What makes a rate
A rate compares two quantities measured in different units.
| Rate | Units |
|---|---|
| 240 km in 3 hours | km and hours |
| $6 for 4 muffins | dollars and muffins |
| 90 heartbeats in 1 minute | beats and minutes |
| 15 L of water in 5 minutes | litres and minutes |
A ratio usually compares quantities with the same units (3 cups to 2 cups); a rate compares different ones.
Writing rates
The word per means "for each", and is often shown by a slash.
- 240 km per 3 h = 240 km/3 h
- 80 km/h = 80 kilometres per hour
- $2.50/kg = $2.50 per kilogram
Always include the units!
Unit rates
A unit rate has 1 as the second quantity. It’s found by dividing.
- 240 km in 3 h → 240 ÷ 3 = 80 km/h
- $6 for 4 muffins → 6 ÷ 4 = $1.50 per muffin
Unit rates make comparison easy. (See the unit rate lesson for more.)
Common rates
| Rate | Usual units |
|---|---|
| Speed | km/h, m/s |
| Price | $/kg, $/L, $/item |
| Wage | $/h |
| Fuel use | L/100 km |
| Typing speed | words per minute |
| Heart rate | beats per minute |
| Population density | people per km² |
| Data speed | megabits per second |
Using a rate to find an amount
If you know the rate, multiply.
- A car drives at 90 km/h for 2.5 h → 90 × 2.5 = 225 km
- Apples cost $3.20/kg. 1.5 kg costs 3.20 × 1.5 = $4.80
Rate tables
| Hours | 1 | 2 | 3 | 5 |
|---|---|---|---|---|
| Pay ($) | 17 | 34 | 51 | 85 |
The rate stays the same ($17/h) in every column.
Worked examples
Example 1: A printer prints 120 pages in 4 minutes. Write the rate as a unit rate.
120 ÷ 4 = 30 pages per minute.
Example 2: A tap fills 50 L in 8 minutes. How much in 20 minutes?
Unit rate 6.25 L/min × 20 = 125 L.
Example 3: Is "3 cups flour to 2 cups milk" a rate or a ratio?
Same units → ratio.
Common mistakes
- Dropping the units, so 80 could mean anything.
- Dividing the wrong way: 3 h ÷ 240 km gives hours per km, not km per hour.
- Mixing up rate and ratio.
Try it yourself
- Write 300 words in 5 minutes as a unit rate.
- A cyclist rides 45 km in 1.5 h. Find the speed.
- Grapes cost $4.40 for 2 kg. Price per kg?
- At 72 beats/min, how many beats in 5 minutes?
Answers: 1) 60 words/min 2) 30 km/h 3) $2.20/kg 4) 360
Quick recap
- A rate compares quantities with different units.
- Per means "for each".
- A unit rate has 1 as its second term.