Conversions and Units
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Revision notes for AQA GCSE Maths Conversions and Units. Open the guide for explanations and worked examples. Written against the AQA GCSE Maths (8300) specification, so the content matches what's examinable rather than general Maths background.

Conversions and Units

What you'll learn

  • How to change from one unit to another using a conversion graph.
  • How to convert between metric units like cm, mm, m and km.
  • Why area and volume conversions use bigger scale factors.
  • How to change speeds from kilometres per hour to metres per second.

1. What is a conversion?

A unit tells you what a measurement is counted in, such as cm, metres, litres, pints, kg or seconds.

A conversion means writing the same amount using a different unit.

For example, 1 metre and 100 cm are the same length, just written in different units.

Definition

Conversion factor

A conversion factor is the number you multiply or divide by to change from one unit to another. For example, because 1 cm = 10 mm, the conversion factor between cm and mm is 10.

Common metric facts to know

  • 1 cm = 10 mm
  • 1 m = 100 cm
  • 1 km = 1000 m
  • 1 hour = 60 minutes = 3600 seconds
Key Idea

Multiply or divide

If you are changing to a smaller unit, you usually multiply. If you are changing to a larger unit, you usually divide.

Example

Changing metres to centimetres

Write 3.7 m in centimetres.

A metre-to-centimetre scale shows that each metre is 100 cm, so 3.7 m corresponds to 370 cm.

  1. Use the fact that 1 m = 100 cm.

  2. Metres are bigger than centimetres, so multiply by 100:

    3.7×100=3703.7 \times 100 = 3703.7×100=370

  3. Write the answer with the new unit: 370 cm.

2. Using conversion graphs

A conversion graph is a straight-line graph that changes one unit into another.

The horizontal line is the x-axis. The vertical line is the y-axis.

For example, a graph might have litres along the bottom and pints up the side.

Reading from the x-axis to the y-axis

If you are given the value on the bottom axis:

  1. Find the number on the x-axis.
  2. Go vertically up to the line.
  3. Go horizontally across to the y-axis.
  4. Read off the converted value.
Example

Using a graph from litres to pints

A conversion graph shows that 20 litres is about 35 pints. Use the graph to estimate 12 litres in pints.

Read 12 litres by going up to the conversion line, then across to about 21 pints.

  1. Find 12 on the litres axis.

  2. Move straight up until you reach the conversion line.

  3. Move across to the pints axis.

  4. Read the value. It should be about 21 pints.

Reading backwards from the y-axis to the x-axis

Sometimes you are given the value on the vertical axis. Then you work backwards:

  1. Find the number on the y-axis.
  2. Go horizontally to the line.
  3. Go vertically down to the x-axis.
  4. Read off the answer.
Example

Using a graph from pints to litres

A conversion graph shows litres on the x-axis and pints on the y-axis. Estimate 70 pints in litres.

Read backwards from 70 pints by going across to the line, then down to about 40 litres.

  1. Find 70 pints on the vertical axis.

  2. Move horizontally across to the conversion line.

  3. Move straight down to the litres axis.

  4. Read the value. Since 35 pints is about 20 litres, 70 pints is about 40 litres.

Tip

Check the axes

Before reading a conversion graph, always check which unit is on each axis. The bottom axis and side axis are often different.

Common Mistake

Going the wrong way

A common mistake is to read from the wrong axis. If the question gives you cm, start on the cm axis. If it gives you inches, start on the inches axis.

3. Estimating from a straight conversion line

Most conversion graphs are straight lines through zero. This means the units are in direct proportion: doubling one measurement doubles the other.

Definition

Direct proportion

Two quantities are in direct proportion if they increase at the same rate. For example, if 5 litres is 8.75 pints, then 10 litres is 17.5 pints.

This helps when the number you want is outside the graph or easier to calculate from a known point.

Example

Using proportion instead of the graph

A conversion graph shows that 10 inches is about 25.4 cm. Estimate 5 inches in centimetres.

Direct proportion means halving 10 inches also halves 25.4 cm.

  1. Notice that 5 inches is half of 10 inches.

  2. So 5 inches is half of 25.4 cm:

    25.4÷2=12.725.4 \div 2 = 12.725.4÷2=12.7

  3. The answer is about 12.7 cm.

4. Converting area units

Area measures the amount of flat space inside a shape. Area units are squared, such as cm², mm² and m².

