You cannot compare, add or simplify two quantities until they are measured in the same unit. Converting units is therefore the very first step in a huge number of GCSE questions.
Every conversion means multiplying or dividing by a conversion factor.
Going to a smaller unit → the number gets bigger → multiply.
Going to a bigger unit → the number gets smaller → divide.
Convert $3.6$ km into metres.
Sense check: 3600 is bigger than 3.6 ✓
The metric system is built on powers of $10$, so all conversions are $\times 10$, $\times 100$ or $\times 1000$.
| Measure | Conversions you must know |
|---|---|
| Length | $1$ cm $= 10$ mm · $1$ m $= 100$ cm $= 1000$ mm · $1$ km $= 1000$ m |
| Mass | $1$ g $= 1000$ mg · $1$ kg $= 1000$ g · $1$ tonne $= 1000$ kg |
| Capacity | $1$ litre $= 1000$ ml · $1$ litre $= 100$ cl · $1$ cl $= 10$ ml |
| Link | $1$ ml $= 1\text{ cm}^3$ · $1$ litre $= 1000\text{ cm}^3$ · $1\text{ m}^3 = 1000$ litres |
Blue = going to a smaller unit (multiply) · Green = going to a bigger unit (divide)
Convert $47\,500$ mm into kilometres.
• the area factor is $n^2$
• the volume factor is $n^3$
| Conversion | Length | Area | Volume |
|---|---|---|---|
| cm ↔ mm | $\times 10$ | $\times 100$ | $\times 1000$ |
| m ↔ cm | $\times 100$ | $\times 10\,000$ | $\times 1\,000\,000$ |
| km ↔ m | $\times 1000$ | $\times 1\,000\,000$ | $\times 10^9$ |
A carpet has area $6.4\text{ m}^2$. Convert this to $\text{cm}^2$.
A tank holds $2\,500\,000\text{ cm}^3$ of water. Convert this to $\text{m}^3$ and then to litres.
Answer: $2.5\text{ m}^3 = 2500$ litres
Time is the one measure that is not metric, so be extra careful.
| From | To | Do this |
|---|---|---|
| hours | minutes | $\times 60$ |
| minutes | seconds | $\times 60$ |
| hours | seconds | $\times 3600$ |
To turn minutes into a decimal of an hour, divide by $60$: $30 \div 60 = 0.5$.
Write $3$ hours $24$ minutes as a decimal number of hours.
And in reverse: $1.75$ hours $= 0.75 \times 60 = 45$ min, so $1$ h $45$ min.
A compound unit is made from two different measures combined, almost always by dividing one by the other. The word "per" (or the slash "/") tells you a division is happening.
| Quantity | Formula | Typical units |
|---|---|---|
| Speed | $\text{speed} = \dfrac{\text{distance}}{\text{time}}$ | m/s, km/h, mph |
| Density | $\text{density} = \dfrac{\text{mass}}{\text{volume}}$ | g/cm³, kg/m³ |
| Pressure | $\text{pressure} = \dfrac{\text{force}}{\text{area}}$ | N/m² (pascals), N/cm² |
| Rate of pay | $\dfrac{\text{money}}{\text{time}}$ | £/hour |
| Unit price | $\dfrac{\text{cost}}{\text{quantity}}$ | p/gram, £/litre |
| Fuel economy | $\dfrac{\text{distance}}{\text{fuel}}$ | miles per gallon, km/litre |
| Population density | $\dfrac{\text{people}}{\text{area}}$ | people/km² |
The unit itself tells you the formula.
A block of oak has mass $840$ g and volume $1200\text{ cm}^3$. Find its density.
To convert a compound unit you must convert both parts. Do it one part at a time.
- Write the quantity as a fraction, e.g. $\dfrac{72\text{ km}}{1\text{ h}}$.
- Convert the top into the unit you want.
- Convert the bottom into the unit you want.
- Divide to get the new value.
Convert $72$ km/h into m/s.
Answer: $20$ m/s
The density of aluminium is $2.7\text{ g/cm}^3$. Convert this to $\text{kg/m}^3$.
Answer: $2700\text{ kg/m}^3$
In other words, to go from g/cm³ to kg/m³ you simply multiply by $1000$.
These approximations are worth learning; harder questions will usually give you the conversion factor.
| Imperial | Metric (approx.) |
|---|---|
| $1$ inch | $2.5$ cm |
| $1$ foot | $30$ cm |
| $1$ mile | $1.6$ km (exactly $1.609\ldots$) |
| $5$ miles | $8$ km |
| $1$ pound (lb) | $450$ g |
| $1$ kg | $2.2$ lb |
| $1$ gallon | $4.5$ litres |
| $1$ pint | $570$ ml |
A road sign says the next town is $56$ km away. Roughly how many miles is this? Use $5$ miles $= 8$ km.
