Units Of Measure

Video coming soon4 worked examples

Theory

In mathematical modelling, paying close attention to units of measure is essential. You must ensure that your units make logical sense and remain consistent throughout your calculations.

🚨 Important Exam Note 🚨

Because this is Applications of Mathematics (not Physics or Chemistry), you are never expected to have prior scientific knowledge of complex units. You simply need to be able to substitute given units into a formula and simplify them.

1. Deriving Suitable Units of Measure

The units of an output variable are determined completely by the units of the input variables.

To find the final unit, simply replace the variables in the formula with their units and simplify (e.g., if measuring Speed = Distance / Time, and distance is in miles and time is in hours, the unit is miles/hours or mph).

2. Consistency in Comparisons

When making a direct comparison between two quantities, the units must be consistent for the comparison to be valid.
Even if two quantities seem related, there are often hidden variables or factors that must be assumed to be constant to make it a completely fair comparison. If these hidden variables change, the comparison becomes invalid.

3. Consistency in Formulae

When using a formula, the units of the variables must be consistent in how they are defined and how they relate to each other.
For example, scaling up a measurement linearly (like doubling a length) does not mean the time taken or the volume will also double exactly. You must look out for these logical traps in exam questions.

Worked examples

Example 1

Example 1: Deriving Units via Substitution

A scientist is using the mathematical model $R = \frac{Q}{P}$ . Given that $P$ is measured in grams (g) and $Q$ is measured in cubic centimetres (cm $^3$ ), deduce the units of measure for $R$ .

Solution: Substitute the units directly into the formula:

R = \frac{\text{cm}^3}{\text{g}}

The units of $R$ are grams per cubic centimetre (or g/cm $^3$ ).

Example 2

Example 2: Deriving Units (With Cancelling)

A factory machine's efficiency is modelled using the formula $E = W \times T$ . Given that $W$ is measured in boxes per minute (boxes/min) and $T$ is measured in minutes (min), deduce the units of measure for $E$ .

Solution: Substitute the units into the formula:

E = \frac{\text{boxes}}{\text{min}} \times \text{min}

The 'minutes' cancel each other out. Therefore, the unit of $E$ is simply boxes.

Example 3

Example 3: Inconsistent Comparisons (Hidden Variables)

A courier company tracks two of its delivery vans. Van A uses 12 litres of diesel during its morning shift. Van B uses 18 litres of diesel during its morning shift. The manager concludes that Van A has a much more fuel-efficient engine. Explain why this conclusion may be invalid.

Solution: The conclusion is invalid because we do not know the distance each van travelled. It is entirely possible that Van B drove a significantly longer distance than Van A during the morning shift, meaning it could actually be the more fuel-efficient van overall.

Example 4

Example 4: Inconsistent Logic in Formulae

A decorator takes 3 hours to paint the floor of a square room that has a length of 4 metres. They are hired to paint the floor of another square room that has a length of 8 metres. Because the length is doubled, the decorator estimates it will take exactly 6 hours to paint the new floor. Explain why this estimation is mathematically flawed.

Solution: The estimation is flawed because the time taken to paint a floor depends on the area of the room, not just the length. (A 4m $\times$ 4m room has an area of 16m $^2$ . An 8m $\times$ 8m room has an area of 64m $^2$ . The area has actually quadrupled, not doubled, so the time estimate will be completely incorrect).

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7 Units of Measure questions with answers and video solutions