Units and Conversions for the ESAT

Standard Units

To solve problems in the ESAT, you must be familiar with the standard units used to measure physical properties. These are categorised as follows:

Mass

Units include milligrams ($mg$), grams ($g$), kilograms ($kg$), and tonnes ($t$).

Force

Force is measured in Newtons ($N$).

Length

Standard measures are millimetres ($mm$), centimetres ($cm$), metres ($m$), and kilometres ($km$).

Area

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

Capacity and Volume

Volume describes the three-dimensional space an object occupies, while capacity often refers to the amount of liquid a container can hold. The common units are cubic millimetres ($mm^3$), cubic centimetres ($cm^3$), cubic metres ($m^3$), millilitres ($ml$), and litres ($l$).

There are specific relationships between these units:

1. $1\text{ ml} = 1\text{ cm}^3$
2. $1\text{ l} = 1\text{ dm}^3 = 1000\text{ cm}^3$
3. $1000\text{ l} = 1\text{ m}^3$

Small quantities of liquid, such as medicine or drinks, are typically measured in $ml$ or $l$. Larger quantities, such as the volume of water in a reservoir or swimming pool, are measured in $m^3$.

Time

Time units range from seconds, minutes, and hours to days, weeks, months, and years.

A year consists of 12 months. A standard year has 365 days, while a leap year has 366 days. Leap years occur nearly every 4 years. Longer periods include the century (100 years) and the millennium (1000 years).

Exercise A: Choosing Standard Units

Consider which units are most appropriate for the following:

1. The volume of water in a swimming pool: Use $m^3$ to avoid excessively large numbers.
2. The area of a kitchen floor: Use $m^2$ as the floor dimensions are usually measured in metres.
3. The volume of liquid in a can of cola: This is commonly measured in $ml$.

Compound Units

Compound units are formed when two different types of measurement are combined, often to describe a rate. For example, average speed is found by dividing distance in $km$ by time in hours, resulting in kilometres per hour ($km/h$).

In mathematical notation, these can also be written using negative indices. For instance, $km/h$ is equivalent to $km\ h^{-1}$.

Exercise B: Identifying Compound Units

Identify the missing units in the following scenarios:

1. The density of a rock ($mass = 600\ g$, $volume = 20\ cm^3$). Using $density = \frac{mass}{volume}$, the density is $30\ g/cm^3$.
2. The average speed of a ball ($distance = 10\ m$, $time = 2\ s$). Using $speed = \frac{distance}{time}$, the speed is $5\ m/s$.
3. The rate of pay for a worker paid $£300$ for 15 hours. The rate is $20\ £/h$.
4. The average speed of a car travelling $200\ km$ in 4 hours. The speed is $50\ km/h$.
5. The pressure exerted by a force of $6\ N$ on an area of $2\ m^2$. Using $pressure = \frac{force}{area}$, the pressure is $3\ N/m^2$.

Unit Cost of an Item

The unit cost is the price of exactly one item. If $x$ items cost $£y$ in total, the unit cost is calculated by dividing the total cost by the number of items: $£\frac{y}{x}$ per item.

Example: Calculating Unit Cost If 50 boxes of sweets cost $£215$, what is the unit cost per box? Total cost is $£215$. Number of items is $50$. Unit cost = $£\frac{215}{50} = £4.30$ per box.

Changing Between Standard Units

When converting between units of different scales, you must apply the correct conversion factor.

Measure	Conversion Factors
Length	$1\ km = 1000\ m$; $1\ m = 100\ cm = 1000\ mm$; $1\ cm = 10\ mm$
Area	$1\ km^2 = 1,000,000\ m^2$; $1\ m^2 = 10,000\ cm^2$; $1\ cm^2 = 100\ mm^2$
Volume	$1\ m^3 = 1,000,000\ cm^3$; $1\ cm^3 = 1000\ mm^3$
Mass	$1\ kg = 1000\ g$; $1\ g = 1000\ mg$
Time	$60\ s = 1\ min$; $60\ min = 1\ hr$; $24\ hr = 1\ day$; $7\ days = 1\ week$

Exercise C: Converting Area

How many $cm^2$ are in $2.65\ m^2$? Since $1\ m = 100\ cm$, it follows that $1\ m^2 = 100^2\ cm^2 = 10,000\ cm^2$. Therefore, $2.65\ m^2 = 2.65 \times 10,000 = 26,500\ cm^2$.

Changing Between Compound Units

To convert compound units, you must convert each component unit separately. There are two primary methods to approach this.

Example: Converting Density Convert a density of $20\ g\ cm^{-3}$ into $kg\ m^{-3}$.

Method 1: Fractional Conversion Write the unit as a fraction: $20\ g\ cm^{-3} = \frac{20\ g}{1\ cm^3}$. Convert grams to kilograms: $20\ g = \frac{20}{1000}\ kg$. Convert $cm^3$ to $m^3$: $1\ cm^3 = \frac{1}{1,000,000}\ m^3$. Now divide the two: $\frac{20}{1000} \div \frac{1}{1,000,000} = \frac{20}{1000} \times \frac{1,000,000}{1} = 20,000\ kg\ m^{-3}$.

Method 2: Step-by-Step Multiplication/Division First, convert $g$ to $kg$. Since kilograms are larger, the numeric value for the same mass will be smaller: $20 \div 1000 = 0.02\ kg/cm^3$. Next, convert $cm^3$ to $m^3$. Since a cubic metre is much larger than a cubic centimetre, there is much more mass in one cubic metre: $0.02 \times 1,000,000 = 20,000\ kg/m^3$.

Example: Units and Problem Solving A car travels $35,000\ m$ in 30 minutes. Calculate the average speed in $km/h$.

1. Convert distance to $km$: $35,000\ m = \frac{35,000}{1000} = 35\ km$.
2. Convert time to hours: $30\ min = \frac{30}{60} = 0.5\ hours$.
3. Calculate speed: $average\ speed = \frac{total\ distance}{total\ time} = \frac{35}{0.5} = 70\ km/h$ (or $70\ km\ h^{-1}$).