AQA GCSE Mathematics 8300: Measures and accuracy – Study Guide

This study guide teaches section 3.1.3, Measures and accuracy (specification references N13 to N16), from Topic 1 Number of the AQA GCSE Mathematics (8300) specification, for teaching from September 2015 with exams from May/June 2017 onwards (version 1.0). Most of it is assessed at both Foundation and Higher tier. Error intervals and limits of accuracy are additional foundation content, so they are on both tiers too. Calculating with upper and lower bounds is Higher tier only and is labelled below. Exams are available every May/June and November.

Paper 1 is non-calculator, so conversions, rounding and estimation must be done by hand there; Papers 2 and 3 allow a calculator.

Use this guide with the Measures and accuracy revision notes and the Measures and accuracy practice questions. For the whole of Topic 1, see the Number study guide, the Number revision notes and the Number practice questions. The course hub is AQA GCSE Mathematics, the printable checklist lists every statement, and the free 10-minute diagnostics show where to start.

What this section covers

Spec	What you must be able to do	Tier
N13	Use standard units of mass, length, time, money and other measures, including standard compound measures, using decimals where appropriate; know metric conversion factors for length, area, volume and capacity	Both
N14	Estimate answers; check calculations (including calculator answers) by approximation and estimation; evaluate the results	Both
N15	Round numbers and measures to an appropriate degree of accuracy (decimal places, significant figures, in context); do not round during intermediate steps	Both
N15	Use inequality notation to write simple error intervals due to truncation or rounding	Both (additional foundation)
N16	Apply and interpret limits of accuracy	Both (additional foundation)
N16	Upper and lower bounds	Higher tier only

Imperial/metric conversion factors are given in the question. You must know the metric ones.

N13: Standard units and conversions

Metric units you must 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
Time:      60 s = 1 min   60 min = 1 hour   24 hours = 1 day
Money:     100p = £1

Area and volume conversions

An area unit is a length unit squared, so the length factor is squared. A volume unit is a length unit cubed, so the factor is cubed.

1 cm² = 10² mm² = 100 mm²          1 m² = 100² cm² = 10 000 cm²
1 cm³ = 10³ mm³ = 1000 mm³         1 m³ = 100³ cm³ = 1 000 000 cm³

Worked example 1 (non-calculator). Convert 0.62 m² to cm², and 2.4 m³ to litres.

0.62 m² = 0.62 × 10 000 cm² = 6200 cm²

2.4 m³ = 2.4 × 1 000 000 cm³ = 2 400 000 cm³
2 400 000 cm³ ÷ 1000 = 2400 litres

A useful link: 1 m³ = 1000 litres.

Time as a decimal

Time is not decimal. 3.4 hours is not 3 hours 40 minutes.

Worked example 2. Write 3.4 hours in hours and minutes. A coach travels 189 km in 2 hours 15 minutes. Find its average speed.

0.4 hours = 0.4 × 60 = 24 minutes, so 3.4 hours = 3 h 24 min

2 h 15 min = 2 + 15/60 = 2.25 hours
speed = 189 ÷ 2.25 = 84 km/h

Speed, density, pressure, rates of pay and unit prices are the standard compound measures, taught in full in the Ratio, proportion and rates of change revision notes.

Money and imperial units

Money answers in pounds go to 2 decimal places: £17.3 should be written £17.30. Cheese at £12.80 per kg costs 1.35 × 12.80 = £17.28 for 1.35 kg.

For an imperial conversion, the question gives the factor. If you are told 1 mile ≈ 1.6 km, then 35 miles ≈ 35 × 1.6 = 56 km. A mile is longer than a kilometre, so the number of kilometres is bigger.

N14: Estimation and checking

To estimate, round each number to 1 significant figure (unless another choice makes the arithmetic easier), then work out the simpler calculation. Show the rounded values: the method mark is for them.

Worked example 3 (non-calculator). Estimate the value of (61.8 × 4.93) ÷ 0.211.

61.8 ≈ 60     4.93 ≈ 5     0.211 ≈ 0.2

(60 × 5) ÷ 0.2 = 300 ÷ 0.2 = 1500

Dividing by 0.2 is the same as multiplying by 5. The exact value is 1443.95…

Checking a calculator answer. Suppose someone keys in the same calculation and writes 144.4. The estimate of 1500 shows the answer is about ten times too small, so a decimal point or a key has gone wrong. This is what N14 means by checking answers obtained using technology.

Over or underestimate?

You are often asked whether an estimate is too big or too small. Look at the direction of each rounding.

Worked example 4. 19 tins cost £3.85 each. Ana estimates the total as 20 × £4 = £80. Is this an overestimate or an underestimate?

Both numbers were rounded up, and both are multiplied, so the estimate is bigger than the true value: an overestimate. (The true total is £73.15.)

Rounding a divisor down makes the answer bigger. If one rounding pushes up and another pushes down, you cannot tell without working it out.

N15: Rounding to an appropriate degree of accuracy

Decimal places and significant figures

• Decimal places count digits after the decimal point.
• Significant figures count from the first non-zero digit. Zeros before it are placeholders and are not significant.

Worked example 5. Round 0.0040372 to 3 significant figures, 58 649 to 2 significant figures and 7.996 to 2 decimal places.

