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Units, rates and conversions
Metric units and prefixes, converting length, area, volume and mass, hectares and millilitres, speed and other rates, and using ratios and map scales.
Units are where easy marks quietly disappear. A nurse who mixes up mg and g, or a builder who orders 45 m³ of concrete instead of 4.5 m³, has a real problem. In exams it is the same story: the maths is right, the unit is wrong, and the mark is gone.
The metric system runs on powers of ten
Every metric unit is a base unit with a prefix in front. The prefix tells you the multiple of ten, so once you know the prefixes you know every conversion.
| Prefix | Symbol | Means | Everyday example |
|---|---|---|---|
| giga | G | $1\,000\,000\,000$ | a 128 GB phone |
| mega | M | $1\,000\,000$ | a 5 MW wind turbine |
| kilo | k | $1000$ | km, kg, kL, kWh |
| centi | c | $\dfrac{1}{100}$ | cm |
| milli | m | $\dfrac{1}{1000}$ | mm, mg, mL |
| micro | µ | $\dfrac{1}{1\,000\,000}$ | µg of vitamin B12 |
A few non-SI units are used alongside these. The hectare (ha) is the area of a $100$ m by $100$ m square, so $1 \text{ ha} = 10\,000 \text{ m}^2$. A tonne is $1000$ kg. Sailors and pilots use the nautical mile, which is $1852$ m.
Area and volume units
Area units are squares, so the conversion factor gets squared too. A square metre is $100$ cm by $100$ cm, which is $100 \times 100 = 10\,000$ cm². A cubic metre is $100 \times 100 \times 100 = 1\,000\,000$ cm³.
Capacity (how much a container holds) links to volume by one fact worth memorising: $1 \text{ cm}^3$ holds exactly $1$ mL. So $1000 \text{ cm}^3 = 1$ L, and $1 \text{ m}^3 = 1000$ L $= 1$ kL.
Convert (a) $0.45$ km to cm (b) $5600$ cm² to m² (c) $3$ L to cm³.
- Pick Area and convert $1$ m² to cm². Look at which steps light up on the ladder and what each one multiplies by.
- Convert $1$ ha to m², then to km². How many hectares make $1$ km²?
- Pick Volume and convert $1$ m³ to L. Then try mL to cm³. Why doesn't the number change?
- Pick Speed and convert $60$ km/h to m/s. Check the note: what single number do you divide by?
Rates
A rate compares two different kinds of quantity: kilometres per hour, litres per minute, dollars per kilogram, beats per minute. To convert a rate, convert the top unit and the bottom unit separately.
The one you will use most is speed. One kilometre per hour is $1000$ m in $3600$ s, which is $\dfrac{1000}{3600} = \dfrac{1}{3.6}$ m/s. So:
Usain Bolt's 100 m world record is $9.58$ s. Find his average speed in m/s and in km/h, correct to one decimal place.
A game update is $1.5$ GB. Your internet downloads at $12$ MB/s. How long will it take?
Ratios and scales
A ratio compares quantities of the same kind, in the same unit, with no unit written. A concrete mix of cement, sand and gravel in the ratio $1 : 2 : 3$ has $6$ parts, so cement is $\dfrac16$ of the mix.
A map scale is a ratio. On a $1 : 50\,000$ map, $1$ cm on paper is $50\,000$ cm on the ground, which is $500$ m.
On a $1 : 50\,000$ bushwalking map, the track from the car park to the lookout measures $7.4$ cm. How far is the walk in kilometres?
Your turn
A community garden in Parramatta is $2400$ m². Write its area in hectares.
Show the working
$1$ ha $= 10\,000$ m². Small unit to big unit, so divide: $2400 \div 10\,000 = 0.24$ ha.
Convert $4.2$ km to metres.
Big unit to small unit, so multiply: $4.2 \times 1000 = 4200$ m.
Convert $3.5$ m² to cm².
$1$ m² $= 10\,000$ cm², so $3.5 \times 10\,000 = 35\,000$ cm².
A farm is $2\,400\,000$ m². How many hectares is that?
$1$ ha $= 10\,000$ m², so $2\,400\,000 \div 10\,000 = 240$ ha.
Convert $72$ km/h to m/s.
Divide by $3.6$: $72 \div 3.6 = 20$ m/s.
Convert $15$ m/s to km/h.
Multiply by $3.6$: $15 \times 3.6 = 54$ km/h.
A water tank holds $1.8$ m³. Its capacity in litres is:
$1$ m³ $= 1000$ L, so $1.8 \times 1000 = 1800$ L.
A $120$ kg concrete mix uses cement, sand and gravel in the ratio $1 : 2 : 3$. How many kilograms of cement are in it?
There are $1 + 2 + 3 = 6$ parts, so one part is $120 \div 6 = 20$ kg. Cement is one part: $20$ kg.
On a $1 : 25\,000$ map, two schools are $6$ cm apart. How far apart are they in kilometres?
$6 \times 25\,000 = 150\,000$ cm. Then $150\,000 \div 100\,000 = 1.5$ km.
Exam style: A leaking tap loses $0.5$ mL of water every second. How many litres does it waste in one day?
One day is $24 \times 60 \times 60 = 86\,400$ s. Water lost $= 0.5 \times 86\,400 = 43\,200$ mL. Divide by $1000$: $43.2$ L.
Exam style: A pool is $25$ m long, $10$ m wide and filled to a depth of $1.2$ m. A hose delivers $20$ L per minute. How many hours does it take to fill the pool?
Volume $= 25 \times 10 \times 1.2 = 300$ m³ $= 300\,000$ L. Time $= 300\,000 \div 20 = 15\,000$ min. Then $15\,000 \div 60 = 250$ h, more than ten days.