Rates
A rate compares two quantities with different units, like 150 km in 3 hours or $42 for 6 kg. In NSW Year 7 maths, students learn how a rate differs from a ratio and how to write it as an amount for one unit (NESA outcome MA4‑RAT‑C‑01).
You will write simplified rates like 9 mm/h, read the slash as “per”, change one unit in a rate, find average speed in km/h, and use a rate to work out an amount or a time. Pay per hour, rainfall, typing speed and your heart rate are all rates you meet every day. This subtopic is part of Year 7 Ratios, Rates and Time.
Every question with a fully worked solution.
- Rates - Rates - Ratios and Unit Rate Examples and Word Problems! Watch
Theory
A rate compares two quantities measured in different units, such as kilometres and hours, or dollars and kilograms. A simplified rate gives the amount for one unit: \(\$35\) for \(5\) kg is \(\$7\)/kg. This Year 7 maths page shows how to write, change and use rates, including speed.
A ratio compares quantities in the same units: \(4\) cm to \(10\) cm. A rate compares quantities in different units: \(150\) km in \(3\) hours.
A simplified rate gives the amount for one unit of the second quantity. Divide to find it: \(45\) mm in \(5\) h is \(45 \div 5 = 9\) mm/h. Read the slash as per: \(9\) mm/h is \(9\) millimetres per hour. Always write the units.
A decimal in the division? Double (or multiply by \(10\)) both numbers first: \(18 \div 1.5 = 36 \div 3 = 12\).
Less than one unit? Scale up: \(12\) beats in \(10\) s is \(12 \times 6 = 72\) beats in \(60\) s, so \(72\) beats/min.
Change one unit with a unit fact: \(2.5\) L/h \(= 2.5 \times 24 = 60\) L/day, and \(0.3\) m/year \(= 0.3 \times 100 = 30\) cm/year.
Speed is a rate: speed \(=\) distance \(\div\) time. For km/h, change minutes to hours first. Using a rate: amount \(=\) rate \(\times\) time, and time \(=\) amount \(\div\) rate.
Common rates and how to read them:
| Rate | Read as |
|---|---|
| km/h | kilometres per hour |
| \(\$\)/kg | dollars per kilogram |
| L/min | litres per minute |
| \(\$\)/h | dollars per hour |
| words/min | words per minute |
How to write a simplified rate
- Check the units are different (otherwise it is a ratio).
- Divide the first quantity by the second, in the order of the units (\(\$\)/kg means dollars \(\div\) kilograms).
- Write the answer with both units, using a slash.
How to use a rate
- Amount \(=\) rate \(\times\) time.
- Time \(=\) amount \(\div\) rate.
- Match the time units to the rate first.
| \(35 \div 5\) | \(=\) | \(7\) |
\(\$7\)/kg
(i) \(210\) km in \(3\) hours
(ii) \(40\) cm to \(2\) m
| \(210 \div 3\) | \(=\) | \(70\) |
(i) A rate (different units): \(70\) km/h.
(ii) A ratio (both lengths): \(40:200 = 1:5\).
| \(27 \div 4.5\) | \(=\) | \(54 \div 9\) |
| \(\) | \(=\) | \(6\) |
\(6\) L/min. Check: \(4.5 \times 6 = 27\).
| \(28 \times 100\) | \(=\) | \(2800\) |
| \(2800 \div 175\) | \(=\) | \(16\) |
\(16\) cm/year
| \(48 \div 60\) | \(=\) | \(0.8\) |
| \(36 \div 0.8\) | \(=\) | \(360 \div 8\) |
| \(\) | \(=\) | \(45\) |
\(45\) km/h. Check: \(45 \times 0.8 = 36\).
(i) Write the rate in L/h.
(ii) Write the rate in L/min.
(iii) How many hours does it take to pump \(585\) L?
| \(360 \div 8\) | \(=\) | \(45\) |
| \(45 \div 60\) | \(=\) | \(0.75\) |
| \(585 \div 45\) | \(=\) | \(13\) |
(i) \(45\) L/h
(ii) \(0.75\) L/min
(iii) \(13\) h
| \(60 \div 15\) | \(=\) | \(4\) |
| \(20 \times 4\) | \(=\) | \(80\) |
\(80\) beats/min
Common pitfalls
Frequently asked questions
What is the difference between a rate and a ratio?
A ratio compares quantities in the same units; a rate compares quantities in different units, such as km and hours.
How do you write a simplified rate?
Divide to find the amount for one unit. \(\$35\) for \(5\) kg is \(\$7\)/kg.
How do you find average speed?
Divide the distance by the time in hours. \(36\) km in \(48\) min (\(0.8\) h) is \(45\) km/h.
What does per mean?
For each one. \(9\) mm/h means \(9\) millimetres in each hour.
How do you use a rate to find a time?
Divide the amount by the rate. \(585\) L at \(45\) L/h takes \(13\) hours.