Units of Measurements

The metric system is built around powers of 10, which makes conversions much easier than working with old units like feet and gallons.

The prefixes you need are:

• kilo- means $1000$,
• centi- means $\frac{1}{100}$ (one hundredth),
• milli- means $\frac{1}{1000}$ (one thousandth).

The same prefixes work for length, mass and capacity:

• Length: $1\text{ km} = 1000\text{ m}$, $1\text{ m} = 100\text{ cm}$, $1\text{ cm} = 10\text{ mm}$.
• Mass: $1\text{ kg} = 1000\text{ g}$, $1\text{ g} = 1000\text{ mg}$, $1\text{ tonne} = 1000\text{ kg}$.
• Capacity: $1\text{ L} = 1000\text{ mL}$, and importantly $1\text{ mL} = 1\text{ cm}^3$.

A simple method that never lets you down:

1. Write down what you have, with its unit.
2. Decide whether the new unit is larger or smaller.
3. Multiply when moving to a smaller unit (the number gets bigger), divide when moving to a larger unit.

For example, $2.5\text{ kg} = 2.5 \times 1000 = 2500\text{ g}$, and $750\text{ mL} = 750 \div 1000 = 0.75\text{ L}$.

Area conversions trip up a lot of students because the conversion factor between linear units gets squared when you switch to area units.

A square that is $1\text{ m}$ along each side has an area of $1\text{ m}^2$. The same square is $100\text{ cm}$ along each side, so its area is $100 \times 100 = 10{,}000\text{ cm}^2$.

The same idea applies to every area conversion:

• $1\text{ cm}^2 = 10 \times 10 = 100\text{ mm}^2$.
• $1\text{ km}^2 = 1000 \times 1000 = 1{,}000{,}000\text{ m}^2$.

So $3.2\text{ m}^2 = 3.2 \times 10{,}000 = 32{,}000\text{ cm}^2$, and $50{,}000\text{ cm}^2 = 50{,}000 \div 10{,}000 = 5\text{ m}^2$.

A reliable check: when you convert to a smaller unit, the number gets bigger. Always glance at the answer to see if it feels right.

Volume conversions take the same idea one step further: the linear conversion factor is cubed.

A cube that is $1\text{ m}$ along each edge has a volume of $1\text{ m}^3$. In centimetres each edge is $100\text{ cm}$, so the volume is $100 \times 100 \times 100 = 1{,}000{,}000\text{ cm}^3$.

So:

• $1\text{ cm}^3 = 10 \times 10 \times 10 = 1000\text{ mm}^3$.
• $1\text{ m}^3 = 1{,}000{,}000\text{ cm}^3$.

A very useful link to remember: $1\text{ mL} = 1\text{ cm}^3$, so $1\text{ L} = 1000\text{ cm}^3$.

A tank with a volume of $0.25\text{ m}^3$ holds $0.25 \times 1{,}000{,}000 = 250{,}000\text{ cm}^3 = 250\text{ L}$ of water.

Compound units combine two measurements into a single rate. The two you meet most often are speed and density.

Speed measures how fast something moves: $\text{speed} = \frac{\text{distance}}{\text{time}}.$

Common units: $\text{km/h}$ (kilometres per hour), $\text{m/s}$ (metres per second). A car at $60\text{ km/h}$ covers $60$ kilometres in $1$ hour.

Density measures how heavy a material is for its size: $\text{density} = \frac{\text{mass}}{\text{volume}}.$

A common unit is $\text{g/cm}^3$. Water has a density of $1\text{ g/cm}^3$; iron is about $7.9\text{ g/cm}^3$.

You can rearrange both formulae. For speed, $\text{distance} = \text{speed} \times \text{time}$ and $\text{time} = \dfrac{\text{distance}}{\text{speed}}$.

The most important habit: make the units match before you calculate. If a speed is given in $\text{km/h}$ but the time is in minutes, convert the time to hours (or the speed to $\text{km/min}$) first.

A train travels $180\text{ km}$ in $2.5$ hours, so its speed is $180 \div 2.5 = 72\text{ km/h}$. To convert to $\text{m/s}$: $72\text{ km/h} = 72 \times 1000 \div 3600 = 20\text{ m/s}$.

Time is the one place where everyday units do not use powers of 10. You need to remember:

• $60$ seconds in a minute,
• $60$ minutes in an hour,
• $24$ hours in a day,
• $7$ days in a week, $365$ (or $366$) days in a year.

The 12-hour clock uses am and pm (with $12$ noon and $12$ midnight). The 24-hour clock runs from $00\!:\!00$ to $23\!:\!59$. So $3\!:\!45\text{ pm}$ is $15\!:\!45$, and $7\!:\!30\text{ am}$ is $07\!:\!30$.

When you add or subtract time, never just subtract the numbers — they roll over at $60$. A reliable method is to step up to the next whole hour, then jump full hours, then add the leftover minutes. From $09\!:\!50$ to $11\!:\!25$: $10$ min + $1$ h + $25$ min = $1$ h $35$ min.

Working with decimal hours also catches people out. $1.5$ hours is $1$ h $30$ min, not $1$ h $50$ min. $2.25$ hours is $2$ h $15$ min. Multiply the decimal part by $60$ to turn it into minutes.

Choosing a sensible unit makes your answers easier to read and to check.

• A door is best measured in metres, not millimetres.
• An ant's length sits in millimetres, not metres.
• A box of cereal weighs in grams; a person in kilograms; a truckload in tonnes.
• A small bottle of medicine is in millilitres; a bottle of water in litres.

When the numbers in your answer look very large or very small, ask yourself if a different unit would make it tidier. For example, $4{,}500\text{ m}$ is more naturally written as $4.5\text{ km}$, and $0.0025\text{ kg}$ as $2.5\text{ g}$.

In real questions the first job is often a quick conversion so that every quantity uses the same unit. A garden $6\text{ m}$ long and $200\text{ cm}$ wide has an area of $6 \times 2 = 12\text{ m}^2$ — but only after you turn $200\text{ cm}$ into $2\text{ m}$. Forgetting that step is a common slip.

Comfort with units pays off in every science and almost every later maths topic:

• Physics is built on compound units — speed, acceleration, force, density, pressure.
• Chemistry uses concentrations (g/L, mol/L) and converts between mL and cm³ constantly.
• Real-life problems in cooking, sports timing, travel and shopping all use these conversions.
• Later algebra problems hide unit conversions inside word problems — spotting them early is half the battle.

Keep this revision handy — when an exam-style question feels confusing, the fix is often to write every quantity with its unit and convert to a single common unit first.