Working with units and dimensional consistency, converting between units, rounding appropriately, and using tolerance, absolute error and percentage error to judge whether a result is acceptable.

What this dot point is asking

The SQA wants you to keep units consistent through a calculation, convert between units correctly, round a result to a sensible accuracy, and judge whether a value is acceptable using tolerance and error. These skills underpin every modelling and finance question, because an answer with the wrong units or unrealistic precision loses marks.

Units and dimensional consistency

A model only makes sense if every quantity is in compatible units. Before substituting into a formula, convert everything to one system, for example all lengths to metres or all times to hours. If a speed is in kilometres per hour and a time is in minutes, you must convert the time to hours first.

A common chain of conversions: $1$ metre $= 100$ cm $= 1000$ mm; $1$ km $= 1000$ m; $1$ hour $= 60$ minutes $= 3600$ seconds; $1$ litre $= 1000$ ml. To convert a rate, convert the numerator and denominator separately.

Rounding to a sensible accuracy

Report a result to an accuracy that matches the data and the context. Money is rounded to the nearest penny, a length measured to the nearest millimetre should not be quoted to thousandths of a millimetre, and a population is a whole number. Carry full precision through the working and round only at the end, because rounding early introduces error that grows through a calculation.

A figure given to $3$ significant figures, such as $14\,100$, claims accuracy to the nearest hundred; quoting $14\,053.27$ from rounded input data overstates how precise the model really is.

Absolute and percentage error

When a measured or modelled value differs from a true or target value, two measures describe the gap.

The percentage error always divides by the true value, not the measured value. A small absolute error can still be a large percentage error if the true value is small: an error of $2$ mm on a $10$ mm part is $20\%$, but the same $2$ mm on a $2$ m beam is only $0.1\%$. Percentage error lets you compare the seriousness of errors on quantities of different sizes.

Tolerance

In manufacturing and engineering a measurement is acceptable if it lies within a stated tolerance of the target. A specification of $50 \pm 0.4$ mm means any value from $49.6$ mm to $50.4$ mm is acceptable.

The lower limit is target minus tolerance and the upper limit is target plus tolerance. A value passes if lower limit $\le$ value $\le$ upper limit. Tolerance is how a real process allows for unavoidable variation while keeping parts usable.

Why this matters in modelling

A model fed inconsistent units gives nonsense, and a result quoted too precisely misleads a reader into trusting it more than the data allows. Stating units, rounding sensibly, and reporting error or tolerance is part of communicating a model honestly, which the SQA rewards directly.

Try this

Q1. Convert $2.5$ litres to millilitres, and a speed of $72$ km/h to metres per second. [2 marks]

• Cue. $2.5 \times 1000 = 2500$ ml; $72 \times 1000 \div 3600 = 20$ m/s.

Q2. A component is specified as $25 \pm 0.6$ mm. State the acceptable range and decide whether $24.3$ mm passes. [3 marks]

• Cue. Range $24.4$ mm to $25.6$ mm; $24.3$ mm is below $24.4$ mm, so it fails.

Q3. A model predicts $340$ but the true value is $325$. Find the percentage error. [2 marks]

• Cue. Absolute error $= 15$; percentage error $= \dfrac{15}{325} \times 100 = 4.6\%$.