Mathematics • Year 7 • Unit 1 • Lesson 14
Rates and Unit Rates
Build the basics of rates: comparing quantities with different units (km/h, $/kg, beats/min). Find a unit rate by dividing total by quantity. Use unit rates to compare best value.
1. I do — fully worked example
Watch a worked "compare best value" example. Two unit prices, decide which is cheaper per unit.
Problem. Which is better value: Pack A — 750 mL for $4.50, or Pack B — 1 L for $5.80?
Step 1 — Identify the rate and the unit you want.
We want $ per L (or $ per mL). Pick one unit and stick to it.
Reason: you can't compare $/L against $/mL directly — both packs must use the same unit.
Step 2 — Convert both quantities to the same unit. 1 L = 1000 mL.
Pack A: 750 mL for $4.50. Pack B: 1000 mL for $5.80.
Reason: now both are measured in mL, so the unit prices will be directly comparable.
Step 3 — Find the unit price ($ per 1 mL) for each.
Pack A: $4.50 ÷ 750 = $0.006/mL = $6.00/L.
Pack B: $5.80 ÷ 1000 = $0.0058/mL = $5.80/L.
Reason: the lower the price per unit, the better the value.
Step 4 — Compare.
$5.80/L < $6.00/L, so Pack B is cheaper per litre.
Answer: Pack B (1 L for $5.80) is better value, at $5.80 per litre vs $6.00 per litre.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. A car travels 240 km in 3 hours. What is its average speed in km/h?
Step 1 — Identify the rate. Speed is measured in km per ______.
Step 2 — Write the formula:
Speed = distance ÷ ______ = _______ ÷ _______.
Step 3 — Calculate:
240 ÷ 3 = _______ km/h.
Step 4 — Check by multiplying back:
_______ km/h × 3 h = _______ km ✓ (matches the original distance)
3. You do — independent practice
Show working under each problem. The first four are foundation, the middle two are standard, and the last two are extension.
Foundation — single step
3.1 A car travels 315 km in 3.5 hours. What is its speed in km/h? 1 mark
3.2 5 kg of apples cost $12.50. What is the price per kg? 1 mark
3.3 A printer prints 60 pages in 4 minutes. What is its rate in pages per minute? 1 mark
3.4 A heart beats 90 times in 1 minute. Express this as a unit rate in "beats per second". 1 mark
Standard — combine two ideas
3.5 A train travels 420 km in 3 hours 30 minutes. Find its average speed in km/h. (Hint: convert 3 h 30 min to a decimal first.) 2 marks
3.6 Which is better value: 3 kg of rice for $8.40, or 4 kg for $11.20? Find the unit price for each and compare. 2 marks
Extension — push your thinking
3.7 Which is better value: 600 g for $7.20, or 800 g for $9.60? Show the price per kg for each. 3 marks
3.8 A cyclist rides at 24 km/h. How far does the cyclist travel in (a) 1 hour, (b) 30 minutes, (c) 15 minutes? Show each calculation. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (240 km in 3 h)
Step 1: km per hour.
Step 2: Speed = distance ÷ time = 240 ÷ 3.
Step 3: 240 ÷ 3 = 80 km/h.
Step 4: 80 km/h × 3 h = 240 km ✓.
3.1 — Car speed
Speed = 315 ÷ 3.5. Shift both 1 place: 3150 ÷ 35 = 90 km/h.
3.2 — Apples price per kg
$12.50 ÷ 5 = $2.50/kg.
3.3 — Printer rate
60 pages ÷ 4 min = 15 pages/min.
3.4 — Heart rate per second
90 beats in 60 sec → 90 ÷ 60 = 1.5 beats/second. (Or: 3 beats every 2 seconds.)
3.5 — Train speed
3 h 30 min = 3.5 h. Speed = 420 ÷ 3.5. Shift dots: 4200 ÷ 35 = 120 km/h. Check: 120 × 3.5 = 420 ✓.
3.6 — 3 kg for $8.40 vs 4 kg for $11.20
Pack A: $8.40 ÷ 3 = $2.80/kg.
Pack B: $11.20 ÷ 4 = $2.80/kg.
Same value — both cost $2.80 per kg.
3.7 — 600 g for $7.20 vs 800 g for $9.60
Pack A: 600 g = 0.6 kg. $7.20 ÷ 0.6 = $12.00/kg.
Pack B: 800 g = 0.8 kg. $9.60 ÷ 0.8 = $12.00/kg.
Same value — both work out to $12.00 per kg.
3.8 — Cyclist at 24 km/h
(a) 1 hour: 24 km.
(b) 30 min = 1/2 hour: 24 × 0.5 = 12 km.
(c) 15 min = 1/4 hour: 24 × 0.25 = 6 km.