Standard Units

Standard Units and Compound Measures

Mathematics uses standardised units so that measurements are unambiguous and comparable. The metric system — based on powers of ten — is used throughout science and GCSE mathematics. Beyond simple measures, many real-world quantities are compound measures: they are defined as a ratio of two other quantities, such as speed (distance per unit time) or density (mass per unit volume).

Metric Conversions

Length

\( 1 ext{ km} = 1000 ext{ m} \)
\( 1 ext{ m} = 100 ext{ cm} \)
\( 1 ext{ m} = 1000 ext{ mm} \)
\( 1 ext{ cm} = 10 ext{ mm} \)

Mass

\( 1 ext{ tonne} = 1000 ext{ kg} \)
\( 1 ext{ kg} = 1000 ext{ g} \)
\( 1 ext{ g} = 1000 ext{ mg} \)

Capacity

\( 1 ext{ litre} = 1000 ext{ ml} \)
\( 1 ext{ litre} = 100 ext{ cl} \)
\( 1 ext{ cl} = 10 ext{ ml} \)

To convert to a smaller unit, multiply (more of them). To convert to a larger unit, divide (fewer of them).

Compound Measures — Formula Triangles

The three main compound measures each involve a pair of other quantities. A formula triangle shows all three relationships at once: cover the quantity you want to find, and the remaining two show the calculation.

D S T Speed · Distance · Time

\( S = D \div T quad D = S \times T quad T = D \div S \)

M D V Mass · Density · Volume

\( D = M \div V quad M = D \times V quad V = M \div D \)

F P A Force · Pressure · Area

\( P = F \div A quad F = P \times A quad A = F \div P \)

Worked Examples

Convert 4.8 km to metres and millimetres. Convert 3750 ml to litres and centilitres.
\[ 4.8 \text{ km} = 4.8 \times 1000 = 4800 \text{ m} = 4800 \times 1000 = 4,800,000 \text{ mm} \] \[ 3750 \text{ ml} = 3750 \div 1000 = 3.75 \text{ litres} = 3750 \div 10 = 375 \text{ cl} \]

Converting speed units: To convert km/h to m/s, multiply by \( \frac{1000}{3600} = \frac{1}{3.6} \).

\[ 72 \text{ km/h} = 72 \div 3.6 = 20 \text{ m/s} \]
A train travels 135 km in 1.5 hours. Find its average speed. Then find how long it takes to travel 270 km at the same speed.
\[ S = D \div T = 135 \div 1.5 = 90 \text{ km/h} \] \[ T = D \div S = 270 \div 90 = 3 \text{ hours} \]

Units: Speed in km/h when distance is in km and time in hours. If mixing units (e.g. distance in m, time in s), the result is in m/s.

A material has density 8.5 g/cm³ and volume 20 cm³. Find its mass. A force of 120 N acts on an area of 0.04 m². Find the pressure.
\[ M = D \times V = 8.5 \times 20 = 170 \text{ g} \] \[ P = F \div A = 120 \div 0.04 = 3000 \text{ N/m}^2 \text{ (pascals)} \]

Units of density: g/cm³ or kg/m³. Units of pressure: N/m² or Pa (pascals). Always check that units are consistent before substituting.

 Key Takeaways

  • Metric prefixes: kilo = ×1000, centi = ÷100, milli = ÷1000. Converting to a smaller unit: multiply. To a larger unit: divide.
  • Speed = Distance ÷ Time; Density = Mass ÷ Volume; Pressure = Force ÷ Area.
  • Formula triangles: cover the unknown quantity — the remaining two show multiply (side by side) or divide (top over bottom).
  • Unit consistency is essential: if distance is in km and time in hours, speed is in km/h. To convert km/h to m/s, divide by 3.6.
  • Density units: g/cm³ (for everyday objects) or kg/m³ (for larger volumes). 1 g/cm³ = 1000 kg/m³.