Converting Units
Converting units means changing a measurement from one unit to another, such as square kilometres to hectares or litres to cubic metres. Area units change by the square of the length factor and volume units by the cube, so \(1~\text{m}^3=10^6~\text{cm}^3\). In NSW Year 10 Advanced maths very large and very small answers are written in scientific notation (NESA outcome MA5‑MAG‑C‑01).
You will convert lengths, areas, volumes, capacities and masses, including micrometres and nanometres, and change rates such as litres per minute into cubic metres per hour and km/h into m/s. Problems include rain on a catchment flowing into a dam, sand for a sandpit, a child's medicine dose, a heart's daily output, the length of a light-year and a train passing through a tunnel. It is part of Year 10 Advanced Measurement.
Theory
To convert units, multiply when changing to a smaller unit and divide when changing to a larger one. For area square the length factor and for volume cube it. This Year 10 Advanced maths page covers hectares, litres and cubic metres, very small units such as nanometres in scientific notation, and converting rates.
A unit is a standard amount used to measure, such as a metre or a litre. Metric units are linked by powers of \(10\), and the prefix tells you which power: kilo means \(1000\), milli means one thousandth.
An area unit is a square, so its conversion factor is the length factor squared: \(1\text{ m}^2 = 100^2\text{ cm}^2 = 10\,000\text{ cm}^2\). A volume unit is a cube, so its factor is cubed: \(1\text{ m}^3 = 100^3\text{ cm}^3 = 1\,000\,000\text{ cm}^3\).
Capacity is how much a container holds: \(1\text{ mL} = 1\text{ cm}^3\), \(1\text{ L} = 1000\text{ cm}^3\) and \(1\text{ m}^3 = 1000\text{ L} = 1\text{ kL}\). A hectare is \(10\,000\text{ m}^2\), the area of a \(100\) m square.
A rate compares two different units, such as km/h or L/min. Convert the top unit and the bottom unit separately.
Length, area and volume:
Capacity, land area and mass:
Prefixes:
| Prefix | Symbol | Meaning | Example |
|---|---|---|---|
| mega | M | \(10^{6}\) | \(1\text{ ML} = 10^{6}\text{ L}\) |
| kilo | k | \(10^{3}\) | \(1\text{ kL} = 1000\text{ L}\) |
| centi | c | \(10^{-2}\) | \(1\text{ cm} = 0.01\text{ m}\) |
| milli | m | \(10^{-3}\) | \(1\text{ mL} = 0.001\text{ L}\) |
| micro | \(\mu\) | \(10^{-6}\) | \(1\,\mu\text{m} = 10^{-6}\text{ m}\) |
| nano | n | \(10^{-9}\) | \(1\text{ nm} = 10^{-9}\text{ m}\) |
How to convert any unit
- Write the link between the two units, e.g. \(1\text{ m} = 100\text{ cm}\).
- Square or cube the factor for area or volume units.
- Multiply to change to a smaller unit (more of them); divide to change to a larger unit.
- Check the size of the answer makes sense, and use scientific notation for very large or very small numbers.
Rates and multi-step problems
- Rates: convert the top unit and the bottom unit one at a time, e.g. L/min to m³/h: multiply by \(60\) for hours, then divide by \(1000\) for cubic metres.
- Chained rates: multiply step by step, e.g. mL per beat \(\times\) beats per minute \(\times\) minutes per day.
- Volume from a depth: rainfall or a layer of sand is a prism, so volume \(=\) area \(\times\) depth, with all lengths in the same unit (Example 3).
- Dosages and concentrations: find the amount per unit first, e.g. \(250\text{ mg}\) in \(5\text{ mL}\) is \(50\text{ mg}\) per mL, then divide.
- Buying whole items: if \(38.3\) bags are needed, buy \(39\). If asking how many full households can be supplied, round down.
- Distance travelled by a long object: a train passing completely through a tunnel travels the tunnel length plus its own length.
- Unknown in the units: write each quantity in the new unit in terms of the pronumeral, e.g. a side of \(x\) cm is \(10x\) mm, then form and solve an equation.
(i) A vineyard covers \(0.85\text{ km}^2\). Write this area in square metres and in hectares.
(ii) A drop of oil has a volume of \(640\text{ mm}^3\). Write this in cubic metres, using scientific notation.
(i) \(1\text{ km} = 1000\text{ m}\), so \(1\text{ km}^2 = 1000^2\text{ m}^2 = 10^6\text{ m}^2\). A hectare is \(10\,000\text{ m}^2\).
| \(0.85\text{ km}^2\) | \(=\) | \(0.85 \times 10^6\text{ m}^2\) |
| \(\) | \(=\) | \(\mathbf{850\,000}\text{ m}^2\) |
| \(\) | \(=\) | \(850\,000 \div 10\,000\text{ ha}\) |
| \(\) | \(=\) | \(\mathbf{85}\text{ ha}\) |
(ii) \(1\text{ m} = 1000\text{ mm}\), so \(1\text{ m}^3 = 1000^3\text{ mm}^3 = 10^9\text{ mm}^3\). Going to a larger unit, divide.
| \(640\text{ mm}^3\) | \(=\) | \(640 \div 10^9\text{ m}^3\) |
| \(\) | \(=\) | \(6.4 \times 10^2 \times 10^{-9}\) |
| \(\) | \(=\) | \(\mathbf{6.4 \times 10^{-7}}\text{ m}^3\) |
Check the direction: a cubic metre is huge compared with a cubic millimetre, so the number must get much smaller.
