083i - Formulating Percentage Problems

Lesson 83 — Mathematical Modelling: Formulating Problems with Percentages

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

Learning Intentions

• To understand the stages of the mathematical modelling cycle: formulate, solve, interpret, report/review.
• To formulate real-world percentage scenarios (discounts, GST, wage changes) as precise mathematical statements.

Success Criteria

I can:

1. Describe each stage of the mathematical modelling cycle in my own words.
2. Identify the “whole” (100%) in a worded percentage scenario.
3. Write a clear mathematical formulation for a percentage scenario before solving it.
4. Distinguish between finding a percentage of an amount, a percentage increase, and a percentage decrease.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

For each scenario, state whether it involves finding a percentage of an amount, a percentage increase, or a percentage decrease.

1. A $60 jumper is reduced by 25% in a sale.
2. GST of 10% is added to an $80 item.
3. A town’s population of 12 000 grows by 3% this year.
4. Finding 15% of a $40 restaurant bill as a tip.

Answers: 1. Decrease; 2. Increase; 3. Increase; 4. Percentage of an amount.

Activities

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

Introduce the four-stage mathematical modelling cycle, drawn as a loop:

• Formulate: translate the real situation into a precise mathematical statement — identify the whole (100%) and what operation is required.
• Solve: carry out the calculation using an efficient strategy.
• Interpret: translate the numeric answer back into the language of the original situation.
• Report / Review: communicate the solution clearly, and check whether the model’s assumptions were reasonable.

This unit (Lessons 83–87) works through this cycle in stages. Today’s focus is formulating only — writing down what calculation is needed, without solving it yet.

I do: “A 80 (100%), GST adds , so the total is of , i.e. the calculation needed is .

We do: Formulate (do not solve) — “A 45; decrease of 30%; calculation needed: .)*

You do: Formulate only, for each of the following:

1. A worker’s $900 weekly wage rises by 4%.
2. A $250 television is discounted by 15%.
3. A charity’s donations of increase by 12% after a campaign.

Activity 2 — Formulating from Messy Real Text (12 min)

Give students richer scenarios containing distractor information. For each, they must identify the whole and write the formulation as an expression, without solving.

1. “A café had 40 tables before renovating. After adding a new outdoor area, it now has 25% more tables. The renovation took six weeks and cost .” (Distractors: weeks, cost. Formulation: .)
2. “Maya bought a 120 \times 1.10$.)*
3. “A school of 640 students expects enrolment to fall by 8% next year due to a new school opening nearby.” (Formulation: .)
4. “A loan attracts a one-off 6% establishment fee, charged in the first week.” (Formulation: — note this asks for the fee itself, not a new total.)

Discuss Q4 as a contrast: sometimes the question asks for the percentage amount, not the new total — formulating correctly means reading precisely what is being asked.

Activity 3 — Inquiry Task: Are Two Discounts the Same as One? (10 min)

Pairs, then whole-class share.

A shop advertises “20% off, then an extra 10% off for club members.” A customer says, “That’s the same as 30% off.” A $150 jacket is being considered. Formulate a plan to check the customer’s claim, then solve it.

Socratic scaffolding:

Prompt	Purpose
Understand: what is being compared?	The result of two successive percentage decreases versus one combined percentage decrease.
What do you know?	The original price ($150), and the two claims: apply 20% then 10%, versus apply 30% once.
Devise a plan	Formulate both calculations as decimal multipliers, then compare the final prices.
Carry out the plan	Successive: . Single 30%: .
Look back — is the customer correct?	No — successive discounts of 20% and 10% leave the price at 105). The effective combined discount is , not 30%.
Look back — does this generalise?	Ask: will successive percentage discounts always give a smaller total discount than simply adding the percentages? (Yes — this previews later lessons on choosing efficient strategies.)

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Write, but do not solve, the calculation needed to find the sale price of a $200 item discounted by 15%.
2. Write, but do not solve, the calculation needed to find the total cost of a $65 item with 10% GST added.
3. A gym’s membership of 480 people falls by 5%. Identify the “whole” and write the formulation.
4. Explain, in one sentence, the difference between formulating “find 20% of 300”.

Answers: 1. ; 2. ; 3. Whole ; formulation ; 4. “Find 20% of 300 \times 0.20300 \times 1.20$).

Common Misconceptions

Misconception	How to pre-empt it
Successive percentage changes can simply be added together (20% then 10% = 30%).	Model the Activity 3 jacket example explicitly and revisit it whenever successive changes appear.
The “whole” (100%) is always the larger number written in the question.	Show counterexamples where the original amount is not the largest number present (e.g. a discount stated in dollars alongside the price).
A percentage increase calculation and a percentage of calculation use the same formulation.	Contrast “20% of = 300 \times 0.20=300 \times 1.20$) side by side.
Formulating means solving quickly in your head rather than writing the plan down.	Insist, during this lesson only, that no calculator or arithmetic is used — the goal is the written expression, not the number.
GST or a fee is calculated on the new total rather than the original amount, unless stated otherwise.	Always ask: “What does the percentage apply to — the original price, or a price that has already changed?”

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A price is increased by 25% and then decreased by 20%. Formulate the combined multiplier and determine whether the final price is greater than, less than, or equal to the original.

Answer

Combined multiplier: .

The final price equals the original price exactly — a neat case where the two changes cancel.

E2 (AMC Junior style). After a 40% discount, a jacket costs P$, and solve it.

Answer

Original price: $90.

E3 (Challenge). A number is increased by and the result is then decreased by . Formulate the combined multiplier in terms of , and explain why the final value can never exceed the original for .

Answer

Multiplier .

Since whenever , the multiplier is always less than 1 — the final value is always less than the original.

Homework

1. Write, but do not solve, the formulation for each: (a) 12% of (b) a 720 laptop discounted by 18%.
2. A library’s collection of 5 400 books grows by 6% after a donation drive. Identify the whole and write the formulation.
3. A $150 fine is reduced by 20% for early payment, then an administration fee of 5% of the reduced fine is added. Write both steps as a formulation (do not solve).
4. Solve the formulations from Question 1.
5. Reasoning. Explain why “a 50% increase followed by a 50% decrease” does not return a price to its original value, using a formulation in terms of a multiplier.
6. Challenge. A retailer wants a 180 after a single discount is applied, and separately wants to know: if instead two equal successive discounts of each were applied to reach the same final price, what would be (to one decimal place)?

Answers: 1. (a) (b) (c) . 2. Whole ; formulation . 3. Step 1: ; Step 2: (result) . 4. (a) (b) (c) . 5. Multiplier , so the price ends at 75% of the original — a net 25% decrease, not the original value. 6. Single discount needed: , i.e. 10% off. For two equal discounts: .