087i - Percentage Modelling Consolidation and Check

Lesson 87 — Consolidation and Check: Modelling with Rational Numbers and Percentages

Strand: Number | Descriptor: AC9M8N05 | Duration: 45 minutes

Learning Intentions

• To run a complete mathematical modelling cycle — formulate, solve, interpret, report/review — on an unfamiliar real-world problem involving rational numbers and percentages.
• To consolidate efficient calculation strategies and apply them confidently to new financial and community contexts.

Success Criteria

I can:

1. Formulate a real-world scenario involving percentages as a precise mathematical statement.
2. Choose and apply an efficient calculation strategy, using a digital tool where appropriate.
3. Interpret a numerical result in a complete, correctly rounded sentence tied to the original context.
4. Review whether a model’s assumptions are reasonable, and explain how the result would change if they didn’t hold.
5. Work backwards through the modelling cycle to find an unknown starting value that meets a target.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

For each statement, identify which stage of the modelling cycle it belongs to: Formulate, Solve, Interpret, or Review.

1. “The cost is — let me work that out.”
2. “So after the donation, the Year 8 fund receives $540.”
3. “Let = the number of tickets sold. Revenue , and cost .”
4. “This all assumed exactly 90 people attend — what actually happens if only 60 turn up?”

Answers: 1. Solve; 2. Interpret; 3. Formulate; 4. Review.

Discussion: Lessons 83–86 built each stage of this cycle separately. Today we run the full loop, start to finish, on scenarios you haven’t seen before.

Activities

Activity 1 — Explicit Instruction: Running the Full Cycle (12 min)

I do — Trivia Night. “The Year 8 committee is running a trivia night. Venue hire costs $180 flat. Catering costs $3.80 per person. Tickets are $14, but the first 30 sold get a 15% early-bird discount. 90 people are expected to attend (30 of them early-bird). The committee donates 20% of the profit to the school library fund, and keeps the rest for the Year 8 fund. How much does the Year 8 fund receive?”

Formulate: Whole = total cost. Revenue = (30 early-bird tickets × discounted price) + (60 full-price tickets). Profit = revenue − cost. Year 8 fund = profit × 0.80.

Solve:

Interpret: “The trivia night raises $675 profit. After donating $135 (20%) to the library fund, the Year 8 fund receives $540.”

Review: The model assumes exactly 90 people attend and that catering costs exactly $3.80/person no matter how many are fed. In reality, attendance is uncertain, and a caterer might offer a bulk discount above a certain number of guests — the model is a reasonable estimate, not a guarantee.

We do — Bake Sale. “Ingredients cost $45 flat plus $0.60 per item baked. Items sell for $2.50 each. Teachers buying 5 or more get a 10% discount — 20 items are sold this way, and 100 are sold at full price. What is the profit?”

Interpret together: “The bake sale raises $178 profit.”

You do — Car Wash. “Cleaning supplies cost $60 flat. Water costs $0.50 per car. Cars are charged $5 each. 45 cars are washed. 25% of the profit is donated to charity. Formulate, then solve for the profit and the amount kept after the donation.”

Activity 2 — Applied Inquiry Task: the Fun Run, and Reviewing the Model (12 min)

Pairs, then whole-class share.

The Year 8 team is organising a sponsored fun run. Runners pay an $8 entry fee. Costs are $220 for marshalling and safety gear (flat), plus $1.20 per runner for a finisher’s ribbon. A local sports store will give the team a store credit worth 50% of the profit, to spend on new equipment. Attendance is uncertain — planning estimates range from 60 runners (if it rains) to 150 runners (if it’s fine).

Formulate a plan to find the store credit at both ends of this range, solve it, interpret the results, and review whether a model using a fixed per-runner ribbon cost is realistic — especially at the higher end of the range.

Solve (60 runners):

Solve (150 runners):

Interpret: “Depending on the weather, the store credit could be anywhere from $94 (poor turnout) to $400 (strong turnout) — a wide range.”

Review — questions for pairs to discuss and record an answer to:

• Is a fixed $1.20/runner ribbon cost realistic at 150 runners? (A supplier may offer a bulk discount at higher volumes, which this model ignores — meaning the true profit at 150 runners could be even higher than $800.)
• Some runners register but don’t show up on the day (entry fee still collected, but no ribbon cost incurred) — how would this affect the model?
• Would it be more useful to report a range (“$94 to $400”) to the sports store than a single guess? Why?

Activity 3 — Harder Problem: Working backwards through the Cycle (10 min)

Individual or pairs, then share strategy.

Returning to the Trivia Night (Activity 1): the committee decides $540 for the Year 8 fund isn’t enough — they want to guarantee at least $600. Everything else stays the same (90 attendees, 30 buying early-bird at 15% off, costs of $180 + $3.80/person, 20% of profit donated). What full ticket price should they charge instead of $14?