0.0040372 -> first significant digit is 4; digits 4, 0, 3; next digit 7 rounds up
          -> 0.00404

58 649    -> digits 5, 8; next digit 6 rounds up -> 59 000 (keep the place value)

7.996     -> 7.99|6 rounds up, carrying -> 8.00 (the zeros show 2 d.p.)

Rounding in context

The context decides the sensible answer. 230 people need coaches with 52 seats each: 230 ÷ 52 = 4.42…, so 5 coaches are needed. Rounding to the nearest whole number would leave people behind. In the same way, you round down to find how many full boxes you can fill.

If no accuracy is given, match the data: eight calculator digits do not suit a length measured to the nearest centimetre.

Do not round in the middle

The specification says you should not round values during intermediate steps. Keep the full calculator value (or use the answer memory) and round once at the end.

Worked example 6 (calculator). A tank measures 1.37 m by 0.84 m by 0.62 m. Find its volume in litres to the nearest litre.

1.37 × 0.84 × 0.62 = 0.713496 m³
0.713496 × 1000 = 713.496 litres  ->  713 litres

If you round 1.37 × 0.84 = 1.1508 to 1.2 first, you get 1.2 × 0.62 = 0.744 m³, which is 744 litres. That is wrong by more than 30 litres.

Error intervals (both tiers)

A rounded value stands for a range of possible values. The error interval shows that range with inequalities.

For a value rounded to the nearest unit u, the true value lies within half a unit either side.

x = 6.3 to 1 decimal place:      6.25 ≤ x < 6.35
n = 3700 to the nearest 100:     3650 ≤ n < 3750
m = 0.48 kg to 2 s.f.:           0.475 ≤ m < 0.485

Use ≤ on the left (6.25 rounds up to 6.3) and < on the right (6.35 rounds to 6.4).

Truncation means cutting off digits without rounding. A truncated value is never more than the true value, so the interval goes from the given value up by one whole unit.

y = 6.3 truncated to 1 decimal place:   6.3 ≤ y < 6.4
age 15 (whole years, truncated):        15 ≤ age < 16

N16: Limits of accuracy

Interpreting limits (both tiers)

The ends of an error interval are the limits of accuracy. Use them to decide whether you can be sure of something.

Worked example 7. A parcel has a mass of 2.0 kg to the nearest 0.1 kg. The cheapest postage rate is for parcels up to 2.04 kg. Can you be sure the parcel qualifies?

1.95 ≤ mass < 2.05

The mass could be anything up to (but not including) 2.05 kg, for example 2.045 kg, which is over 2.04 kg. So you cannot be sure. A good answer quotes the limit and compares it.

Upper and lower bounds in calculations (Higher tier only)

The lower bound (LB) and upper bound (UB) are the limits of accuracy. When rounded values are combined, choose the bounds that make the answer as large or as small as possible.

Calculation	Maximum (UB)	Minimum (LB)
a + b	UB a + UB b	LB a + LB b
a − b	UB a − LB b	LB a − UB b
a × b	UB a × UB b	LB a × LB b
a ÷ b	UB a ÷ LB b	LB a ÷ UB b

For subtraction and division the second value crosses over: taking away less, or dividing by less, gives a bigger answer.

Worked example 8 (Higher, calculator). A plank is 2.4 m long, to the nearest 0.1 m. A piece 85 cm long, to the nearest centimetre, is cut off. Find the greatest and least possible lengths left, in metres.

Plank:  2.35 ≤ P < 2.45        Piece:  0.845 ≤ Q < 0.855

Greatest left = UB P − LB Q = 2.45 − 0.845 = 1.605 m
Least left    = LB P − UB Q = 2.35 − 0.855 = 1.495 m

Convert to the same unit before you use the bounds.

Worked example 9 (Higher, calculator). A cyclist rides 1500 m, to the nearest 10 m, in 97.6 seconds, to 1 decimal place. Work out her average speed to a suitable degree of accuracy. Give a reason.

Distance: 1495 ≤ d < 1505        Time: 97.55 ≤ t < 97.65

UB speed = UB d ÷ LB t = 1505 ÷ 97.55 = 15.427... m/s
LB speed = LB d ÷ UB t = 1495 ÷ 97.65 = 15.309... m/s

To 2 s.f. both bounds round to 15.
To 3 s.f. they differ (15.4 and 15.3).

Average speed = 15 m/s (2 s.f.), because the upper and lower bounds agree to 2 significant figures. This “round both bounds until they agree” method is how you answer “to an appropriate degree of accuracy” questions on bounds.

You use the upper bound in calculations even though the interval ends with <: use 1505, not 1504.99…

Common errors

• Using 100 instead of 10 000 to change m² to cm², or 1000 instead of 1 000 000 for m³ to cm³.
• Estimating without writing down the rounded values, so the method mark is lost.
• Rounding 0.0040372 to 3 s.f. as 0.004 (that is 1 s.f.) or counting the leading zeros.
• Giving 4 coaches for 230 people because 4.42 “rounds to 4”.
• Rounding an intermediate answer, then rounding again at the end.
• Writing an error interval with ≤ at both ends, or forgetting that truncation only goes upwards.
• (Higher) Using UB ÷ UB for the maximum of a division, or UB − UB for the maximum of a subtraction.

Where to go next

Condense this page with the Measures and accuracy revision notes, then test yourself with the Measures and accuracy practice questions. Area and volume formulas used with these units are in the Geometry and measures study guide.

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.