Converting Rates
Converting a rate means changing both of its units, such as metres per second to kilometres per hour. This NSW Year 9 Advanced maths subtopic revises the Stage 4 method (NESA outcome MA4‑RAT‑C‑01): convert the top unit, convert the bottom unit, then divide.
You will convert speeds, flow rates such as \(120\) L/min to mL/s, prices in cents per gram, and spreading rates in kg/ha. You will use converted rates to find a car's stopping distance, the petrol cost of a trip, the water wasted by a dripping tap, a download time and the mass of a block of ice, and find how long a leaking tank takes to fill. It is part of Year 9 Advanced Working with Numbers.
Every question with a fully worked solution.
- Converting Rates - Video - Converting Rates Watch
Theory
To convert a rate, change the top unit and the bottom unit one at a time, checking whether the number should get bigger or smaller. This Year 9 Advanced maths page revises the skill from earlier years and extends it to km/h and m/s, rates per area and per volume, and multi-step problems that combine rates given in different units.
A rate compares two quantities in different units, such as \(60\) L/min. The first unit is the top unit and the unit after "per" is the bottom unit.
To convert a rate, write it as a fraction with its units, replace each unit with its equivalent, then simplify. Change one unit at a time.
Top unit: a smaller unit means more of them, so multiply (\(3\) kg is \(3000\) g). Bottom unit: it works the other way. Per hour is a longer time than per minute, so more is done in it and you multiply by \(60\).
Area and volume units square or cube the length conversion: \(1\text{ m}^2 = 10\,000\text{ cm}^2\), \(1\text{ ha} = 10\,000\text{ m}^2\), \(1\text{ m}^3 = 1\,000\,000\text{ cm}^3 = 1000\) L.
Write the rate as a fraction and substitute each unit:
| Quantity | Conversions |
|---|---|
| Time | \(1\) min \(= 60\) s, \(1\) h \(= 3600\) s, \(1\) day \(= 24\) h |
| Mass | \(1\) kg \(= 1000\) g, \(1\) t \(= 1000\) kg |
| Capacity | \(1\) L \(= 1000\) mL, \(1\) kL \(= 1000\) L, \(1\) ML \(= 1\,000\,000\) L |
| Area | \(1\text{ m}^2 = 10\,000\text{ cm}^2\), \(1\text{ ha} = 10\,000\text{ m}^2\) |
| Volume | \(1\text{ m}^3 = 1\,000\,000\text{ cm}^3 = 1000\text{ L}\), \(1\text{ cm}^3 = 1\text{ mL}\) |
| Money | \(\$1 = 100\) cents |
How to convert a rate
- Write the rate as a fraction with its units showing.
- Change the top unit and decide: multiply or divide?
- Change the bottom unit and decide again. It works the opposite way to the top.
- Check the sense: a bigger unit on top means a smaller number.
Using converted rates
- Compare two rates only once they are in the same units, such as two speeds both in km/h.
- Combine rates (a tap and a leak) only in the same units. Add rates that help, subtract rates that work against, then time \(=\) amount \(\div\) rate.
- Stopping distance \(=\) reaction distance \(+\) braking distance. Change the speed to m/s, then reaction distance \(=\) speed \(\times\) reaction time.
- Fuel: a rate of \(6.5\) L/100 km means litres used \(= \dfrac{\text{distance}}{100} \times 6.5\). Change cents per litre to dollars before finding the cost.
- Long time spans: build up seconds to days and years step by step (\(86\,400\) s in a day) and convert units at the end.
- Data: \(1\) byte \(= 8\) bits, so a speed in megabits per second must be compared with a file size in megabits.
- Rain: volume \(=\) area \(\times\) depth, all in metres, then \(1\text{ m}^3 = 1000\) L. Round down for whole days of use.
(i) \(54\) km/h to m/s
(ii) \(2.4\) kg/min to g/s.
Change the top unit, then the bottom unit. Ask each time whether the number should get bigger or smaller.
(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)
| \(2.4\text{ kg/min}\) | \(=\) | \(2400\text{ g/min}\quad(\times 1000)\) |
| \(\) | \(=\) | \(\dfrac{2400}{60}\text{ g/s}\quad(\div 60)\) |
| \(\) | \(=\) | \(\mathbf{40}\text{ g/s}\) |
Check (i): \(54 \div 3.6 = 15\).
Change kilograms to grams on top, and hectares to square metres underneath.
| \(\dfrac{300\text{ kg}}{1\text{ ha}}\) | \(=\) | \(\dfrac{300 \times 1000\text{ g}}{10\,000\text{ m}^2}\) |
| \(\) | \(=\) | \(\dfrac{300\,000}{10\,000}\text{ g/m}^2\) |
| \(\) | \(=\) | \(\mathbf{30}\text{ g/m}^2\) |
Check the sense: a square metre is tiny next to a hectare, so the amount on each square metre is much smaller.
The two rates must be in the same units before they can be combined. Change the leak to L/min.
| \(50\text{ mL/s}\) | \(=\) | \(50 \times 60\text{ mL/min}\) |
| \(\) | \(=\) | \(3000\text{ mL/min}\) |
| \(\) | \(=\) | \(3\text{ L/min}\) |
| \(\text{Net rate}\) | \(=\) | \(9 - 3 = 6\text{ L/min}\) |
| \(\text{Time}\) | \(=\) | \(120 \div 6\) |
| \(\) | \(=\) | \(\mathbf{20}\text{ min}\) |
The leak works against the tap, so its rate is subtracted.
Volume \(=\) area \(\times\) depth, with every length in metres.
| \(\text{Area}\) | \(=\) | \(1.5 \times 10\,000 = 15\,000\text{ m}^2\) |
| \(\text{Depth}\) | \(=\) | \(18 \div 1000 = 0.018\text{ m}\) |
| \(V\) | \(=\) | \(15\,000 \times 0.018 = 270\text{ m}^3\) |
| \(\) | \(=\) | \(270 \times 1000 = 270\,000\text{ L}\) |
| \(\) | \(=\) | \(\mathbf{0.27}\text{ ML}\) |
Changing the depth to metres first keeps every unit matching, so the volume comes out in cubic metres.
Common pitfalls
Frequently asked questions
How do you convert km/h to m/s?
Divide by 3.6. This is the same as multiplying by 1000 to change kilometres to metres and dividing by 3600 to change hours to seconds. For example, 45 km/h is 12.5 m/s.
How do you convert m/s to km/h?
Multiply by 3.6. For example, 10 m/s is 36 km/h. The km/h number is always the larger one.
How do you convert a rate with two units?
Write the rate as a fraction with its units, then change the top unit and the bottom unit one at a time. Replace each unit with its equivalent, such as 1 minute equals 60 seconds, and simplify the fraction.
Why do you multiply when changing per minute to per hour?
An hour is 60 times as long as a minute, so 60 times as much happens in it. A tap that runs 5 litres per minute runs 300 litres per hour.
How many litres are in a cubic metre?
There are 1000 litres in 1 cubic metre, and 1 cubic centimetre holds 1 millilitre. To turn rainfall into litres, multiply the area in square metres by the depth in metres and then by 1000.
How do you work out reaction distance?
Change the speed to metres per second by dividing the km/h value by 3.6, then multiply by the reaction time in seconds. At 108 km/h, which is 30 m/s, a 1.5 second reaction covers 45 metres.