HookThe airliner that ran out of fuel at 41,000 feet
On 23 July 1983 Air Canada Flight 143, a brand-new Boeing 767, ran out of fuel in the cruise at roughly 41,000 feet over Ontario. Both engines fell silent; the crew glided the powerless jet about 100 km and landed it, without engines, on a disused airstrip at Gimli, Manitoba. Everyone survived. The cause was not the weather or the aircraft — it was a units mistake. Canada was switching to metric, the 767 measured fuel in kilograms, and the ground crew worked out the load using a conversion factor meant for pounds. They ended up loading roughly half the fuel the aircraft needed, and nobody's rough mental estimate caught it before take-off.
Every idea in this section is a piece of the defence the Gimli crew were missing. You will handle standard units of mass, length, time and money — including compound measures like speed and density that stitch two units together — and do it with the decimal quantities that real measurement throws at you. You will use estimation as a lie-detector, rounding numbers to something friendly to check that an answer is even the right size. You will round to an appropriate degree of accuracy, neither more nor less than a situation deserves. And you will read the limits of accuracy hiding inside any rounded measurement, writing them as error intervals with inequality signs. None of it is hard arithmetic; all of it is the habit that keeps a 767 in the sky.
ModelStandard units and compound measures
Metric units convert by powers of ten, which is why the system exists: 1 kg = 1000 g, 1 m = 100 cm, 1 litre = 1000 ml. Move up a unit and you divide; move down and you multiply. The one place this breaks is time, which runs in sixties, not tens — and that is the source of the most common measures error on the whole paper. 2.25 hours is not 2 hours 25 minutes; the 0.25 is a quarter of an hour, and a quarter of 60 minutes is 15, so 2.25 hours = 2 hours 15 minutes.
A compound measure is built from two other units, and the unit itself tells you the calculation. Speed is measured in kilometres per hour, and 'per' means divide, so speed = distance ÷ time. Density is grams per cubic centimetre, so density = mass ÷ volume. Rates of pay are pounds per hour; unit prices are pence per gram. In every case the wording of the unit is the formula, which means you never have to memorise a triangle — read the 'per' and divide.
Working with decimal quantities is where accuracy is won or lost. Keep the units consistent before you calculate (do not divide kilometres by minutes and call it km/h), and keep the full decimal precision through the working, rounding only at the very end.
A train covers 84 km in 0.7 hours — how fast is it going? Speed = distance ÷ time = 84 ÷ 0.7 = 120 km/h. Now a pay calculation with decimals: a shift of 3.5 hours at £11.44 per hour earns 3.5 × 11.44 = £40.04. And a density: a metal block of mass 240 g and volume 30 cm³ has density 240 ÷ 30 = 8 g/cm³. Finally the time trap — a job logged as 2.25 hours is 2 hours 15 minutes (because 0.25 × 60 = 15), not 2 hours 25 minutes.
MechanismEstimation — the lie-detector for every answer
To estimate, round every number in a calculation to one significant figure, then do the easy sum in your head or on the page. The point is not to get the exact answer — it is to find out what size the exact answer should be, so that a slipped decimal point or a mis-keyed calculator screams at you. The Gimli crew needed exactly this: a one-figure estimate of the fuel would have shown the load was roughly half of what a transatlantic-class jet burns, and the number would have looked obviously wrong.
The most famous version of this failure is NASA's Mars Climate Orbiter, lost in 1999 when one team worked in imperial units and another in metric; the spacecraft, reportedly a $327 million mission, flew too low and burned up. A sanity-check estimate is cheap insurance against errors that expensive. On a calculator paper, AQA specifically asks you to check answers 'obtained using technology' — the estimate is how you prove the calculator's answer is believable, not just copied down.
Estimation also handles awkward division. Rounding a divisor to one significant figure can turn an ugly sum into a clean one, and dividing by a number below 1 (like 0.5) makes the answer bigger, which catches students out — 180 ÷ 0.5 = 360, because you are asking how many halves fit into 180.
Estimate 5.87 × 31.2 ÷ 0.48. Round each to one significant figure: 6 × 30 ÷ 0.5. Work left to right: 6 × 30 = 180, then 180 ÷ 0.5 = 360. So the answer should be around 360. The true value is 5.87 × 31.2 ÷ 0.48 ≈ 381.6 — comfortably the same size, which tells you a calculator answer of 38 or 3800 would be a keying error. The estimate did its job: it fixed the order of magnitude.
