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Lesson 12 ~25 min Unit 1 · Financial Maths +85 XP

Introduction to Rates

Speed, fuel use, pay-per-hour — comparing two different units to describe how fast or how much.

Today's hook: A car uses 8.5 L per 100 km. A bus uses 25 L per 100 km. Which is more fuel-efficient per person if the bus carries 40 passengers?
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Think First
warm-up

A car uses 8.5 L per 100 km. A bus uses 25 L per 100 km. Which is more fuel-efficient per person if the bus carries 40 passengers? Jot down your first reaction — then we'll see who's right.

Record your answer in your workbook.
1
The Big Idea
+5 XP

A rate compares two quantities with DIFFERENT units, like km per hour or dollars per kilogram. A unit rate is a rate where the second amount is "1" — like $\$5$/kg.

A rate is a comparison of two different units. $60$ km/h means $60$ km PER hour. $\$8$/kg means $\$8$ per kilogram. The "/" is read "per". A unit rate has just one unit in the denominator — like "$60$ km per 1 hour" — which makes comparisons easy.

Rate $= \dfrac{\text{quantity}_1}{\text{quantity}_2}$ with units attached
Different units
Rates compare TWO different things: km with hours, $\$$ with kg.
Read the "/"
"/" means "per". $\$2/\text{L}$ = "two dollars per litre".
Unit rate = 1
A rate where the denominator unit is 1 (per kg, per hour).
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What You'll Master
objectives

Know

  • A rate compares two quantities with different units
  • Common examples: speed (km/h), price ($/kg), wages ($/h)
  • Unit rate has 1 in the denominator
  • "Per" means divide

Understand

  • Why rates use different units (otherwise they're ratios)
  • How to convert a rate to a unit rate
  • Why unit rates make comparison fair

Can Do

  • Recognise rates in real-world contexts
  • Convert any rate to a unit rate
  • Use unit rates to compare options
3
Words You Need
vocabulary
RateA comparison of two quantities with different units.
Unit rateA rate where the denominator quantity is 1 (e.g., $\$5$ per kg).
SpeedA rate of distance per unit time, often km/h or m/s.
Fuel efficiencyHow much fuel per distance: L/100 km.
PerMathematical word for "divided by".
DensityMass per unit volume — another rate (kg/m$^3$).
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Spot the Trap
heads-up

Wrong: "A ratio of 60 km : 1 h is the same as a rate of $60$ km/h." — Actually different ideas: a ratio compares like with like, a rate compares unlike units.

Right: Ratios are unitless or same-unit comparisons; rates have different units attached.

Wrong: "L/100 km is just a number." — NO. The units L per 100 km are essential to its meaning.

Right: Always include units when writing a rate: $8.5$ L/100 km, not just $8.5$.

5
Common Real-World Rates
+5 XP

Rates appear everywhere. Most of the numbers you see in shops, on cars, and in sport are rates.

Speed is a rate (km/h). Price per kg in a supermarket is a rate ($/kg). A wage like $\$22$/hour is a rate. Even your heart rate (beats/minute) is a rate. Once you spot them, you see them everywhere — and you can use them to compare options fairly.

Speed $= \dfrac{\text{distance}}{\text{time}}$,   Price $= \dfrac{\text{cost}}{\text{weight}}$
Speed
$60$ km/h is distance per time.
Cost
$\$3/$L is price per unit volume.
Heart rate
Beats per minute — a rate!
6
Unit Rates
+5 XP

A unit rate has "1" in the denominator. Almost all useful rates are unit rates — they make comparisons trivial.

Suppose 3 pens cost $\$4.50$. The unit rate is $\$4.50 \div 3 = \$1.50$ per pen. Now you can quickly answer ANY question: 7 pens? $7 \times \$1.50 = \$10.50$. The unit rate scales up or down trivially.

Unit Rate $= \dfrac{\text{cost}}{\text{units}}$
Divide to find unit
Always divide the total by the quantity.
Scale to find any
Multiply unit rate by however many you need.
Lower unit rate = cheaper
$\$3$/kg beats $\$4$/kg.

Watch Me Solve It · 3 examples

Watch Me Solve It · Fuel efficiency
+15 XP per step
Q1
PROBLEM
A car uses 8.5 L per 100 km. A bus uses 25 L per 100 km but carries 40 passengers. Which is more efficient per person?
  1. 1
    Car: L per person per 100 km
    Car carries 1 driver, uses 8.5 L for 100 km
    $8.5$ L per person per 100 km
  2. 2
    Bus: L per person per 100 km
    $25 \div 40 = 0.625$ L per person per 100 km
    Bus is more efficient per passenger!
  3. 3
    Compare
    $0.625 < 8.5$
    Bus wins by a factor of $\approx 13.6$.
AnswerBus — $\$0.625$ L/person/100 km
Watch Me Solve It · Speed rate
+15 XP per step
Q2
PROBLEM
A bike covers 18 km in 45 minutes. What is its speed in km/h?
  1. 1
    Convert time to hours
    $45$ min $= \tfrac{45}{60} = 0.75$ hours
    Same units for hours.
  2. 2
    Speed = distance/time
    $\tfrac{18}{0.75}$
    Divide.
  3. 3
    Compute
    $18 \div 0.75 = 24$
    $24$ km/h.
Answer$24$ km/h
Watch Me Solve It · Unit price
+15 XP per step
Q3
PROBLEM
A 2.5 kg bag of rice costs $\$8.75$. Find the price per kg.
  1. 1
    Set up
    Price per kg = $\$8.75 \div 2.5$ kg
    Divide total cost by total weight.
  2. 2
    Compute
    $8.75 \div 2.5 = \$3.50$
    That's per kg.
  3. 3
    Use it
    To buy 7 kg: $7 \times 3.50 = \$24.50$
    Unit rate makes scaling trivial.
Answer$\$3.50/$kg