(i) Write \(54\) km/h in metres per second.
(ii) A pump moves \(2.4\text{ m}^3\) of water per hour. Write this rate in millilitres per second, correct to \(1\) decimal place.
Convert the top unit and the bottom unit separately, then divide. \(1\text{ h} = 60 \times 60 = 3600\text{ s}\).
(i)
| \(54\text{ km/h}\) | \(=\) | \(\dfrac{54 \times 1000\text{ m}}{3600\text{ s}}\) |
| \(\) | \(=\) | \(\dfrac{54\,000}{3600}\text{ m/s}\) |
| \(\) | \(=\) | \(\mathbf{15}\text{ m/s}\) |
(ii) \(1\text{ m}^3 = 1000\text{ L} = 1\,000\,000\text{ mL}\).
| \(2.4\text{ m}^3\) | \(=\) | \(2.4 \times 1\,000\,000\text{ mL} = 2\,400\,000\text{ mL}\) |
| \(\text{Rate}\) | \(=\) | \(\dfrac{2\,400\,000\text{ mL}}{3600\text{ s}}\) |
| \(\) | \(=\) | \(666.666\ldots\text{ mL/s}\) |
| \(\) | \(\approx\) | \(\mathbf{666.7}\text{ mL/s}\) |
Shortcut for speeds: to change km/h to m/s, multiply by \(\dfrac{1000}{3600} = \dfrac{5}{18}\). To go back, multiply by \(3.6\).
(i) How many litres of water run off the roof?
(ii) Does the tank overflow?
The rain forms a thin layer over the roof: a prism whose base is the roof and whose height is the rainfall. Put every length in metres first.
(i)
| \(25\text{ mm}\) | \(=\) | \(0.025\text{ m}\) |
| \(V\) | \(=\) | \(15 \times 12 \times 0.025\) |
| \(\) | \(=\) | \(4.5\text{ m}^3\) |
| \(\) | \(=\) | \(4.5 \times 1000\text{ L}\) \(= \mathbf{4500}\text{ L}\) |
(ii) \(10\text{ kL} = 10\,000\text{ L}\).
| \(\text{Water already in tank}\) | \(=\) | \(0.4 \times 10\,000\) \(= 4000\text{ L}\) |
| \(\text{After the rain}\) | \(=\) | \(4000 + 4500\) \(= 8500\text{ L}\) |
\(8500\text{ L} \lt 10\,000\text{ L}\), so the tank does not overflow. It has room for another \(1500\) L.
Write both lengths in the same unit, metres, in scientific notation. Then divide.
| \(120\text{ nm}\) | \(=\) | \(120 \times 10^{-9}\text{ m} = 1.2 \times 10^{-7}\text{ m}\) |
| \(1.5\text{ mm}\) | \(=\) | \(1.5 \times 10^{-3}\text{ m}\) |
| \(\text{Number}\) | \(=\) | \(\dfrac{1.5 \times 10^{-3}}{1.2 \times 10^{-7}}\) |
| \(\) | \(=\) | \(1.25 \times 10^{-3 - (-7)}\) |
| \(\) | \(=\) | \(\mathbf{1.25 \times 10^{4}}\) |
That is \(12\,500\) virus particles. Divide the numbers and subtract the powers: \(1.5 \div 1.2 = 1.25\) and \(-3 - (-7) = 4\).
Common pitfalls
Frequently asked questions
How do you convert square metres to square centimetres?
Multiply by \(10\,000\), because \(1\text{ m} = 100\text{ cm}\) and a square metre is \(100\text{ cm} \times 100\text{ cm}\). So \(3.5\text{ m}^2 = 35\,000\text{ cm}^2\).
How many litres are in a cubic metre?
\(1000\) litres. A cubic metre is \(1\,000\,000\text{ cm}^3\) and each litre is \(1000\text{ cm}^3\). That is also \(1\) kilolitre.
How do you convert km/h to m/s?
Multiply by \(1000\) to change kilometres to metres and divide by \(3600\) to change hours to seconds. That is the same as multiplying by \(\dfrac{5}{18}\), or dividing by \(3.6\). For example, \(90\) km/h is \(25\) m/s.
How many square metres are in a hectare?
\(10\,000\text{ m}^2\), the area of a square \(100\) m long and \(100\) m wide. There are \(100\) hectares in a square kilometre.
What is a nanometre in metres?
A nanometre is one billionth of a metre, \(10^{-9}\) m. A micrometre is \(10^{-6}\) m, so \(1\,\mu\text{m} = 1000\) nm.