Unit Conversions

What you will learn

• Choose whether to multiply or divide when changing units.
• Use unit fractions to convert in several steps.
• Convert between common metric and imperial units and between Celsius and Fahrenheit.

Before you start: you should know the metric prefixes and how to multiply and divide by $10$, $100$ and $1000$. See Units of Measurement.

The idea

To convert a measurement, you multiply or divide by a conversion factor, the number of small units in one big unit. The trick is knowing which to do.

Big to small, small to big

• Going from a bigger unit to a smaller unit, you need more of them, so multiply. $3$ m $= 3 \times 100 = 300$ cm.
• Going from a smaller unit to a bigger unit, you need fewer, so divide. $4500$ g $= 4500 \div 1000 = 4.5$ kg.

The unit-fraction method

A more powerful approach writes the conversion factor as a fraction equal to one, such as $\frac{1000 \text{ m}}{1 \text{ km}}$. Multiplying by one doesn’t change the amount, and you choose which way up the fraction goes so that the unwanted unit cancels. This method, sometimes called dimensional analysis, works for any conversion, including chains of several steps.

Metric and imperial

Imperial units don’t follow a neat pattern ($12$ inches in a foot, $3$ feet in a yard, $1760$ yards in a mile), so it helps to remember a few links to metric. One inch is exactly $2.54$ cm, a mile is about $1.6$ km, a kilogram is about $2.2$ pounds, and a litre is about $1.76$ UK pints.

Worked examples

Example 1: A metric conversion

Convert $2.75$ km to centimetres.

1. km to m: multiply by $1000$, giving $2750$ m.
2. m to cm: multiply by $100$, giving $275\,000$ cm.

Answer: $275\,000$ cm

Example 2: Unit fractions in a chain

A car travels at $90$ km/h. What is this in metres per second?

1. Write the speed with unit fractions: $\dfrac{90 \text{ km}}{1 \text{ h}} \times \dfrac{1000 \text{ m}}{1 \text{ km}} \times \dfrac{1 \text{ h}}{3600 \text{ s}}$.
2. The km and h cancel, leaving m/s.
3. Calculate: $\dfrac{90 \times 1000}{3600} = 25$.

Answer: $25$ m/s

Example 3: Metric to imperial

A suitcase limit is $23$ kg. What is this in pounds?

1. Use $1 \text{ kg} \approx 2.205 \text{ lb}$.
2. Multiply: $23 \times 2.205 \approx 50.7$.

Answer: about $50.7$ lb, which is why airlines often quote “50 lb or 23 kg”.

Example 4: Temperature

Convert $68^\circ\text{F}$ to Celsius.

1. Subtract $32$: $68 - 32 = 36$.
2. Multiply by $\frac{5}{9}$: $36 \times \frac{5}{9} = 20$.

Answer: $20^\circ\text{C}$

Tip: area and volume conversions use the factor squared or cubed: $1 \text{ m}^2 = 100^2 = 10\,000 \text{ cm}^2$, not $100$.

Why it matters

Unit conversions are part of travel (miles vs kilometres, Fahrenheit vs Celsius), cooking from foreign recipes (cups vs grams), DIY (inches vs millimetres), health (stones and pounds vs kilograms) and science, where every formula expects consistent units. Our unit converter does the arithmetic, but understanding the method lets you sense-check its answers.