Calculations, estimation and sensible rounding

1. Approximation as a check on practical calculations

Calculations are a fundamental topic integrated throughout QCAA Essential Mathematics Units 3 and 4. The current syllabus includes correct arithmetic order, checking accuracy and reasonableness, leading-digit approximation, multi-step calculator use, decimal place value, and rounding up or down to the requested number of decimal places. These skills support measurement, probability, graphs, data and financial models and should be practised in those contexts.

An estimate gives a comparison for a more detailed calculation. If a supplier charges about $20 for each of about 30 items, a total near $600 is plausible; a display of $60 or $6000 needs investigation. The estimate does not replace accurate working when an exact result is requested. It helps detect a misplaced decimal point, an incorrect operation or a unit conversion in the wrong direction.

Technology supports these checks. The CIA requires technology and provides the QCAA Essential Mathematics formula book. Use the technology arrangements specified by your school and the task, and record the quantities and operations so another reader can follow the method. Entering an expression twice in the same way is less informative than checking it using a different approximation or relationship.

Reporting precision is a separate decision. Keep intermediate values until the requested final rounding unless the task specifies period-by-period rounding or a whole-item decision. A material order may need rounding upward to buy enough, while the number of complete items obtainable from a fixed supply may need rounding downward. Significant figures, interval bounds and percentage error appear later as related revision techniques; they are not presented as an extra official list of compulsory CIA topics.

2. Rounding rules, and the one rule that matters most

To round to a stated place, look at the next digit along. If it is 5 or more, round the digit you are keeping up by one; if it is 4 or less, leave it unchanged. Then drop the remaining digits (or replace them with zeros if they are to the left of the decimal point, to preserve place value).

Examples: 47.362 to one decimal place is 47.4, because the next digit is 6. 47.362 to two decimal places is 47.36, because the next digit is 2. 8492 to the nearest hundred is 8500. $63.487 to the nearest cent is $63.49. $63.487 to the nearest dollar is $63.

Two mechanical details catch people out. When rounding up causes a carry, the carry propagates: 9.98 to one decimal place is 10.0, not 9.9 or 10. And trailing zeros must be kept when they are needed to show the stated precision: 4.30 to two decimal places is written 4.30, not 4.3, and $12.00 is not written $12 when two decimal places are required.

A useful default for multi-step calculations is round only at the final step, unless the task specifies a different rounding rule. Carry every intermediate value at full calculator precision — using ANS or memory rather than rewriting a rounded figure — and apply the rounding instruction once, when you report the answer. Rounding early accumulates error. Consider a cost of 7 items at $4.286 each: rounding the unit price to $4.29 first gives $30.03, while the exact calculation gives $30.002, which is $30.00 to the nearest cent. Over a longer chain, or where a value is squared or cubed, the drift is larger and will regularly cost you the last digit the question asked for.

The exception is where a question explicitly instructs an intermediate rounding — for example, a pay calculation that says round the hourly rate to the nearest cent before multiplying, because that is what really happens. If the stem says it, do it; otherwise do not.

3. Significant figures: a different question from decimal places

Extension for interpreting measurement precision. The required calculations topic explicitly specifies decimal-place rounding. Significant figures are included here as a useful related convention, not an extra stated CIA requirement.

Decimal places count digits after the point. Significant figures count digits that carry information about the size and precision of the number, starting from the first non-zero digit. The two answers are usually different, and questions specify which they want.

The counting rules are: all non-zero digits are significant; zeros between non-zero digits are significant; leading zeros are never significant, because they only place the decimal point; trailing zeros after a decimal point are significant, because writing them is a claim about precision.

So 4831 has four significant figures. 0.00427 has three — the three zeros after the point are placeholders. 5.60 has three, because the final zero is a deliberate statement of precision. 40 900 is ambiguous in isolation, which is why a question will normally tell you what to round to rather than asking you to count trailing zeros in a whole number.