ModelRounding to an appropriate degree of accuracy
There are two rounding languages, and Foundation expects both. Decimal places count digits after the point; significant figures count from the first non-zero digit, wherever it sits. The rule is the same for both: look at the next digit along — 5 or more rounds up, 4 or less rounds down. What changes is where you start counting, and that is the source of most rounding errors.
Significant figures are the trickier of the two because leading zeros do not count. In 0.04963 the first significant figure is the 4, not either zero, so rounding to 2 significant figures keeps the 4 and the 9, looks at the next digit (6), rounds the 9 up — and that carry turns 0.049 into 0.050. Students who miscount and give 0.05 have rounded to one significant figure by accident.
'Appropriate' is a judgement, not a fixed number of places, and it is marked. Money is quoted to the nearest penny, so an answer of £12.678 becomes £12.68; a length from a ruler marked in millimetres is sensibly given to the nearest millimetre; a population is not given to the nearest person. And you round once, at the end — rounding partway through a calculation and then carrying the rounded figure onward drags error into every later step, a habit the mark scheme quietly punishes.
Round 3.14159 to 2 decimal places: the third decimal is 1, which is below 5, so it stays — 3.14. Round the same number to 2 significant figures: the first two significant figures are 3 and 1, the next digit is 4, so it stays — 3.1. Now the leading-zero case: round 0.04963 to 2 significant figures. Skip the zeros; the significant figures are 4 and 9; the next digit is 6, so the 9 rounds up to 10, carrying to give 0.050. Writing 0.05 would be only one significant figure — a common and costly slip.
MechanismError intervals — what a rounded number is hiding
Every rounded measurement is a lie of a controlled size, and this leaf is about measuring that size. If a length is given as 46 cm 'to the nearest centimetre', the true length could be anything that rounds to 46: as low as 45.5 cm and anything below 46.5 cm. We capture that with inequality notation — 45.5 ≤ L < 46.5 — and the two different signs matter. The lower bound uses ≤ because 45.5 itself rounds up to 46 and so is allowed; the upper bound uses < because 46.5 would round up to 47 and so is not allowed. This is an error interval, and it is exactly why section N1.1 made such a fuss about the difference between ≤ and <.
Truncation is rounding's blunt cousin: instead of rounding to the nearest, it simply chops off the unwanted digits. A value truncated to 8 could be anything from 8 up to just below 9, because truncation never rounds up — so the interval is 8 ≤ x < 9, sitting entirely above the stated figure rather than straddling it. Reading whether a measurement was rounded or truncated tells you which interval to write.
At Foundation you interpret and state these limits of accuracy; you write the interval and you understand that the real value lies inside it. Combining bounds in a further calculation — finding the largest possible area from the largest possible sides, for instance — is Higher-tier only, so a Foundation question will ask you for the interval itself, not for arithmetic performed on its ends.
A concert crowd is reported as 2000 'to the nearest 100'. Half of 100 is 50, so the true figure runs from 2000 − 50 up to 2000 + 50: the error interval is 1950 ≤ n < 2050 (the lower end included, the upper end excluded, because 2050 would round up to 2100). Separately, a plank measured as 46 cm to the nearest centimetre satisfies 45.5 ≤ L < 46.5. And a score that has been truncated to a whole 8 satisfies 8 ≤ x < 9 — the interval sits above 8 because truncation only ever chops downward.
VocabularyKey terms the mark scheme pays for
TrapsMisconceptions that cost marks
ExamWhat examiners want
A one-significant-figure estimation question is close to guaranteed on Paper 1, and the marks are for the method: write each rounded number, then the easy calculation, then the estimate — even a candidate whose final arithmetic slips keeps marks for a correct rounded set-up. On the calculator papers you may be asked to estimate specifically to check a technology answer, so show the estimate alongside the exact value and state that they agree in size.
For units and compound measures, convert everything into consistent units before you calculate and let the 'per' in the unit tell you to divide. Treat time with suspicion: base-60, never base-10, so convert decimal hours to minutes with ×60. Keep full precision through the working and round only once, at the end, to an accuracy the context justifies — money to the penny, and never round mid-calculation.
Error-interval questions are marked on the inequality signs as much as the numbers: the lower bound takes ≤ (it rounds up into the value) and the upper bound takes < (it would round up to the next value). Halve the rounding unit to find how far each way the true value can sit. Remember the tier boundary — Foundation states and interprets the interval, while calculating with upper and lower bounds is Higher-only, so you will never be asked to multiply two bounds together on this paper.