Definition

Squared unit

A squared unit means a unit multiplied by itself. For example, 1 cm² is a square that is 1 cm by 1 cm.

One square centimetre is the same square as a 10 mm by 10 mm grid, making 100 square millimetres.

This is important: if 1 cm = 10 mm, then 1 cm² is not 10 mm².

A 1 cm by 1 cm square is the same as a 10 mm by 10 mm square:

10×10=10010 \times 10 = 10010×10=100

So 1 cm² = 100 mm².

Key Idea

Area conversion

For area, square the length conversion factor. If the length scale factor is 10, the area scale factor is 100.

Example

Changing square metres to square centimetres

Write 2.8 m² in cm².

Area conversion squares the length scale factor: each metre side becomes 100 cm.

  1. Use the length fact: 1 m = 100 cm.

  2. Because this is area, square the conversion factor:

    100×100=10000100 \times 100 = 10000100×100=10000

  3. Multiply by 10000:

    2.8×10000=280002.8 \times 10000 = 280002.8×10000=28000

  4. Write the answer: 28000 cm².

Example

Changing square millimetres to square centimetres

Write 450 mm² in cm².

  1. Use the fact that 1 cm = 10 mm.

  2. So 1 cm² = 100 mm².

  3. You are changing from smaller units to larger units, so divide by 100:

    450÷100=4.5450 \div 100 = 4.5450÷100=4.5

  4. Write the answer: 4.5 cm².

Common Mistake

Only multiplying by 10

For area, do not just multiply by the length scale factor. From cm² to mm², multiply by 100, not 10.

5. Converting volume units

Volume measures the amount of space inside a 3D object. Volume units are cubed, such as cm³ and mm³.

Definition

Cubed unit

A cubed unit means a unit multiplied by itself three times. For example, 1 cm³ is a cube that is 1 cm by 1 cm by 1 cm.

One cubic centimetre has side length 1 cm, which is 10 mm in each of the three dimensions.

If 1 cm = 10 mm, then:

1 cm3=10×10×10=1000 mm31\text{ cm}^3 = 10 \times 10 \times 10 = 1000\text{ mm}^31 cm3=10×10×10=1000 mm3
Key Idea

Volume conversion

For volume, cube the length conversion factor. If the length scale factor is 10, the volume scale factor is 1000.

Example

Changing cubic centimetres to cubic millimetres

Write 24 cm³ in mm³.

  1. Use the length fact: 1 cm = 10 mm.

  2. Because this is volume, cube the conversion factor:

    10×10×10=100010 \times 10 \times 10 = 100010×10×10=1000

  3. Multiply by 1000:

    24×1000=2400024 \times 1000 = 2400024×1000=24000

  4. Write the answer: 24000 mm³.

6. Converting speed units

Speed tells you how far something travels in a certain time.

Definition

Compound unit

A compound unit combines two units. For example, kilometres per hour, written km/h, combines distance and time.

To change kilometres per hour into metres per second:

  • Change kilometres to metres by multiplying by 1000.
  • Change hours to seconds by dividing by 3600.
Example

Changing km/h to m/s

Change 72 kilometres per hour into metres per second.

A speed conversion changes both the distance unit and the time unit before dividing metres by seconds.

  1. Start with 72 km/h.

  2. Change kilometres into metres:

    72×1000=7200072 \times 1000 = 7200072×1000=72000

  3. Change 1 hour into seconds: 1 hour = 3600 seconds.

  4. Divide by 3600:

    72000÷3600=2072000 \div 3600 = 2072000÷3600=20

  5. Write the answer: 20 m/s.

Tip

Quick shortcut for km/h to m/s

To change km/h to m/s, divide by 3.6. For example, 72 ÷ 3.6 = 20.

Exam technique

In the exam

  1. Check which unit you are starting with before you do anything.
  2. For graphs, start on the axis with the given unit, move to the line, then read the other axis.
  3. For area and volume, remember to square or cube the conversion factor.
  4. Always include the final unit in your answer.
Self review

Check yourself

  • If 1 m = 100 cm, what is 1 m² in cm²?
  • On a conversion graph, which axis do you start from if the question gives you pints?
  • What two changes are needed to convert km/h into m/s?

Recap questions

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