Answer: $35$ miles
Sense check: miles are longer than km, so the number of miles should be smaller ✓
Petrol in the UK costs £$1.48$ per litre. In the USA it costs $$3.60$ per US gallon. Given $1$ US gallon $= 3.79$ litres and £$1$ = $$1.25$, which country's petrol is cheaper?
USA petrol is cheaper — about half the UK price.
Direction rule
Smaller unit → bigger number → multiply. Bigger unit → smaller number → divide.
Length
mm $\to$ cm $\div10$, cm $\to$ m $\div100$, m $\to$ km $\div1000$.
Area & volume
Square the length factor for area, cube it for volume.
Capacity link
$1\text{ cm}^3 = 1$ ml, $1000\text{ cm}^3 = 1$ litre, $1\text{ m}^3 = 1000$ litres.
Time
Minutes $\to$ decimal hours: $\div 60$. Never write $2$ h $30$ as $2.3$.
Speed shortcut
km/h $\div 3.6 =$ m/s. m/s $\times 3.6 =$ km/h.
Density shortcut
g/cm³ $\times 1000 =$ kg/m³.
Imperial
$5$ miles $= 8$ km, $1$ kg $= 2.2$ lb, $1$ gallon $= 4.5$ litres.
Convert: (a) $2.7$ km to metres (b) $845$ g to kilograms (c) $0.36$ litres to millilitres.
▶ Show solution
(a) km → m is $\times 1000$: $2.7 \times 1000 = 2700$ m
(b) g → kg is $\div 1000$: $845 \div 1000 = 0.845$ kg
(c) litres → ml is $\times 1000$: $0.36 \times 1000 = 360$ ml
A rectangle measures $2.4$ m by $150$ cm. Find its area in $\text{m}^2$.
▶ Show solution
Make the units match first. $150$ cm $= 150 \div 100 = 1.5$ m.
Area $= 2.4 \times 1.5 = 3.6$
Answer: $3.6\text{ m}^2$
Convert $0.045\text{ m}^2$ into $\text{cm}^2$.
▶ Show solution
Length factor m → cm is $\times 100$, so the area factor is $100^2 = 10\,000$.
$0.045 \times 10\,000 = 450$
Answer: $450\text{ cm}^2$
A fish tank is a cuboid measuring $80$ cm by $40$ cm by $50$ cm. How many litres of water does it hold when full?
▶ Show solution
Volume $= 80 \times 40 \times 50 = 160\,000\text{ cm}^3$
$1000\text{ cm}^3 = 1$ litre, so divide by $1000$:
$160\,000 \div 1000 = 160$
Answer: $160$ litres
Write (a) $4$ hours $12$ minutes as a decimal number of hours, and (b) $2.85$ hours in hours and minutes.
▶ Show solution
(a) $12 \div 60 = 0.2$, so the time is $4.2$ hours.
(b) $0.85 \times 60 = 51$ minutes, so $2.85$ h $= 2$ hours $51$ minutes.
A cyclist travels at $9$ m/s. Convert this speed to km/h.
▶ Show solution
In $1$ second: $9$ m. In $1$ hour ($3600$ s): $9 \times 3600 = 32\,400$ m.
$32\,400$ m $= 32\,400 \div 1000 = 32.4$ km.
Answer: $32.4$ km/h (check: $9 \times 3.6 = 32.4$ ✓)
Mercury has a density of $13\,600\text{ kg/m}^3$. Convert this into $\text{g/cm}^3$.
▶ Show solution
kg/m³ → g/cm³ is $\div 1000$ (the reverse of the shortcut in Section 6).
$13\,600 \div 1000 = 13.6$
Answer: $13.6\text{ g/cm}^3$
Full working: $13\,600$ kg $= 13\,600\,000$ g; $1\text{ m}^3 = 1\,000\,000\text{ cm}^3$; $13\,600\,000 \div 1\,000\,000 = 13.6$ ✓
A plumber charges £$34$ per hour. She works from $08{:}45$ to $14{:}15$. How much does she charge?
▶ Show solution
Time worked: $08{:}45 \to 14{:}15$ is $5$ hours $30$ minutes.
As a decimal: $30 \div 60 = 0.5$, so $5.5$ hours.
Charge $= 34 \times 5.5 = £187$
A car's fuel tank holds $12$ gallons. Using $1$ gallon $= 4.5$ litres, and given that fuel costs £$1.52$ per litre, find the cost of filling the tank from empty.
▶ Show solution
Capacity in litres: $12 \times 4.5 = 54$ litres.
Cost $= 54 \times 1.52 = £82.08$
A goods lorry may not exceed $7.5$ tonnes. It weighs $4200$ kg when empty and is loaded with $180$ boxes each of mass $16.5$ kg. Is the lorry within the limit?
▶ Show solution
Mass of boxes $= 180 \times 16.5 = 2970$ kg.
Total mass $= 4200 + 2970 = 7170$ kg.
Limit $= 7.5$ tonnes $= 7.5 \times 1000 = 7500$ kg.
$7170 < 7500$, so yes, the lorry is within the limit — with $330$ kg to spare.