Common Pitfalls

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Common Pitfalls
heads-up
Dropping the units
Calling a rate just “$60$” instead of “$60$ km/h”.
Fix: Always include the units.
Confusing rate with ratio
Treating $60$ km/h as a ratio.
Fix: Rates compare different units; ratios compare the same.
Reversing the rate
Saying $\$1$ per $5$ pens when you mean $\$5$ per pen.
Fix: Read the question carefully — what's per what?
Copy Into Your Books

Definition

  • Rate compares two DIFFERENT units
  • Unit rate has 1 in denominator
  • "/" means "per"

Common Rates

  • Speed: km/h
  • Price: $/kg, $/L
  • Wages: $/hour
  • Heart rate: beats/min

Finding Unit Rate

  • Divide total quantity by units
  • $\$15$ for 3 kg = $\$5/$kg
  • Use to scale up or down

Comparing Rates

  • Convert each to a unit rate
  • Compare directly
  • Lower price/kg = cheaper

How are you completing this lesson?

Brain Trainer · 4 problems

D
Brain Trainer · Introduction to Rates
4 problems

Four drill problems to sharpen your skills. Work each, then reveal the answer.

  1. 1 $\$60$ for 4 hours of work. Hourly rate?

    $60 \div 4 = \$15$/h.$\$15$/h

  2. 2 $3.5$ kg of cheese at $\$28$. Unit rate?

    $28 \div 3.5 = \$8/$kg.$\$8/$kg

  3. 3 Travel 240 km in 3 hours. Speed?

    $240 \div 3 = 80$ km/h.$80$ km/h

  4. 4 Heart beats 75 times in 30 seconds. Rate per minute?

    $75 \times 2 = 150$ bpm.$150$ bpm
Complete in your workbook.

Quick Check · 5 questions

1
Which of these is a rate?
+10 XP
2
A car drives 150 km in 2.5 hours. Speed:
+10 XP
3
5 kg of apples for $\$18.50$. Unit price:
+10 XP
4
Lucia earns $\$84$ in 6 hours. Her hourly rate is:
+10 XP
5
A car uses $7.2$ L per 100 km. To travel 350 km, fuel needed is:
+10 XP

Show Your Working · 3 questions

Show Your Working
9 marks total
Apply Medium 3 MARKS

Q6. Calculate the unit rate for each: (a) A 3.2 kg watermelon costs $\$11.20$. (b) A truck travels 240 km on 30 L of fuel. (c) Mia's heart beats 245 times in 3.5 minutes.

Answer in your workbook.
Understand Easy 2 MARKS

Q7. A swimmer trains for 90 minutes and swims 4.5 km. What is the average speed in km/h?

Answer in your workbook.
Reason Hard 4 MARKS

Q8. Two delivery vans are compared. Van A travels 150 km on 9.6 L of petrol. Van B travels 200 km on 12 L of petrol. (a) Compute the unit rate (L/100 km) for each. (b) Which is more fuel-efficient? (c) Petrol costs $\$1.85/$L. Find the petrol cost for each van to travel 500 km, and the difference.

Answer in your workbook.
Comprehensive Answers

Quick Check

1. C — rate has different units.

2. B — $60$ km/h.

3. C — $\$3.70/$kg.

4. C — $\$14$/h.

5. A — $25.2$ L.

Show Your Working Model Answers

Q6 (3 marks): (a) $11.20 \div 3.2 = \$3.50$/kg [1]. (b) $240 \div 30 = 8$ km/L [1]. (c) $245 \div 3.5 = 70$ beats/min [1].

Q7 (2 marks): $90$ min $= 1.5$ h [1]. Speed $= 4.5 \div 1.5 = 3$ km/h [1].

Q8 (4 marks): (a) Van A: $\tfrac{9.6}{150} \times 100 = 6.4$ L/100 km. Van B: $\tfrac{12}{200} \times 100 = 6$ L/100 km [2]. (b) Van B more efficient [1]. (c) Van A: $\tfrac{6.4}{100} \times 500 \times 1.85 = \$59.20$. Van B: $\tfrac{6}{100} \times 500 \times 1.85 = \$55.50$. Difference $= \$3.70$ [1].

Stretch Challenge · +25 XP, +10 coins

The Better Bargain

A grocery store sells the same brand of olive oil in three sizes: $500$ mL for $\$8.40$, $1$ L for $\$15.80$, $1.5$ L for $\$22.50$. (a) Find the unit rate ($\$$/L) for each. (b) Which size is the cheapest per litre? (c) If you only need $750$ mL, what is the cheapest way to buy it?

Reveal solution

(a) $500$ mL: $\$16.80$/L. $1$ L: $\$15.80$/L. $1.5$ L: $\$15.00$/L. (b) The $1.5$ L is cheapest per litre. (c) For $750$ mL, either buy one $500$ mL plus another or one $1$ L. Two $500$ mL = $\$16.80$. One $1$ L = $\$15.80$. Cheapest is one $1$ L bottle ($\$15.80$), leaving $250$ mL extra.

R
Quick Review

Rate

Comparison of two DIFFERENT units

"/" = per

60 km/h = 60 km per 1 hour

Unit rate

Denominator = 1

Speed

Distance ÷ time

$/kg

Price ÷ weight

Compare fairly

Use unit rates

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