Mathematics • Year 8 • Unit 1 • Lesson 12
Rates in the Real World
Use rates where they actually show up: fuel efficiency for two delivery vans, hourly wages, swim training pace, supermarket pricing per 100 g. Then explain why a rate without units is meaningless.
1. Word problems
Each problem uses the idea that a rate compares two different units. Show your working — a single final answer with no working only earns half marks.
1.1 — Delivery vans. 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 van.
(b) Which van is more fuel-efficient?
(c) Petrol costs $\$1.85$/L. Find the petrol cost for EACH van to travel 500 km, and the difference. 4 marks
1.2 — Pool training. A swimmer trains for 90 minutes and swims 4.5 km.
(a) What is the average speed in km/h?
(b) At the same pace, how far would the swimmer travel in 2 hours?
(c) How long would 6 km take? 3 marks
1.3 — Supermarket shelf. A 3.2 kg watermelon is priced at $\$11.20$. The shelf label shows the unit rate “per kg”.
(a) What unit rate appears on the shelf label?
(b) How much would 5 kg of the same watermelon cost?
(c) If you only have $\$5$, the biggest piece you could buy weighs how many grams (to the nearest 10 g)? 3 marks
1.4 — Hourly pay. Mia is offered two part-time jobs. Job A pays $\$60$ for 4 hours of work. Job B pays $\$135$ for 9 hours of work.
(a) Find the hourly rate for each job.
(b) Which job has the higher pay rate?
(c) Mia is offered a 6-hour shift. How much would she earn at each job? 3 marks
1.5 — Car vs bus per person. A car uses 8.5 L per 100 km and carries 1 driver. A bus uses 25 L per 100 km but carries 40 passengers.
(a) Find the litres per person per 100 km for the car.
(b) Find the litres per person per 100 km for the bus.
(c) About how many times more fuel-efficient is the bus PER PERSON? 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate writes “My family's car does 6” and shows you a number with no units. In your own words, explain (i) why “6” on its own is meaningless for fuel use, (ii) what the two MOST common Year 8 rates for cars are (with units), (iii) which one “6” is more likely to mean, and (iv) what “6” in that unit tells you about how the car compares with one that does “10” in the same unit. Use the phrase “a rate must have units” in your answer.
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What I'll revisit before next class:
1.1 — Delivery vans
(a) Van A: $(9.6 / 150) \times 100 = \textbf{6.4 L/100 km}$. Van B: $(12 / 200) \times 100 = \textbf{6 L/100 km}$.
(b) Van B is more fuel-efficient (uses fewer L per 100 km).
(c) For 500 km: Van A uses $5 \times 6.4 = 32$ L, costing $32 \times 1.85 = \textbf{\$59.20}$. Van B uses $5 \times 6 = 30$ L, costing $30 \times 1.85 = \textbf{\$55.50}$. Difference = $\textbf{\$3.70}$.
1.2 — Pool training
(a) 90 min = 1.5 h. Speed = $4.5 \div 1.5 = \textbf{3 km/h}$.
(b) In 2 h: $2 \times 3 = \textbf{6 km}$.
(c) For 6 km: $6 \div 3 = \textbf{2 hours}$.
1.3 — Supermarket shelf
(a) $\$11.20 \div 3.2 = \textbf{\$3.50/kg}$.
(b) 5 kg costs $5 \times 3.50 = \textbf{\$17.50}$.
(c) $5 \div 3.50 \approx 1.428$ kg, which rounds DOWN to about 1.42 kg or 1420 g (you can't go over the $5).
1.4 — Hourly pay
(a) Job A: $\$60 \div 4 = \textbf{\$15/h}$. Job B: $\$135 \div 9 = \textbf{\$15/h}$.
(b) Same hourly rate.
(c) 6-hour shift at either: $6 \times 15 = \textbf{\$90}$.
1.5 — Car vs bus per person
(a) Car: 8.5 L per person per 100 km (only 1 person).
(b) Bus: $25 \div 40 = \textbf{0.625 L}$ per person per 100 km.
(c) $8.5 \div 0.625 = \textbf{13.6 times}$ more efficient per person on the bus.
2.1 — Explain your thinking (sample response)
“6” on its own is meaningless for fuel use because a rate must have units — the same number can describe very different cars in different unit systems. The two most common rates for cars are km per litre (kilometres on each litre of fuel, where bigger is better) and L per 100 km (litres used to drive 100 km, where smaller is better). “6” is most likely to mean 6 L per 100 km, because most modern small cars use somewhere between 5 and 10 L per 100 km, while “6 km per litre” would be a very thirsty car. Read as 6 L/100 km, the car is more efficient than one rated 10 L/100 km — the second car uses 10 − 6 = 4 extra litres for every 100 km, costing roughly $\$7$ more in petrol per 100 km at $\$1.85$/L.
Marking: 1 mark for stating “a rate must have units”; 1 mark for naming both common car rates with units; 1 mark for identifying L/100 km as the more likely meaning of “6”; 1 mark for a sensible comparison with the “10” car.