Mathematics • Year 7 • Unit 1 • Lesson 14
Rates in the Real World
Use rates and unit rates to plan a road trip, compare supermarket packs, work out paid hourly work, and read sports speeds. Anywhere two different units appear together, a rate is hiding.
1. Word problems
Each problem uses one of the rate ideas from Lesson 14: speed, price per unit, best-value comparison, or scaling a recipe. Show working — a final answer with no working only earns half marks.
1.1 — Road trip. A family drives from Sydney to Newcastle, a distance of 162 km. The trip takes 2 hours.
(a) What is their average speed in km/h?
(b) If they then drive 90 km in the next 1.5 hours, what is the average speed for the second leg?
(c) What is the average speed for the whole trip (total distance ÷ total time)? 3 marks
1.2 — Best deal on milk. The supermarket offers three sizes:
- 1 L for $2.40
- 2 L for $4.20
- 3 L for $6.30
(a) Find the price per litre for each size.
(b) Which size is the best value, and which is the worst? 3 marks
1.3 — Casual job. Mia earns $18.50 per hour at her after-school job.
(a) How much does she earn for a 4-hour shift?
(b) Last week she earned $129.50. How many hours did she work? 2 marks
1.4 — Sprint times. Two athletes complete a 100 m sprint. Athlete A runs it in 12.5 s. Athlete B runs 200 m in 26 s.
(a) What is Athlete A's speed in m/s?
(b) What is Athlete B's speed in m/s?
(c) Who is faster, and by how many m/s? 3 marks
1.5 — Scaling a recipe. A muffin recipe for 12 muffins needs 360 g of flour and 200 mL of milk.
(a) How much flour and milk are needed for 1 muffin (unit rate)?
(b) How much flour and milk are needed for 18 muffins? 3 marks
2. Explain your thinking
This question is about communication. Use full sentences. 4 marks
2.1 Two shops advertise the same brand of cereal. Shop A: 500 g for $4.50. Shop B: 750 g for $6.30. A classmate says "Shop B is cheaper because $6.30 sounds like a lot but you're getting much more cereal." In your own words explain (i) why "more cereal for more money" is not enough to decide which is better value, (ii) calculate the unit price ($/100 g) for each shop, and (iii) say which shop is actually the better value and by how much per 100 g. End with one sentence on why unit rates are the fairest way to compare prices.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Road trip
(a) 162 ÷ 2 = 81 km/h.
(b) 90 ÷ 1.5 = 60 km/h.
(c) Total distance = 162 + 90 = 252 km. Total time = 2 + 1.5 = 3.5 h. Average speed = 252 ÷ 3.5 = 72 km/h. (Note: it's not the simple average of 81 and 60 because they spent different amounts of time at each speed.)
1.2 — Best deal on milk
1 L: $2.40 ÷ 1 = $2.40/L.
2 L: $4.20 ÷ 2 = $2.10/L.
3 L: $6.30 ÷ 3 = $2.10/L.
Best value: 2 L and 3 L are equally cheap at $2.10/L. Worst value: the 1 L at $2.40/L.
1.3 — Casual job
(a) 4 × $18.50 = $74.00.
(b) $129.50 ÷ $18.50 = 7 hours. (Check: 7 × 18.50 = 129.50 ✓.)
1.4 — Sprint times
(a) Athlete A: 100 ÷ 12.5 = 8 m/s.
(b) Athlete B: 200 ÷ 26 = 7.69 m/s (to 2 d.p.) — exact: 7.692… m/s.
(c) Athlete A is faster, by 8 − 7.69 ≈ 0.31 m/s.
1.5 — Scaling a recipe
(a) Per muffin: flour = 360 ÷ 12 = 30 g; milk = 200 ÷ 12 = 16.67 mL (to 2 d.p.).
(b) For 18 muffins: flour = 30 × 18 = 540 g; milk = 16.67 × 18 = 300 mL (exact: 200 × 18/12 = 200 × 1.5 = 300 mL).
2.1 — Explain your thinking (sample response)
"More cereal for more money" is not enough information by itself because both the amount of cereal and the price change at the same time. The classmate is comparing apples and oranges. To compare fairly, we need both shops measured in the same unit, like dollars per 100 g.
Shop A: $4.50 ÷ 500 g = $0.009/g = $0.90 per 100 g.
Shop B: $6.30 ÷ 750 g = $0.0084/g = $0.84 per 100 g.
Shop B is actually the better value, by $0.90 − $0.84 = $0.06 per 100 g cheaper.
Unit rates are the fairest comparison because they reduce both products to the same "per 1 unit" basis — once both are measured per 100 g (or per kg), the cheaper one is unambiguously the better value.
Marking: 1 for explaining that price + quantity change together; 1 each for the two correct unit prices; 1 for the correct conclusion and per-100 g difference.