.p.	9.2 ≤ r < 9.3

N16: Limits of accuracy

The two ends of the error interval are the limits of accuracy. To decide “can you be sure?”, compare the relevant limit with the target, and quote it in your answer.

(Higher) Upper and lower bounds in calculations

Operation	Maximum	Minimum
Add	UB + UB	LB + LB
Subtract	UB − LB	LB − UB
Multiply	UB × UB	LB × LB
Divide	UB ÷ LB	LB ÷ UB

(Higher) Worked reminder. A rectangle is 6.5 m by 3.2 m, both to 1 d.p. Area bounds:

UB = 6.55 × 3.25 = 21.2875 m²
LB = 6.45 × 3.15 = 20.3175 m²

(Higher) Method: answer to an appropriate degree of accuracy

1. Find the UB and LB of the answer.
2. Round both to 1 s.f., then 2 s.f., then 3 s.f. ...
3. Stop at the most figures where they still agree.  That is your answer.
4. Give the reason: "both bounds round to ... to ... s.f."

For the rectangle: to 1 s.f. both bounds round to 20; to 2 s.f. they are 21 and 20. So the area is 20 m² (1 s.f.).

Must-know distinctions

• Rounding vs truncation: rounding goes half a unit either way; truncation goes a whole unit upwards only.
• Error interval vs bounds: an error interval is one measurement; bounds in a calculation (Higher) combine several.
• d.p. vs s.f.: 0.0503 is 4 d.p. but 3 s.f.
• Length factor vs area/volume factor: 100 for cm to m; 10 000 for cm² to m²; 1 000 000 for cm³ to m³.
• Decimal time vs clock time: 1.3 hours is 1 h 18 min, not 1 h 30 min.

Quick self-test

1. Convert 4.5 cm² to mm².
2. Convert 3750 cm³ to litres.
3. Write 3 hours 36 minutes in hours, as a decimal.
4. Round 0.30219 to 2 significant figures.
5. Round 64 970 to 2 significant figures.
6. Estimate 397 × 0.512 ÷ 19.6.
7. Write the error interval for y = 23.6, rounded to 1 decimal place.
8. w = 4.7, truncated to 1 decimal place. Write the error interval.
9. n = 2800 to the nearest 100. Write the error interval.
10. A rope is 12 m to the nearest metre. A hook can hold a rope up to 12.4 m long. Can you be sure the rope is short enough?
11. (Higher) a = 7.4 and b = 2.6, both to 1 d.p. Find the upper bound of a − b.
12. (Higher) Using the same a and b, find the upper bound of a ÷ b to 3 s.f.

Answers

1. 4.5 × 100 = 450 mm²
2. 3750 ÷ 1000 = 3.75 litres
3. 36 ÷ 60 = 0.6, so 3.6 hours
4. 0.30
5. 65 000
6. 400 × 0.5 ÷ 20 = 10 (the exact value is 10.37…)
7. 23.55 ≤ y < 23.65
8. 4.7 ≤ w < 4.8
9. 2750 ≤ n < 2850
10. 11.5 ≤ length < 12.5. The rope could be longer than 12.4 m (for example 12.45 m), so no, you cannot be sure.
11. 7.45 − 2.55 = 4.9
12. 7.45 ÷ 2.55 = 2.9215… = 2.92

Where marks are usually lost

• Converting m² to cm² with × 100, or m³ to litres with × 1 000 000 instead of × 1000.
• Treating 0.4 hours as 40 minutes.
• Writing only the final estimate, with no rounded values shown.
• Dropping a significant trailing zero (writing 0.07 for 2 s.f.) or losing place value (writing 35 for 350 000).
• Rounding to the nearest whole number when the context needs rounding up (or down).
• Rounding part-way through and then rounding again at the end.
• Using ≤ at both ends of an error interval, or half a unit for a truncated value.
• Answering “can you be sure?” with yes/no but no limit quoted.
• (Higher) Using UB − UB or UB ÷ UB for a maximum.
• (Higher) Stating an answer to “appropriate accuracy” with no reason.

Official syllabus

AQA GCSE Mathematics (8300) specification, for teaching from September 2015 onwards, for exams in May/June 2017 onwards, version 1.0 (AQA), section 3.1.3 Measures and accuracy, references N13 to N16.