Mathematics • Year 8 • Unit 2 • Lesson 15
Linear Models in the Real World
Model and interpret real situations: hire vans, ride-share fares, currency conversion, school fundraisers, mobile data caps. Build the equation, use it, and explain m and c in plain English.
1. Word problems
For each scenario: identify m and c, write the equation, then answer the question with units.
1.1 — Hire van. A removalist charges $80 to hire the van plus $1.20 per kilometre driven.
(a) Write a cost equation C = mx + c for x km driven.
(b) What is the total cost for a 50 km move?
(c) Explain in one sentence what m and c each mean. 3 marks
1.2 — Ride-share fare. A ride-share trip has a $3 booking fee plus $2 per km. Maya's bill came to $19.
(a) Write the cost equation.
(b) Use it to find how many km she travelled. 3 marks
1.3 — Currency conversion. A graph converts AUD to USD. It passes through (0, 0) and (100 AUD, 65 USD).
(a) Find m (the exchange rate). Write the equation.
(b) Use it to convert 250 AUD to USD. 3 marks
1.4 — School fundraiser. A bake sale starts with $30 cash float and earns $4 per cake sold.
(a) Write F = mn + c for the amount of money in the cash box after n cakes are sold.
(b) How much is in the box after 25 cakes?
(c) How many cakes need to be sold to reach $150? 3 marks
1.5 — Mobile data cap. Talia's plan gives her 10 GB of data per month. She uses 0.4 GB per day for streaming.
(a) Write D = mt + c where D is remaining data after t days.
(b) State and interpret the value of m (why is it negative?).
(c) How many days until she runs out? 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate models a parking lot that charges $5 entry plus $3 per hour as C = 5h + 3, where h is hours parked. In your own words explain (i) which parameter (m or c) they have mixed up, (ii) write the correct equation, and (iii) compare both equations at h = 0 to show why their version is wrong. Use the phrase "the gradient is the rate per hour" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Hire van
(a) C = 1.20x + 80.
(b) C = 1.20 × 50 + 80 = 60 + 80 = $140.
(c) m = 1.20 is the cost per kilometre (gradient — how fast cost grows with each km). c = 80 is the fixed hire fee paid even if you drive 0 km.
1.2 — Ride-share fare
(a) C = 2d + 3.
(b) 19 = 2d + 3 → 16 = 2d → d = 8 → 8 km.
1.3 — Currency conversion
(a) m = 65/100 = 0.65. Equation: y = 0.65x (USD = 0.65 × AUD).
(b) y = 0.65 × 250 = 162.50 USD.
1.4 — School fundraiser
(a) F = 4n + 30.
(b) F = 4(25) + 30 = 100 + 30 = $130.
(c) 150 = 4n + 30 → 120 = 4n → n = 30 → 30 cakes.
1.5 — Mobile data cap
(a) D = −0.4t + 10.
(b) m = −0.4 GB/day. It's negative because her remaining data DECREASES over time as she uses it up.
(c) 0 = −0.4t + 10 → 0.4t = 10 → t = 25 → 25 days until she runs out.
2.1 — Explain your thinking (sample response)
The classmate has swapped m and c. They wrote C = 5h + 3, treating $5 (the entry fee) as the gradient and $3 (the per-hour rate) as the y-intercept. But the gradient is the rate per hour, so it should be 3, and the fixed entry fee should be c = 5. The correct equation is C = 3h + 5. To see the error, compare both at h = 0: the classmate's version gives C = $3 (the rate, which is meaningless when you've parked for 0 hours), while the correct version gives C = $5 (exactly the entry fee — which matches reality).
Marking: 1 mark for spotting m and c were swapped; 1 mark for the correct equation C = 3h + 5; 1 mark for the h = 0 comparison; 1 mark for a clear full-sentence answer that uses "the gradient is the rate per hour".