Real-world linear modelling & finance

Many real-world situations grow or shrink at a constant rate, which makes them a perfect match for a linear model — an equation like y = mx + c, where m is the rate of change and c is the starting amount. This is especially useful for financial contexts: a savings plan with a fixed weekly deposit, a phone plan with a flat fee plus a per-GB charge, or a car's value depreciating by a fixed amount each year. Building a mathematical model means turning a word problem into an equation, then using it to answer questions — like when two plans cost the same, or how much you'll have saved after a certain number of weeks.

Key terms

Mathematical model:

An equation built to represent and solve a real-world problem.

Rate of change:

How much a quantity changes for each unit of another (e.g. per week).

Questions

1. 1. In y = mx + c, "m" represents:

   • The rate of change
   • The starting amount always
   • A fixed unrelated number
   • The final answer

2. 2. In y = mx + c, "c" represents:

   • The starting amount
   • The rate of change always
   • A random variable
   • The final answer

3. 3. A savings plan with a fixed weekly deposit is an example of:

   • A linear model
   • A situation that cannot be modelled
   • A random, unpredictable pattern
   • A shape, not a model

4. 4. For savings = 15w + 50, the starting amount is:

   • $50
   • $15
   • $65
   • $0

5. 5. For savings = 15w + 50, the weekly rate added is:

   • $15
   • $50
   • $65
   • $0

6. 6. Building a mathematical model means:

   • Turning a word problem into an equation
   • Guessing an answer with no working
   • Ignoring the problem entirely
   • Drawing a picture only

7. 7. A car depreciating by a fixed amount each year can be modelled as:

   • A linear relationship
   • An unmodellable situation
   • Only a percentage with no equation
   • A random number each year

8. 8. For savings = 15w + 50, how much is saved after 4 weeks?

   • $110
   • $65
   • $60
   • $100

9. 9. A phone plan costs $25 flat fee plus $2 per GB. Model this as an equation (cost, g = GB used):

   • cost = 2g + 25
   • cost = 25g + 2
   • cost = 27g
   • cost = 2 + 25g

10. 10. Using cost = 2g + 25, what is the cost for 10GB?

    • $45
    • $27
    • $250
    • $20

11. 11. Two savings plans: A = 10w + 100 and B = 20w + 40. At w = 6, which has more money?

    • Plan B ($160 vs $160 — actually equal)
    • Plan A always
    • Plan B always regardless of w
    • Neither plan can be compared

12. 12. Why is setting two linear models equal to each other useful (e.g. 15w+50 = 20w+20)?

    • It finds the point where both situations produce the same result
    • It never produces a useful answer
    • Equations can never be set equal to each other
    • This only works for shapes, not finance

13. 13. A gym membership costs $80 to join plus $10 per week. Model the total cost after w weeks:

    • cost = 10w + 80
    • cost = 80w + 10
    • cost = 90w
    • cost = 10 + 80

14. 14. Using cost = 10w + 80, how many weeks until the total cost reaches $180?

    • 10 weeks
    • 8 weeks
    • 18 weeks
    • 26 weeks

15. 15. Solve: 15w + 50 = 20w + 20 for w:

    • w = 6
    • w = 3
    • w = 10
    • w = 30

16. 16. A car worth $30,000 depreciates by $2,500 per year. Model its value after y years, and find its value after 4 years:

    • value = 30000 − 2500y; after 4 years = $20,000
    • value = 2500y − 30000; after 4 years = $10,000
    • value = 30000 + 2500y; after 4 years = $40,000
    • value = 2500 − 30000y; negative, impossible

17. 17. Two phone plans: A costs $20 + $1/GB, B costs $10 + $2/GB. At what usage do they cost the same?

    • 10GB (both cost $30)
    • 5GB
    • 20GB
    • They never cost the same

18. 18. Why might a business use a linear model to decide when a new machine "pays for itself" through savings?

    • Setting the cost of the machine equal to accumulated savings over time reveals the break-even point
    • Linear models cannot be used for business decisions
    • Break-even points cannot be calculated mathematically
    • This decision requires no maths at all

19. 19. A savings model predicts $50 after 0 weeks and grows by $15/week. After how many whole weeks will savings first exceed $200?

    • 11 weeks (15×10+50=200, so week 11 exceeds it)
    • 10 weeks exactly
    • 13 weeks
    • 15 weeks

20. 20. Why is it important to check whether a real-world situation is genuinely linear before applying a linear model?

    • Situations that don't change at a constant rate would be inaccurately represented by a straight-line model
    • All real-world situations are always perfectly linear
    • Linear models work correctly for every possible situation
    • Checking is unnecessary since the model will adjust automatically

21. 21. Understanding real-world linear modelling mainly helps you to:

    • Represent and solve practical problems (like finance) using equations
    • Avoid ever applying maths to real situations
    • Assume every situation grows randomly with no pattern
    • Ignore rates of change in real contexts