To round to a number of significant figures, find the first significant digit, count along to the position you need, then apply the ordinary rounding rule to the next digit — and preserve place value with zeros where necessary. 4831 to two significant figures is 4800, not 48. 0.004273 to two significant figures is 0.0043. 27.68 to three significant figures is 27.7. 0.09996 to three significant figures is 0.100, and the trailing zeros must be written.

Significant figures are the natural language of measurement, because they express relative precision: 4800 to two significant figures says the value is known to within about a hundred, whereas 4.8 to two significant figures says it is known to within about a tenth. Decimal places express absolute precision, which is why money uses them — a cent is a cent regardless of the size of the amount. Use the precision actually requested. Decimal-place rounding is specified in the QCAA calculations topic; significant figures are a related measurement convention included here for revision, not a prediction of a CIA question.

4. Leading-digit approximation and fast mental estimates

Leading-digit approximation is the technique of replacing each number by its first significant digit followed by zeros, then doing the resulting easy calculation mentally. It is designed for speed, not accuracy, and its purpose is to tell you the order of magnitude of an answer before or after you calculate it exactly.

To estimate 38 × 612 by leading digits, replace with 40 × 600 = 24 000. The exact value is 23 256, so the estimate is close enough to confirm the size. To estimate 4873 ÷ 61, use 5000 ÷ 60, which is about 83; the exact value is 79.9. To estimate the cost of 47 items at $19.60, use 50 × $20 = $1000; the exact figure is $921.20.

Two conventions exist and you should say which you used. Rounding to the leading digit sends 38 to 40 and gives a balanced estimate. Truncating to the leading digit sends 38 to 30 and deliberately produces an under-estimate, which is useful when you want to be sure you have not overspent. Either is acceptable in an answer provided your working shows the rounded values you used; what is not acceptable is presenting an estimate as though it were exact.

You can also improve an estimate by compensating, and saying so earns credit. If you rounded 38 up to 40 and 612 up to 620 down to 600, note whether your estimate is likely to be a little high or a little low. In 40 × 600 you rounded one factor up and the other down, so the errors partly cancel and the estimate is good. In 50 × $20 you rounded both up, so the estimate is an over-estimate — which is exactly the useful information for a budgeting question.

This estimation method has two practical uses. Sometimes the command word is literally Estimate, and then the estimate is the answer and your working must show the approximate values you substituted. More often, estimation is your private check: before accepting 15 132 from the calculator for 4.85 × 312, note that 5 × 300 = 1500 and that your screen is a factor of ten out. This check is worth building into every calculation in the paper, and it costs about three seconds.

5. Floor and ceiling values: when the context, not the arithmetic, decides

The floor of a number is the largest whole number less than or equal to it — round down, always. The ceiling is the smallest whole number greater than or equal to it — round up, always. Neither depends on whether the next digit is 5 or more; they depend entirely on the situation.

Use a ceiling when a requirement must be met or exceeded, and part of a unit is no use. Transporting 213 people in 45-seat buses needs 213 ÷ 45 = 4.73 buses, so 5 buses. Covering a 38 m² wall with paint that covers 12 m² per tin needs 3.17 tins, so 4 tins. Fencing a perimeter of 74 m with 2.4 m panels needs 30.8 panels, so 31 panels. In each case, ordinary rounding would give the wrong answer and the shortfall would be real.

Use a floor when a budget, a supply or a capacity limits you and you cannot exceed it. With $500 and tickets at $37.50, you can buy 13.33 tickets, so 13 tickets. From a 6 m length of timber you can cut 6 ÷ 0.85 = 7.06 pieces of 85 cm, so 7 pieces. If a lift is rated to 900 kg and the average load is 78 kg, its capacity is 11.5 people, so 11 people.

The reasoning to write down is short and it is what earns the mark: state the division, state the exact quotient, then state the whole-number answer with the reason. 213 ÷ 45 = 4.73, so 5 buses are needed, because 4 buses would carry only 180 people. That final clause converts a rounded number into a justified decision.

