Standard and Compound Units for the ESAT and TMUA

Q: How do I decide whether to multiply or divide when converting units?

When moving from a large unit (like $km$) to a smaller unit (like $m$), you multiply because there will be more of the smaller units. When moving from a small unit to a larger one, you divide.

Updated August 2025

Master standard measures of mass, length, time, and money essential for UK university admissions tests. This lesson teaches how to perform calculations with compound units like speed, density, and pressure, alongside rigorous methods for unit conversions and problem solving with real world measures.

Core concept

Measurement involves both standard units for single quantities and compound units for relationships between them. Accurate calculations require consistent units and the precise application of conversion factors across linear, area, and volume dimensions.

Standard Units

Success in university admission tests requires a high degree of fluency with standard units. You must be able to recognise and apply the following measures correctly.

Mass

Mass is measured using milligrams ( $mg$ ), grams ( $g$ ), kilograms ( $kg$ ), and tonnes ( $t$ ).

Force

Force is measured in Newtons ( $N$ ).

Length

Standard lengths include millimetres ( $mm$ ), centimetres ( $cm$ ), metres ( $m$ ), and kilometres ( $km$ ).

Area

Area is expressed in square units: square millimetres ( $mm^2$ ), square centimetres ( $cm^2$ ), square metres ( $m^2$ ), and square kilometres ( $km^2$ ).

Capacity and Volume

Capacity refers to the amount of liquid a container holds, measured in millilitres ( $ml$ ) and litres ( $l$ ). Volume refers to the 3 dimensional space occupied, measured in cubic millimetres ( $mm^3$ ), cubic centimetres ( $cm^3$ ), and cubic metres ( $m^3$ ).

There are vital relationships between these measures that you must memorise:

$1 ext{ ml} = 1 ext{ cm}^3$
$1 ext{ l} = 1 ext{ dm}^3 = 1000 ext{ cm}^3$
$1000 ext{ l} = 1 ext{ m}^3$

As a general rule, small quantities of liquid (such as medicines) use $ml$ and $l$ . Larger quantities (such as reservoirs) use $m^3$ .

Time

Standard units of time include seconds, minutes, hours, days, weeks, months, and years. A standard year consists of 12 months and 365 days. A leap year occurs nearly every 4 years and contains 366 days. For longer periods, a century is 100 years and a millennium is 1000 years.

Exercise A: Selecting Units

Which units are commonly used to measure:

The volume of water in a swimming pool?
The area of a kitchen floor?
The volume of liquid in a can of cola?

Compound Units

Compound units are formed by combining two or more quantitative measures, such as metres and seconds into metres per second. To find average speed, we divide the distance travelled in $km$ by the time taken in hours. Since kilometres are divided by hours, the units are $km/h$ , also written as $km h^{-1}$ .

Exercise B: Identifying Compound Units

Find the missing units for these scenarios:

The density of a rock with mass $600 ext{ g}$ and volume $20 ext{ cm}^3$ is $30 ext{ …}$
The average speed of a ball travelling $10 ext{ m}$ in $2 ext{ seconds}$ is $5 ext{ …}$
The average rate of pay for a worker paid $£300$ for $15 ext{ hours}$ is $20 ext{ …}$
The average speed of a car travelling $200 ext{ km}$ in $4 ext{ hours}$ is $50 ext{ …}$
The pressure exerted by a force of $6 ext{ N}$ on a board with an area of $2 ext{ m}^2$ is $3 ext{ …}$

Unit Cost of an Item

If $x$ items cost $£y$ in total, the unit cost is the price of exactly 1 item. This is found by dividing the total cost by the number of items, expressed as $£ rac{y}{x}$ per item.

Worked Example: Unit Cost 50 boxes of sweets cost $£215$ . What is the unit cost of one box?

Answer: $ext{Unit cost} = rac{215}{50} = £4.30 ext{ per box}.$

Changing Between Standard Units

Converting between units requires multiplying or dividing by a specific factor. Note that area and volume factors are the square and cube of the linear factors respectively.

Measure	Conversion Factors
Length	$1 ext{ km} = 1000 ext{ m}$ ; $1 ext{ m} = 100 ext{ cm} = 1000 ext{ mm}$ ; $1 ext{ cm} = 10 ext{ mm}$
Area	$1 ext{ km}^2 = 1,000,000 ext{ m}^2$ ; $1 ext{ m}^2 = 10,000 ext{ cm}^2$ ; $1 ext{ cm}^2 = 100 ext{ mm}^2$
Volume	$1 ext{ m}^3 = 1,000,000 ext{ cm}^3$ ; $1 ext{ cm}^3 = 1000 ext{ mm}^3$
Capacity	$1 ext{ l} = 1000 ext{ ml} = 1000 ext{ cm}^3$ ; $1000 ext{ l} = 1 ext{ m}^3$
Mass	$1 ext{ kg} = 1000 ext{ g}$ ; $1 ext{ g} = 1000 ext{ mg}$
Time	$60 ext{ s} = 1 ext{ min}$ ; $60 ext{ mins} = 1 ext{ hour}$ ; $24 ext{ hours} = 1 ext{ day}$ ; $7 ext{ days} = 1 ext{ week}$

Exercise C: Unit Conversion

How many $cm^2$ are in $2.65 ext{ m}^2$ ?

Changing Between Compound Units

To change compound units, treat each component unit separately. For example, to convert density from $g ext{ cm}^{-3}$ to $kg ext{ m}^{-3}$ , convert the grams to kilograms and the cubic centimetres to cubic metres.

