Rates and Unit Rates

A rate compares two quantities measured in different units. A unit rate expresses a rate with a denominator of 1, making comparisons straightforward.

Examples of Unit Rates

80 km/h

Speed: kilometres per hour

$3.50/L

Price: dollars per litre

150 bpm

Heart rate: beats per minute

To find a unit rate, divide the first quantity by the second. For example, if a car travels 240 km in 3 hours, the unit rate is 240/3 = 80 km/h.

Proportional Reasoning and Scale Drawings

Proportional reasoning uses the fact that two ratios are equivalent. If a/b = c/d, then a × d = b × c (cross-multiplication).

A scale drawing is a proportional representation of a real object. The scale factor is the ratio of the drawing measurement to the actual measurement.

Scale Drawing Formula

Scale factor = Drawing length/Actual length

A scale of 1 : 200 means 1 cm on the drawing represents 200 cm (2 m) in reality.

For area, the scale factor is squared. If the linear scale is 1 : k, then the area scale is 1 : k². For volume, it is cubed: 1 : k³.

Practical Applications

Rates and ratios arise constantly in everyday life and professional contexts:

● Best-buy problems: Compare unit prices to determine the better deal (e.g. $4.80 for 2 L vs $7.50 for 3 L).
● Speed, distance, time: Use d = s × t to solve travel problems, converting units as needed.
● Medication dosage: Doses are often given in mg per kg of body weight.
● Architecture and maps: Scale drawings translate real-world dimensions into workable representations.

Key Vocabulary

Unit Rate

A rate expressed with a denominator of 1, such as km/h or $/kg, used for direct comparison.

Proportional Reasoning

Using equivalent ratios and cross-multiplication to find unknown values in proportion problems.

Scale Factor

The ratio of a measurement on a drawing or model to the corresponding actual measurement.

Rate of Change

How quickly one quantity changes relative to another, often expressed as a gradient or slope.

Worked Examples

1

Brand A sells 750 mL of juice for $4.50. Brand B sells 1.2 L for $6.00. Which is the better buy?

Step 1: Find the unit price for Brand A: $4.50 / 0.75 L = $6.00 per litre.

Step 2: Find the unit price for Brand B: $6.00 / 1.2 L = $5.00 per litre.

Answer: Brand B is the better buy at $5.00/L compared to Brand A at $6.00/L.

2

On a map with scale 1 : 25,000, two towns are 8.4 cm apart. What is the actual distance in kilometres?

Step 1: Actual distance = map distance × scale factor = 8.4 cm × 25,000 = 210,000 cm.

Step 2: Convert to km: 210,000 cm = 2,100 m = 2.1 km.

Answer: The actual distance between the towns is 2.1 km.

3

A tap fills a 300 L tank in 2 hours 30 minutes. What is the flow rate in litres per minute?

Step 1: Convert time to minutes: 2 h 30 min = 150 minutes.

Step 2: Flow rate = volume / time = 300 L / 150 min = 2 L/min.

Answer: The flow rate is 2 litres per minute.

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

A car travels 420 km in 5 hours. What is the average speed in km/h?

80 km/h 84 km/h 85 km/h 90 km/h

Question 2

A floor plan uses a scale of 1 : 50. A room on the plan measures 6 cm by 8 cm. What is the actual area of the room in square metres?

48 m² 24 m² 12 m² 2,400 m²

Question 3

If 5 workers can build a wall in 8 days, how many days would it take 10 workers (at the same rate)?

16 days 2.5 days 8 days 4 days

Question 4

A recipe for 4 people uses 600 g of flour. How much flour is needed for 10 people?

1,500 g 1,200 g 1,000 g 2,400 g

Question 5

A model car is built at a scale of 1 : 18. The model is 25 cm long. What is the length of the actual car in metres?

3.6 m 4.5 m 1.8 m 0.45 m

Rates and Ratios in Context