Use units of measurement and convert between them, including length, area, volume, capacity, mass and compound units such as speed, multiplying or dividing by the correct power of 10

What this dot point is asking

NESA wants you to work confidently in the metric (SI) system. You need to pick a sensible unit for a real quantity. You also need to convert between units of length, mass, area, volume and capacity, and the compound units of speed (units like km/h that combine two measurements), by multiplying or dividing by the correct power of $10$. The arithmetic is not the hard part. The marks are lost on two decisions: choosing whether to multiply or divide, and remembering that area and volume do not convert by the same factor as length. Get those two right and every conversion in the course becomes routine.

The answer

The metric system is built on powers of $10$. So every conversion between related units is a multiplication or a division by a power of $10$. The trick is reading the direction. Moving to a smaller unit makes the number bigger, so you multiply. Moving to a larger unit makes the number smaller, so you divide. The conversion ladder below captures this for length, mass and capacity, all of which share the same factors.

The SI system and its prefixes

Australia uses the SI (International System of Units) metric system, which is built on multiples of $10$. Each quantity has a base unit - the metre for length, the gram for mass, the litre for capacity - and the prefix in front of the unit names the power of $10$. The three prefixes you meet most are:

• kilo ($\text{k}$) means $1000$, so $1$ km $= 1000$ m and $1$ kg $= 1000$ g,
• centi ($\text{c}$) means one hundredth, so $1$ cm $= \tfrac{1}{100}$ m, that is $100$ cm $= 1$ m,
• milli ($\text{m}$) means one thousandth, so $1$ mm $= \tfrac{1}{1000}$ m, that is $1000$ mm $= 1$ m.

Because length, mass and capacity all use the same prefixes, the single ladder above does all three. Converting $5.2$ kg to grams is the same step as converting $5.2$ km to metres: multiply by $1000$ to get $5200$.

The direction rule

Every conversion needs you to decide multiply or divide. The reliable rule is to think about the size of the unit, not the size of the number:

• changing to a smaller unit, the number gets bigger, so multiply, and
• changing to a larger unit, the number gets smaller, so divide.

A metre is smaller than a kilometre, so a distance has more metres than kilometres, and $4250$ m $= 4250 \div 1000 = 4.25$ km. A gram is smaller than a kilogram, so $2.4$ kg $= 2.4 \times 1000 = 2400$ g. If your answer comes out the wrong way (a few metres turning into thousands of kilometres), you have multiplied where you should have divided.

Area: square the factor

This is where most marks are dropped. Area is a length multiplied by a length, so when you convert the units you convert both lengths, which squares the conversion factor. Since $1$ m $= 100$ cm, a square one metre on each side is $100$ cm by $100$ cm:

The diagram makes this visible: the same square is described once in metres and once in centimetres, and counting the small squares gives $10\,000$, not $100$.

The same logic gives the other area factors. Since $1$ km $= 1000$ m, we get $1$ km$^2 = 1000^2 = 1\,000\,000$ m$^2$. The hectare is the area unit for land: $1$ ha $= 10\,000$ m$^2$, which is exactly a square $100$ m by $100$ m. So $0.8$ ha $= 0.8 \times 10\,000 = 8000$ m$^2$, and a paddock of $45\,000$ m$^2$ is $45\,000 \div 10\,000 = 4.5$ ha.

Volume: cube the factor

Volume is a length multiplied by a length multiplied by a length, so converting the units cubes the factor. Since $1$ m $= 100$ cm,

So $0.45$ m$^3 = 0.45 \times 1\,000\,000 = 450\,000$ cm$^3$. The pattern across length, area and volume is the whole idea: the conversion factor is raised to the power $1$ for length, $2$ for area and $3$ for volume.

Capacity and the litre link

Capacity measures how much a container holds, and it ties to volume through one fact you should commit to memory:

That chain lets you turn any volume into a capacity. A volume of $2500$ cm$^3$ is $2500$ mL, which is $2500 \div 1000 = 2.5$ L. A tank of $0.75$ m$^3$ holds $0.75 \times 1000 = 750$ L, which is $750 \div 1000 = 0.75$ kL. Notice $1$ m$^3 = 1$ kL, so cubic metres and kilolitres carry the same number, a handy check.

Compound units: km/h to m/s

A compound unit combines two units. Speed is the one NESA examines: kilometres per hour and metres per second. To convert a speed you convert the distance unit and the time unit. Going from km/h to m/s, turn kilometres into metres ($\times 1000$) and hours into seconds ($\div 3600$, since $1$ hour $= 60 \times 60 = 3600$ seconds):

Combining the two factors, $\dfrac{1000}{3600} = \dfrac{1}{3.6}$, which gives the rule worth memorising: to go from km/h to m/s, divide by $3.6$; to go back from m/s to km/h, multiply by $3.6$. So a car at $100$ km/h is travelling $100 \div 3.6 \approx 27.78$ m/s, and a runner at $12$ m/s is moving $12 \times 3.6 = 43.2$ km/h.

How exam questions ask about units

The wording varies, but each version points to the same decision about factor and direction:

• "Convert / express ... in [unit]" is the plain conversion: identify the factor, then decide multiply or divide by the size rule.
• "... in square ... / square metres / hectares" is an area conversion, so square the factor (and remember $1$ ha $= 10\,000$ m$^2$).
• "... in cubic ... / how much does it hold / capacity in litres" is a volume or capacity conversion, so cube the factor or use the litre chain $1$ m$^3 = 1000$ L.
• "... in metres per second" or "... in km/h" is a speed conversion, so divide or multiply by $3.6$.
• "Which unit is most appropriate ..." asks you to choose a sensible unit: millimetres for a credit card's thickness, kilometres for a road trip, tonnes for a truck, hectares for a farm.
• "Find the total ... giving your answer in [unit]" means do the arithmetic first, then convert the final total once into the requested unit, rather than converting partway.