Rates & Speed
Understand what a rate is, explore unit rates, and use the Distance-Speed-Time triangle to solve real-world problems involving movement and comparison.
What Is a Rate?
A rate compares two quantities of different units. For example, "$3 per litre", "80 km per hour", or "12 words per minute". The word "per" means "for every one" and is shown with the symbol /.
$1.90/litre
Cost per litre of petrol
100 km/h
Kilometres per hour
8 L/min
Litres per minute (tap flow)
Unit Rate
A unit rate expresses the rate with a denominator of 1. It makes comparing rates easy.
Example: Best value juice
Brand A: $3.60 for 1.2 L = $3.00/L
Brand B: $2.80 for 0.8 L = $3.50/L
Brand A is better value!
Calculating a unit rate
If you type 240 words in 4 minutes:
240 ÷ 4 = 60 words per minute
Divide both quantities by the denominator.
Speed, Distance, and Time
Speed is a special rate — it measures how far something travels in a given time. The three quantities are linked by the DST triangle:
Cover the value you want to find:
Find Distance
Speed = 60 km/h, Time = 3 h
D = 60 × 3 = 180 km
Find Speed
Distance = 150 km, Time = 2 h
S = 150 ÷ 2 = 75 km/h
Find Time
Distance = 240 km, Speed = 80 km/h
T = 240 ÷ 80 = 3 hours
Converting Rates
Sometimes we need to convert between units to compare rates or solve problems.
km/h to m/s
1 km = 1000 m 1 hour = 3600 seconds
Convert 90 km/h to m/s:
90 × 1000 ÷ 3600 = 25 m/s
Cost per Unit
Convert cost per large unit to per small unit.
Apples cost $4.80 per kg.
Per 100g: $4.80 ÷ 10 = $0.48/100g
Key Vocabulary
Rate
A comparison of two quantities with different units, such as km/h, $/kg, or L/min.
Unit Rate
A rate with a denominator of 1, making it easy to compare. E.g. $2.50 per kilogram.
Speed
A rate measuring distance travelled per unit of time. Common units: km/h and m/s.
DST Triangle
A memory tool: D = S × T, S = D ÷ T, T = D ÷ S. Cover the one you want to find.
Worked Examples
A car travels 360 km in 4 hours. What is its average speed?
Formula: Speed = Distance ÷ Time
Speed = 360 km ÷ 4 h
Answer: Average speed = 90 km/h
A cyclist rides at 24 km/h for 2.5 hours. How far do they travel?
Formula: Distance = Speed × Time
Distance = 24 km/h × 2.5 h
Answer: Distance = 60 km
Oranges cost $3.60 for 600 g. What is the unit rate per 100 g?
Step 1: 600 g ÷ 100 = 6 units of 100 g
Step 2: $3.60 ÷ 6 = $0.60
Answer: Oranges cost $0.60 per 100 g.
Knowledge Check
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Key Concepts Summary
- ●A rate compares two different quantities — the units are always different (e.g. km/h, $/kg).
- ●A unit rate has a denominator of 1, making comparisons straightforward.
- ●Speed = Distance ÷ Time. Use the DST triangle to rearrange the formula.
- ●Check units carefully when converting — km/h to m/s requires multiplying by 1000 and dividing by 3600.
- ●To find the best value, convert to the same unit rate (e.g. $/100 g) before comparing.