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.