Introduction to Units and Compound Measures

In your GCSE journey, you will find that numbers rarely stand alone. Whether you are measuring the length of a football pitch, the weight of an apple, or how fast a car is moving, you are using units. Compound measures are simply what happens when we mix two different units together, like "miles per hour." This chapter is part of the Ratio, proportion and rates of change section because these measures show how one quantity changes in relation to another.

1. Standard Units of Measure

Before we can master complex measures, we must be "fluent" in standard units. The Edexcel syllabus requires you to convert freely between these:

Length

The metric system uses base \(10\), which makes it easier once you remember the prefixes:

\(10 \text{ millimetres (mm)} = 1 \text{ centimetre (cm)}\)
\(100 \text{ centimetres (cm)} = 1 \text{ metre (m)}\)
\(1000 \text{ metres (m)} = 1 \text{ kilometre (km)}\)

Mass

Mass tells us how much "stuff" is in an object:

\(1000 \text{ milligrams (mg)} = 1 \text{ gram (g)}\)
\(1000 \text{ grams (g)} = 1 \text{ kilogram (kg)}\)
\(1000 \text{ kilograms (kg)} = 1 \text{ tonne (t)}\)

Capacity (Volume)

This is how much liquid a container can hold:

\(10 \text{ millilitres (ml)} = 1 \text{ centilitre (cl)}\)
\(1000 \text{ millilitres (ml)} = 1 \text{ litre (l)}\)

Quick Tip: To go from a large unit (like km) to a small unit (like m), you multiply. To go from a small unit to a large unit, you divide.

2. The "Time" Trap

Time is the only measure that doesn't use base \(10\), which is a common place for mistakes! Don't worry if this seems tricky at first; just remember that there are \(60\) minutes in an hour, not \(100\).

\(0.5 \text{ hours}\) is not \(50 \text{ minutes}\). It is half of \(60\), which is \(30 \text{ minutes}\).
To convert minutes into decimal hours, divide by \(60\). For example, \(15 \text{ minutes} = 15 \div 60 = 0.25 \text{ hours}\).

3. Compound Measures

A compound measure involves two or more different units. The most common ones you will see are speed, density, and pressure. You can use formula triangles to help you rearrange these equations.

Speed, Distance, and Time

Speed is the rate at which distance changes over time.

Formula: \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\)

Example: If a cyclist travels \(30 \text{ miles}\) in \(2 \text{ hours}\), their speed is \(30 \div 2 = 15 \text{ mph}\).

Density, Mass, and Volume

Density tells us how "heavy" an object is for its size.

Formula: \(\text{Density} = \frac{\text{Mass}}{\text{Volume}}\)

Example: A block of wood has a mass of \(200 \text{ g}\) and a volume of \(250 \text{ cm}^3\).
\(\text{Density} = 200 \div 250 = 0.8 \text{ g/cm}^3\).

Pressure, Force, and Area

Pressure is the amount of force applied over a certain area.

Formula: \(\text{Pressure} = \frac{\text{Force}}{\text{Area}}\)

Example: A force of \(100 \text{ Newtons (N)}\) is applied to an area of \(2 \text{ m}^2\).
\(\text{Pressure} = 100 \div 2 = 50 \text{ N/m}^2\).

Key Takeaway: The units of the answer tell you the formula! If speed is in km/h, you must be doing km (distance) divided by h (time).

4. Unit Pricing and Rates of Pay

In the real world, we use compound units to compare value for money or calculate earnings.

Unit Pricing

To find out which shop has the better deal, calculate the cost per unit (e.g., cost per gram or cost per \(100 \text{ ml}\)).

\(\text{Unit Price} = \frac{\text{Total Cost}}{\text{Quantity}}\)

Rates of Pay

This is the amount of money earned per unit of time worked (e.g., \(\text{\pounds} 10.50 \text{ per hour}\)).

\(\text{Total Pay} = \text{Rate of Pay} \times \text{Time Worked}\)

5. Area and Volume Conversions (The Big Pitfall!)

This is a common "trick" question on Edexcel papers. Converting area and volume is not the same as converting length.

Imagine a square that is \(1 \text{ m}\) by \(1 \text{ m}\). Its area is \(1 \text{ m}^2\).
In centimetres, that same square is \(100 \text{ cm}\) by \(100 \text{ cm}\).
Its area is \(100 \times 100 = 10,000 \text{ cm}^2\)!

The Rule:
To convert Area, square the conversion factor: \(1 \text{ m}^2 = 100^2 \text{ cm}^2 = 10,000 \text{ cm}^2\).
To convert Volume, cube the conversion factor: \(1 \text{ m}^3 = 100^3 \text{ cm}^3 = 1,000,000 \text{ cm}^3\).

6. Common Mistakes to Avoid

1. Mixing Units: Always check if the units in the question match. If distance is in \(km\) but speed is in \(m/s\), you must convert one of them first!
2. Incorrect Time Decimals: Never write \(2 \text{ hours and } 30 \text{ minutes}\) as \(2.3\) in a calculator. It must be \(2.5\).
3. Forgetting the Formula Triangle: If you need to find Distance, cover 'D' in the triangle to see that it is \(\text{Speed} \times \text{Time}\).

Quick Review

  • Standard Units: Use base \(10\) (except for time!).
  • Speed: Distance over Time.
  • Density: Mass over Volume.
  • Pressure: Force over Area.
  • Area/Volume Conversion: Remember to square or cube the length conversion factor.

Note: For Higher tier students, you may also see these compound measures in algebraic contexts or be asked to interpret them as the gradient of a graph (e.g., the gradient of a distance-time graph is speed). See the "Direct and inverse proportion" chapter for more on these relationships.