Two extensions appear. Sometimes a question asks for the leftover: with 5 buses there are 225 seats for 213 people, so 12 seats are spare; with 4 tins covering 48 m² there is 10 m² of coverage unused. Sometimes it asks for the cost, and the cost is based on the whole units you must buy, not on the fractional need: 4 tins at $58.90 is $235.60, not 3.17 tins' worth. Charging for the fractional amount is one of the most common errors in materials-and-cost questions.

6. Interval estimates and judging reasonableness

An interval estimate gives a range within which the true value must lie, rather than a single figure. It is useful when inputs are uncertain. Interval calculations are included here to extend reasoning about estimates; the QCAA calculations list explicitly names leading-digit approximation, decimal-place rounding and checking reasonableness, not a separate interval-estimation requirement.

The method is to calculate twice: once using the lowest plausible values of the inputs, and once using the highest plausible values. The two results bracket the true answer. If a function will have between 80 and 95 guests and catering costs between $22 and $26 per head, the cost lies between 80 × 22 = $1760 and 95 × 26 = $2470. You would report this as between about $1800 and about $2500, which is genuinely useful for planning in a way that a single fabricated figure is not.

Interval estimates also arise from measurement precision. A length recorded as 4.7 m to the nearest tenth of a metre lies between 4.65 m and 4.75 m. Feed those bounds through a calculation and you get bounds on the result. A rectangular slab measured as 4.7 m by 3.2 m, each to the nearest tenth, has an area somewhere between 4.65 × 3.15 = 14.6475 m² and the upper bound 4.75 × 3.25 = 15.4375 m², so 15.04 m² is the product of the rounded dimensions, not an exact area. State the measurement assumptions when reporting that estimate.

Judging reasonableness connects the calculation to its context and is a stated syllabus objective. The routine is to ask three questions of any answer. Is the magnitude plausible — is a weekly grocery bill of $12 or $12 000 believable? Is the unit right — should this be litres or millilitres, square metres or cubic metres? Is the direction right — should applying a discount have made the number bigger?

When a question asks you to comment on reasonableness, quote a specific comparison rather than asserting. If a task states that a container holds at most 200 L, a calculated fill of 340 L exceeds that stated capacity. Check the dimensions, units and requested fill level before accepting it. Naming the likely cause, not just the fact of the error, is what lifts these responses.

7. Percentage error: measuring how wrong an approximation is

Related revision technique. Percentage error can describe a comparison when the task supplies a reference value. It is not separately named in the current Unit 3 calculations list.

Absolute error is the size of the difference between an approximate value and the true or accepted value, taken as a positive quantity. Percentage error expresses that difference as a percentage of the true value:

percentage error = (|approximate value − true value| ÷ true value) × 100

The denominator is the true or accepted value, not the estimate, for the same reason that percentage change uses the original value. Getting this the wrong way round produces a slightly different number that is not what was asked for.

Worked example: a builder estimates a room's floor area as 24 m²; measurement shows it is 22.4 m². The absolute error is |24 − 22.4| = 1.6 m². The percentage error is 1.6 ÷ 22.4 × 100 = 7.14%, to two decimal places. Note the units: absolute error carries the unit of the quantity, percentage error carries no unit but the per cent sign.

Second example, linking to leading-digit estimation: estimating 47 × $19.60 as 50 × $20 = $1000 against the exact $921.20 gives an absolute error of $78.80 and a percentage error of 78.80 ÷ 921.20 × 100 = 8.55%. This is the natural way to answer a part that asks how good was your estimate.

Percentage error is the right tool because it is relative. An error of 2 cm is trivial on a 50 m running track (0.04%) and serious on a 15 cm phone screen (13%). A question asking whether an error is acceptable is really asking you to compute the percentage and compare it with a stated tolerance — an error of 1.2% against a stated tolerance of 2% is within tolerance, and saying so explicitly is the required conclusion.

Watch the wording carefully. Percentage error compares an approximation with a true value. Percentage change compares a later value with an earlier one and can be signed. Percentage difference between two measurements with no accepted true value is a different question again, and if a stem uses that phrase it will tell you which value to divide by.