M
hscscience Maths Std · Y12
0/100daily goal
0
0
0 due
0
L1 · 0 XP
KJ
Your weak spots
Insights load after your first practice round.
MST-12-S2-05 · Lesson 1 50–55 min ⚡ +95 XP available

Rates

A rate compares two quantities with different units. Master the unitary method, find the rate per one unit, and you can solve any rate problem by simple multiplication.

Today's hook, Supermarket A sells 2 kg of flour for $\$3.80$. Supermarket B sells 5 kg for $\$8.75$. Which is better value? How would you decide?
0/5QUESTS
Your guided path0 of 7 done · keep going
1

Orient and prepare

Capture your first idea, review the formulas and preview the lesson language.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question
01
Recall, your gut answer first
+5 XP warm-up

Supermarket A sells 2 kg of flour for $\$3.80$. Supermarket B sells 5 kg for $\$8.75$. Which is better value? How would you decide? What calculation would you do?

Without calculating make a prediction and explain your reasoning.

auto-saved
02
The formulas you need to own
+5 XP to read

Rate problems hinge on two ideas: expressing everything per one unit (the unitary method), and the distance–speed–time triangle. Get these right and every rate question becomes routine.

A rate compares two quantities with different units, e.g. 60 km/h, $\$4.50$/kg, 15 L/min. The unitary method converts any rate to "per one unit", making different rates directly comparable. For speed: $D = S \times T$, cover the unknown to read the formula.

Cover the unknown in the D-S-T triangle to read off its formula.
$D = S \times T$, cover the unknown to reveal the formula
Rate, a comparison of two units
Examples: 60 km/h, $\$4.50$/kg, 12 L/min. Always write units, they tell you the operation.
Unitary method
Divide total value by number of units to get rate per 1 unit. Then scale up by multiplying.
Fuel consumption
Measured in L/100 km. Litres used = consumption × distance ÷ 100. Lower number = more economical.
03
What you'll master
Know

Key facts

  • A rate compares two different types of quantities
  • The unitary method finds the rate per one unit
  • $D = ST$ for speed problems; rearranges to $S = D/T$ and $T = D/S$
  • Fuel consumption expressed as L/100 km
Understand

Concepts

  • Why expressing rates per one unit makes comparison easy
  • How unit labels guide the calculation method
  • The difference between average speed and instantaneous speed
Can do

Skills

  • Apply the unitary method to find best value
  • Solve speed/distance/time problems
  • Calculate fuel consumption and cost
  • Compare unit rates in practical contexts
04
Key terms
RateA comparison of two quantities with different units, e.g. 60 km/h, $\$4.50$/kg, 12 L/min.
Unitary methodFinding the value of one unit first, then multiplying or dividing to find the required amount.
Average speedTotal distance divided by total time for a journey; does not reflect varying speed during the trip.
Fuel consumptionThe number of litres of fuel used per 100 km travelled; a lower number means more economical.
ScaleA ratio that compares measurements on a map or diagram to real-world measurements.
2

Use rates and the unitary method

Build the core rate method, then check the average-speed warning.

05
Rate per one unit, the unitary method
core concept

The unitary method converts any rate into "per one unit", making different rates directly comparable. Divide the total quantity by the number of units.

For best-value comparisons, find the cost per gram or per unit, then the cheapest rate wins.

$$\text{Rate per 1 unit} = \frac{\text{total value}}{\text{total number of units}}$$
Method. Rate per 1 unit = (total value) ÷ (total number of units). Then scale up or down by multiplying.

Average speed warning: Average speed = total distance ÷ total time. It is NOT the arithmetic mean of two speeds unless the time spent at each speed is equal.

The unitary method finds the rate per one unit: divide the given quantity by its associated number. Then scale up or down by multiplying. Always check units: the unit rate has 'per 1 [unit]' in its description.

Pause, copy the unitary method steps: (1) divide the given quantity by its number to find the rate per one unit; (2) multiply the unit rate by the required number to scale up or down, and note always check units: the unit rate has "per 1 [unit]" in its label into your book.