Socratic scaffolding:

Prompt	Purpose
Understand: what is being asked?	Find the ticket price, not the profit — this reverses the direction of Activity 1.
What do you know?	90 attendees (30 early-bird at 15% off, 60 full price); fixed costs $180 + $3.80×90; 20% of profit is donated; the rest must be at least $600.
What don’t you know?	The full ticket price — represent it with a pronumeral, .
Devise a plan	Write revenue in terms of , subtract costs to get profit in terms of , then require 80% of that profit to be at least $600.
Carry out the plan	(see working below)
Look back — is the answer sensible?	Compare with Activity 1: at , the fund received only $540. The new price should be a little higher than $14 — check that it is.
Look back — does it generalise?	If attendance were higher, would the required price change? This is the same review process as Activity 2, applied in reverse.

Interpret: Since ticket prices are set in whole or 10-cent amounts, the committee should charge at least $14.90. Checking: at $14.90, revenue , profit , Year 8 fund 601.56600 target.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. State, in order, the four stages of the mathematical modelling cycle.
2. A car wash charges $6 per car; costs are $50 flat plus $0.40 per car; 40 cars are expected. Formulate the calculation needed to find the profit (do not solve).
3. Using your formulation from Q2, solve for the profit.
4. The car wash donates 30% of the profit to charity. Interpret this result in a complete sentence, stating both the donation and the amount kept.
5. Reasoning. Give one reason the “40 cars expected” assumption in Q2 might not hold in reality, and explain how this would affect the reliability of the reported profit.

Answers: 1. Formulate → Solve → Interpret → Report/Review. 2. Profit . 3. Revenue , cost , profit 174174 profit, of which $52.20 (30%) is donated to charity, leaving $121.80 for the organisers.” 5. E.g. weather could reduce turnout below 40 cars, which would lower revenue but leave the $50 flat cost unchanged — meaning the true profit could be significantly lower than modelled; a single point estimate hides this risk.

Common Misconceptions

Misconception	How to pre-empt it
”Solving” the calculation is the whole task; the cycle ends once a number is produced.	Insist every activity in this lesson is marked complete only once all four stages have been written down, not just the arithmetic.
A single attendance figure (e.g. “90 people”) is a certainty rather than a planning assumption.	Use Activity 2’s range (60–150 runners) to show why real models are often reported as a range, not one number.
Fixed costs are believed to scale with attendance, just like the per-person costs.	Highlight in every scenario which cost is flat (paid regardless of numbers) versus per person (scales up or down).
Confusing “profit” with “revenue” when applying a donation or margin percentage.	Always require profit revenue cost to be calculated and labelled before any percentage is taken of it.
Rounding a required value (like a target ticket price) down instead of up, causing the target to be missed.	Model Activity 3 explicitly: rounding $14.877 down to $14.80 would leave the fund short of $600 — always check which direction of rounding is safe.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A market stall sells bracelets for $6 each. Materials cost $2 per bracelet, and stall hire is $40 for the day. How many bracelets must be sold to break even?

Answer

10 bracelets must be sold to break even.

E2 (AMC Junior style). A fundraiser sells raffle tickets at $4 each and spends $150 on prizes. What is the least number of tickets they must sell for the profit to be at least $250?

Answer

At least 100 tickets.

E3 (Challenge). Last year, a charity event’s revenue exactly equalled its costs (profit ). This year, revenue increased by 20% and costs increased by 10%. Using multipliers, show whether this year’s event made a profit, and express it as a percentage of last year’s revenue.

Answer

Let last year’s revenue and cost both equal .

This year’s profit equals 10% of last year’s revenue — a profit was made, even though both revenue and costs rose.

Homework

1. A canteen sells sandwiches for $4.50 each. Ingredient costs are $70 flat plus $1.20 per sandwich. Formulate (but do not solve) the profit for 85 sandwiches sold.
2. Solve your formulation from Q1.
3. The canteen donates 15% of the profit from Q2 to the school garden fund. Interpret the result in a complete sentence, stating both the donation and the amount kept.
4. A market stall’s costs are $90 flat plus $0.80 per item; each item sells for $5.00. Find the number of items needed to break even, rounding up to a whole item.
5. Reasoning. Explain why a modelling cycle should return to “Formulate” if the “Review” stage reveals the model’s assumptions were unrealistic, using an example from this lesson.
6. Challenge. A fundraiser’s revenue is (where is the number of tickets sold) and its costs are . Find the smallest whole number of tickets needed for the profit to exceed $500.

Answers: 1. Profit . 2. Revenue , cost , profit 210.50210.50 profit, of which $31.58 (15%) is donated to the garden fund, leaving $178.92 for the canteen.” 4. , so 22 items are needed to break even. 5. E.g. Activity 2’s fun run model assumed a fixed $1.20/runner ribbon cost; if reviewing revealed a bulk discount applied at high attendance, the formulation itself would need updating (a variable, not fixed, ribbon cost), not just the numbers recalculated. 6. Profit , so the smallest whole number is .