Converting Units
Converting Units
Unit conversion is the process of expressing a measurement in a different unit while preserving its value. Conversion always involves multiplying or dividing by a conversion factor — the ratio that relates the two units. The key discipline is knowing whether to multiply or divide, and keeping track of which unit you are moving between.
Key Metric Conversions
| Quantity | Conversion |
|---|---|
| Length | 1 km = 1000 m | 1 m = 100 cm | 1 cm = 10 mm |
| Mass | 1 kg = 1000 g | 1 tonne = 1000 kg |
| Volume | 1 litre = 1000 ml | 1 m³ = 1 000 000 cm³ | 1 cm³ = 1 ml |
| Area | 1 m² = 10 000 cm² | 1 km² = 1 000 000 m² |
| Time | 1 hour = 60 min | 1 min = 60 s | 1 day = 24 hours |
Key Imperial–Metric Conversions
| Imperial | Metric (approx.) |
|---|---|
| 1 inch | 2.54 cm |
| 1 foot (12 inches) | 30.48 cm |
| 1 mile | 1.6 km (or 8 km ≈ 5 miles) |
| 1 pound (lb) | 454 g (≈ 0.45 kg) |
| 1 gallon | 4.5 litres |
| 1 pint | 568 ml (≈ 0.57 litres) |
Compound Units
Compound units combine two base units, such as speed (distance ÷ time) or density (mass ÷ volume). Converting compound units requires converting each component unit separately.
Example — convert 72 km/h to m/s:
\[ 72 \text{ km/h} = 72 \times \frac{1000 \text{ m}}{3600 \text{ s}} = \frac{72{,}000}{3600} = 20 \text{ m/s} \]The general rule: km/h → m/s, divide by 3.6. m/s → km/h, multiply by 3.6.
Algebraic Contexts
Unit conversions also appear in algebraic settings — for example, expressing a formula for area (in cm²) when lengths are given in mm, or converting a rate expressed as a fraction.
\[ \text{If } v = \frac{d}{t} \text{ with } d \text{ in km and } t \text{ in hours, then } v \text{ is in km/h.} \]To change to m/s, substitute \( d \) in metres and \( t \) in seconds, or apply the ÷ 3.6 conversion factor to the result.
Worked Examples
Convert 3.75 km to metres. Convert 850 ml to litres. Convert 2.4 m² to cm².
\[ 3.75 \text{ km} = 3.75 \times 1000 = 3{,}750 \text{ m} \] \[ 850 \text{ ml} = 850 \div 1000 = 0.85 \text{ litres} \] \[ 2.4 \text{ m}^2 = 2.4 \times 10{,}000 = 24{,}000 \text{ cm}^2 \]A road sign says 45 miles. Approximately how many kilometres is this? A bag weighs 5 lbs. Convert to kg.
\[ 45 \text{ miles} \times 1.6 = 72 \text{ km} \] \[ 5 \text{ lbs} \times 0.45 = 2.25 \text{ kg} \]A car travels at 54 km/h. Convert to m/s. A density is 8 g/cm³ — convert to kg/m³.
\[ 54 \text{ km/h} \div 3.6 = 15 \text{ m/s} \] \[ 8 \frac{\text{g}}{\text{cm}^3} = 8 \times \frac{0.001 \text{ kg}}{0.000001 \text{ m}^3} = 8 \times 1000 = 8000 \text{ kg/m}^3 \]Key Takeaways
- Larger units → smaller units: multiply. Smaller units → larger units: divide.
- Area conversions square the linear factor: 1 m = 100 cm, so 1 m² = 100² = 10,000 cm².
- Volume conversions cube the linear factor: 1 m³ = 100³ = 1,000,000 cm³.
- Compound units: convert each component separately — distance and time for speed, mass and volume for density.
- km/h ↔ m/s: divide by 3.6 (km/h to m/s) or multiply by 3.6 (m/s to km/h).