078i - Formulating Financial Problems

Lesson 78 — Mathematical Modelling: Formulating Financial Problems

Strand: Number | Descriptor: AC9M7N09 | Duration: 45 minutes

Learning Intentions

• To formulate practical financial situations as mathematical problems.
• To choose representations and identify assumptions in money contexts.

Success Criteria

I can:

1. Extract the financially relevant quantities from a situation.
2. Represent income, costs and profit with expressions or calculations.
3. State the assumptions a financial model makes.
4. Distinguish fixed costs from per-item costs.

Warmup

(6 minutes — money vocabulary sort, pairs)

Sort these under income, fixed cost or per-item cost for a school cake stall: ingredients per cake; stall hire; sale price per cake; a donated banner; electricity for the oven (paid per batch); the $40 float.

Answers: Income — sale price per cake. Fixed — stall hire (and the float is not a cost at all — it returns; the banner is free). Per-item — ingredients; electricity is per batch, an in-between case worth discussing.

The structure to name (familiar from Lesson 23):

Activities

Activity 1 — Explicit Instruction: Representing a Financial Situation (14 min)

I do — the car wash. “A team plans a car wash. Detergent and sponges will cost about 35$0.80$8$ per car.”

Formulate aloud:

• Variable: let be the number of cars washed.
• Income: .
• Costs: fixed , per-car .
• Profit:

• Assumptions: every customer pays 8$ (no discounts), supplies estimate holds, labour is free (volunteers).

Interrogate the model with questions before solving anything:

1. What does the mean? (Profit contributed by each car — the margin.)
2. What does mean? (A loss — too few cars.)
3. Roughly how many cars to break even? (About .)

We do — the ticketed movie night. Hall hire 120$60$2.50$7$.

Break-even estimate: attendees exactly.

Activity 2 — Formulation Practice (14 min)

Pairs. For each, produce: variable definition, profit expression, one assumption, and an estimated break-even. No full solving yet — that is Lesson 79.

Scenario 1 — The plant sale. Seedlings cost 1.40$0.60$25$5$ per potted plant.

Scenario 2 — The zine. Printing costs 0.90$48$4$.

Scenario 3 — The subscription choice. A streaming service costs 12$115$ per year paid once. (Not a profit model — a comparison model. What is the variable? What question does the model answer?)

Socratic scaffolding for Scenario 3:

Prompt	Purpose
What varies here?	The number of months you keep the service.
Write both costs as expressions.	Monthly: . Annual: (for ).
What question does comparing them answer?	From which month is the annual plan cheaper?
Estimate before solving.	— so from months.
What assumptions matter?	You keep the service the whole time; prices don’t change; no promotional discounts.
Looking back	Not every financial model is about profit — many are comparisons between options.

Reference formulations:

1. ; break-even plants.
2. ; break-even copies.
3. As scaffolded — annual wins from month .

Activity 3 — Inquiry: what the Model Leaves out (11 min)

Pairs, then whole class.

Take the car wash model .

1. List things the model ignores that could matter on the day.
2. Sort your list: which would change the numbers, and which would change the structure of the model?
3. Pick one structural omission and write the improved expression.

Discussion targets:

• Number-changers: water costing more than 0.80n8$).
• Structure-changers: a second pricing tier (e.g. 12P = 8n + 12m - 0.8(n+m) - 35+dn \le 40$.
• The model is a choice about what to include. More detail is not automatically better — each addition must earn its complexity.

Checks for Understanding

(5 minutes — exit ticket)

1. A stall has fixed costs 52$3.50$ profit per item before fixed costs. Write the profit expression.
2. For : (a) what does the represent? (b) what does mean? (c) estimate the break-even.
3. A gym costs 15$150$/year. Write both cost expressions and state the comparison question.
4. Reasoning. Give one assumption behind almost every school-fundraiser model, and say when it fails.
5. Reasoning. Why is “add more detail” not always good advice for a model?

Answers: 1. ; 2. (a) profit per item (margin) (b) break-even — income exactly covers costs (c) items; 3. versus ; “from how many months is annual cheaper?” ( — from the th month they tie, cheaper beyond); 4. E.g. “everything made will sell” — fails in bad weather or poor placement; 5. Extra terms add data demands and error sources; a model should be as simple as its purpose allows.

Common Misconceptions

Misconception	How to pre-empt it
Mixing fixed and per-item costs into one number.	The warmup sort; the structure equation stays displayed.
Forgetting to subtract per-item costs from the price.	Name the margin explicitly: price minus unit cost.
Treating the float as a cost.	Discussed in the warmup — it returns.
Believing every financial model is a profit model.	Scenario 3 is deliberately a comparison model.
Counting labour as free without stating it.	It is an assumption; volunteers make it true, wages make it false.
Adding detail indiscriminately.	The inquiry’s “earn its complexity” principle.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A stall’s profit is . What is the profit from selling items?

Answer

40$.

E2 (AMC Junior style). Two phone plans: 25$0.10$18$0.24$ per call. Write both monthly cost expressions and estimate the number of calls at which they tie.

Answer

and ; tie when , i.e. calls.

E3 (Challenge). A lemonade stand sells cups at 2$0.45$0.15$1260$ cups.

Answer

; at : 72$.

E4 (Challenge). A cinema offers a 2520%$16$ ticket. Write the total-cost expressions with and without membership, and find how many tickets make membership worthwhile.

Answer

Without: . With: . Membership wins when , i.e. , — from the 8th ticket.

Homework

1. For a fudge stall: ingredients 0.75$0.25$30$3$ per piece. (a) Define the variable. (b) Write the profit expression. (c) State two assumptions. (d) Estimate the break-even.
2. For each expression, say what the two numbers represent: (a) (b) .
3. A bus company charges schools either 400$6.50$ per student. (a) Write both cost expressions. (b) For what group sizes is each option cheaper? Estimate, then check exactly.
4. Sort into fixed and per-item for a T-shirt fundraiser: screen-printing setup 80$7$1.20$25$ total.
5. Improve this model and explain your change: “Sausage sizzle: ”, where is sausages sold at 3$.
6. Reasoning. Explain the difference between a profit model and a comparison model, with one example of each.
7. Challenge. A market stall pays either 40$1015%T$, and find the takings level at which they are equal.

Answers: Q1 — (b) (d) pieces. Q2 — (a) 2.60$42$45$0.084006.5ss = 61.56162P = 3n - c n - F4010 + 0.15T0.15T = 30T = $200$200$.