Worked Example: Converting Density The density of a metal is $20 ext{ g cm}^{-3}$ . What is this in $kg ext{ m}^{-3}$ ?

Method 1: Fraction Approach Write the density as a fraction: $rac{20 ext{ g}}{1 ext{ cm}^3}$ . Convert grams to kg: $20 ext{ g} = rac{20}{1000} ext{ kg}$ . Convert $cm^3$ to $m^3$ : $1 ext{ cm}^3 = rac{1}{1,000,000} ext{ m}^3$ . $ext{Density} = rac{20}{1000} imes rac{1,000,000}{1} = 20,000 ext{ kg m}^{-3}$ .

Method 2: Sequential Approach

Convert $g$ to $kg$ : Divide by $1000$ because $1 ext{ kg}$ is $1000 ext{ g}$ . $20 ext{ g/cm}^3 = 0.02 ext{ kg/cm}^3$ .
Convert $cm^3$ to $m^3$ : Since $1 ext{ m}^3$ contains $1,000,000 ext{ cm}^3$ , the density in $kg/m^3$ must be larger. Multiply by $1,000,000$ . $0.02 imes 1,000,000 = 20,000 ext{ kg/m}^3$ .

Units and Problem Solving

Worked Example: Speed Calculation A car travels $35,000 ext{ m}$ in $30 ext{ minutes}$ . Calculate its average speed in $km/h$ .

Answer: First, convert the distance to kilometres: $35,000 ext{ m} = rac{35000}{1000} ext{ km} = 35 ext{ km}$ . Next, convert the time to hours: $30 ext{ mins} = rac{30}{60} ext{ hours} = 0.5 ext{ hours}$ . Finally, use the formula $ext{speed} = rac{ ext{distance}}{ ext{time}}$ : $ext{speed} = rac{35}{0.5} = 70 ext{ km h}^{-1}$ .

Answers to Exercises

Exercise A: 1. $m^3$ (to avoid large numbers); 2. $m^2$ (matches scale of a room); 3. $ml$ (standard for small drinks).

Exercise B: 1. $g/cm^3$ ; 2. $m/s$ ; 3. $£/h$ ; 4. $km/h$ ; 5. $N/m^2$ .

Exercise C: $1 ext{ m}^2 = 10,000 ext{ cm}^2$ , so $2.65 imes 10,000 = 26,500 ext{ cm}^2$ .

Key takeaways

One cubic centimetre ( $1 ext{ cm}^3$ ) is exactly equal to one millilitre ( $1 ext{ ml}$ ), and $1000 ext{ litres}$ is equivalent to $1 ext{ m}^3$ .
When converting area or volume units, you must square or cube the linear conversion factor respectively.
Compound units like density ( $mass/volume$ ) or pressure ( $force/area$ ) are calculated by dividing the constituent measures.
Unit pricing is determined by dividing total cost by the number of items ( $£/item$ ).

Tips

Always check the final required units in a TMUA question before you begin your calculation. It is often more efficient to convert your initial data into the required units (e.g., converting metres to kilometres) at the start rather than converting your final answer at the end.

Cautions

Be careful with capacity conversions. A common mistake is to assume $1 ext{ litre}$ equals $1 ext{ m}^3$ ; however, $1 ext{ m}^3$ actually contains $1000 ext{ litres}$ . Always remember that $1 ext{ litre}$ is equal to $1000 ext{ cm}^3$ .

Insight

Compound units are a powerful tool for dimensional analysis. If you forget a formula for density or pressure, looking at the standard units (like $g/cm^3$ or $N/m^2$ ) tells you exactly which operations to perform on the values provided in the question.

Worked Examples

Example 1

A solid cube with sides of length 20 cm is made from material with density

2000\,\text{kg}\,\text{m}^{-3}

. The cube is suspended, in equilibrium, from an initially unstretched spring, and this results in the spring gaining strain energy of

3.2\,\text{J}

.

What is the spring constant of the spring?

(gravitational field strength =

10\,\text{Nkg}^{-1}

; the spring obeys Hooke's law)

40\,\text{Nm}^{-1}

80\,\text{Nm}^{-1}

400\,\text{Nm}^{-1}

800\,\text{Nm}^{-1}

4000\,\text{Nm}^{-1}

8000\,\text{Nm}^{-1}

Practice Questions

Practice Question 1

An athlete's training session consists of several complete repetitions of a three-part programme:

1. Walk 100 m at an average speed of

6\,km\,h^{-1}

2. Jog 200 m at an average speed of

10\,km\,h^{-1}

3. Run 100 m at an average speed of

20\,km\,h^{-1}

What is the athlete's average speed for the complete training session, in

km\,h^{-1}

A:7.2

B:9.6

C:11.5

D:12

E:14.4

Frequently asked questions

How do I decide whether to multiply or divide when converting units?

When moving from a large unit (like $km$ ) to a smaller unit (like $m$ ), you multiply because there will be more of the smaller units. When moving from a small unit to a larger one, you divide.

Why is $1 ext{ m}^2$ equal to $10,000 ext{ cm}^2$ and not $100 ext{ cm}^2$ ?

A square with side length $1 ext{ m}$ is $100 ext{ cm}$ by $100 ext{ cm}$ . The area is $100 imes 100 = 10,000 ext{ cm}^2$ . The conversion factor must be squared for area.

What does the notation $ms^{-1}$ mean?

This is index notation for metres per second. The negative exponent $-1$ indicates that the unit is in the denominator of a fraction, equivalent to $m/s$ .

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Conversion Of Standard And Compound Units