Did you get this? True or false: to find average speed for a whole journey, you add the two speeds and divide by 2.

Worked examples · 3 in a row, reveal as you go

3

Work through rate examples

Follow best-value, speed and fuel-consumption solutions step by step.

PROBLEM 1 · BEST VALUE, UNITARY METHOD

Store A sells orange juice: 1.25 L for $\$2.80$. Store B sells 2 L for $\$4.20$. Which is better value?

1
$\text{Store A: } \$2.80 \div 1.25 = \$2.24 \text{ per litre}$
Find cost per litre for Store A using the unitary method.
1 of 3
PROBLEM 2 · SPEED–DISTANCE–TIME

A car travels 270 km in 3 hours and 15 minutes. Find the average speed in km/h.

1
$T = 3 + \dfrac{15}{60} = 3.25 \text{ h}$
Convert time to hours: 15 min = 15/60 h = 0.25 h.
1 of 2
PROBLEM 3 · FUEL CONSUMPTION

A car has fuel consumption of 8.5 L/100 km. Petrol costs $\$2.05$ per litre. Find the cost of driving 420 km.

1
$\text{Fuel} = 8.5 \times \dfrac{420}{100} = 8.5 \times 4.2 = 35.7 \text{ L}$
Multiply consumption rate by number of 100 km units.
1 of 2

Quick check: A car travels at 80 km/h for 2.5 hours. What distance does it cover?

4

Avoid common rate errors

Check units, time conversions and average-speed reasoning.

Cover the unknown in the D = S × T triangle to read its formula.

Common errors · the traps that cost marks

Trap 01
Average speed ≠ mean of two speeds
Wrong: Average speed = (speed1 + speed2) ÷ 2. Right: Average speed = total distance ÷ total time. The arithmetic mean only works when equal time is spent at each speed.
Trap 02
Forgetting to convert minutes to hours
If speed is in km/h, time must be in hours. Divide minutes by 60 before substituting. 45 min = 0.75 h, not 45 h.
Trap 03
Comparing prices without finding per-unit rate
A larger pack is not automatically cheaper per unit. Always divide price by quantity to get the rate per gram, litre, or unit before comparing.

Fill the gap: A car uses 9 L/100 km. For a 350 km trip: litres used = 9 × (350 ÷ ) = L.

5

Build fluency with rates

Complete the quick-fire calculations and explain the key misconception.

Quick-fire practice · 12 rate calculations

1

A printer produces 420 pages in 7 minutes. Find the rate in pages per minute.

2

A tap fills a 180 L bath in 12 minutes. Find the flow rate in L/min.

3

Pack A: 400 g of coffee for $\$7.60$. Pack B: 600 g for $\$11.10$. Which is better value?

4

A worker earns $\$336$ for 8 hours. Another earns $\$270$ for 6 hours. Who has the better hourly rate?

5

A cyclist travels at 24 km/h for 2.5 hours. Find the distance.

6

A train travels 480 km at an average speed of 120 km/h. Find the time in hours and minutes.

7

A runner completes a 10 km race in 52 minutes and 30 seconds. Find the average speed in km/h (to 2 d.p.).

8

A car travels 140 km in 1 h 45 min. Find the average speed.

9

A car uses 9.2 L/100 km. How much fuel is needed for a 350 km journey?

10

Using Q9 above, if petrol costs $\$2.15$/L, find the total fuel cost.

11

A tap drips at 0.3 L/min. How many litres are wasted in 24 hours?

12

A pump drains a 2400 L tank in 40 minutes. Find the drainage rate in L/min. How long to drain a 3600 L tank at the same rate?

Odd one out: Three of these are correct statements about rates. Which one is wrong?

6

Show what you can do

Answer the exam-style questions, then compare your working with the model answers.

02
Short answer
ApplyBand 43 marks

SA 4. A family drives from Sydney to Melbourne, a distance of 880 km. They drive at an average speed of 100 km/h for the first 400 km, then stop for 45 minutes, then continue at 90 km/h for the rest. (a) Find the time for the first 400 km. (1 mark) (b) Find the time for the remaining distance. (1 mark) (c) Find the total travel time including the rest stop, in hours and minutes. (1 mark)

auto-saved
ApplyBand 43 marks

SA 5. A supermarket sells three sizes of olive oil: 375 mL for $\$6.45$, 750 mL for $\$11.40$, and 1.5 L for $\$24.00$. (a) Find the cost per 100 mL for each size. (2 marks) (b) State which size offers the best value. (1 mark)

auto-saved
AnalyseBand 54 marks

SA 6. Riley drives a car with fuel consumption of 7.8 L/100 km. Petrol costs $\$2.10$/L. (a) How many litres are needed for a 520 km trip? (1 mark) (b) What is the fuel cost for the trip? (1 mark) (c) Riley's tank holds 60 L and starts full. After the trip, how many litres remain? (1 mark) (d) At the same consumption rate, how far could Riley travel on a full tank? Give your answer to the nearest km. (1 mark)

auto-saved
📖 Comprehensive answers (click to reveal)

Drill 1: $420 \div 7 = 60$ pages/min  ·  2: $180 \div 12 = 15$ L/min  ·  3: A: $7.60 \div 400 = \$0.019$/g; B: $11.10 \div 600 = \$0.0185$/g → Pack B  ·  4: W1: $336 \div 8 = \$42$/h; W2: $270 \div 6 = \$45$/h → Worker 2  ·  5: $D = 24 \times 2.5 = 60$ km  ·  6: $T = 480 \div 120 = 4$ h  ·  7: $T = 52.5/60$ h; $S = 10 \div (52.5/60) \approx 11.43$ km/h  ·  8: $T = 1.75$ h; $S = 140 \div 1.75 = 80$ km/h  ·  9: $9.2 \times 3.5 = 32.2$ L  ·  10: $32.2 \times 2.15 = \$69.23$  ·  11: $0.3 \times 60 \times 24 = 432$ L  ·  12: Rate $= 2400 \div 40 = 60$ L/min; Time $= 3600 \div 60 = 60$ min

SA 4 (3 marks): (a) $T_1 = 400 \div 100 = 4$ h [1]. (b) $T_2 = 480 \div 90 = 5\frac{1}{3}$ h $= 5$ h 20 min [1]. (c) Total $= 4 + 5$ h 20 min $+ 45$ min $= 10$ h 5 min [1].

SA 5 (3 marks): (a) 375 mL: $\$6.45/3.75 = \$1.72$/100 mL; 750 mL: $\$11.40/7.5 = \$1.52$/100 mL; 1.5 L: $\$24.00/15 = \$1.60$/100 mL [2]. (b) 750 mL at $\$1.52$/100 mL is best value [1].

SA 6 (4 marks): (a) $7.8 \times 5.2 = 40.56$ L [1]. (b) $40.56 \times 2.10 = \$85.18$ [1]. (c) $60 - 40.56 = 19.44$ L [1]. (d) $D = 60 \div 7.8 \times 100 \approx 769$ km [1].

7

Retrieve, reflect and finish

Revisit your opening idea, then use the topic challenge and mark the lesson complete.

10
Revisit your thinking

Earlier you predicted which supermarket had better value flour. Let's check:

Supermarket A: $\$3.80$ ÷ 2 = $\$1.90$/kg

Supermarket B: $\$8.75$ ÷ 5 = $\$1.75$/kg

Supermarket B is cheaper per kilogram despite having the higher total price. The unitary method, find the rate per one unit, removes all ambiguity from comparison problems.

auto-saved
01
Take the full module quiz
quiz

A full module quiz covering every lesson in this module, not just this one. Set aside a decent block of time and treat it like a real assessment.

Start the module quiz →

Mark lesson as complete

Tick when you've finished the practice and review.

Ratios and Rates · Year 12 · Maths Standard