093i - Solving Ratio and Rate Models in Financial Contexts

Lesson 93 — Mathematical Modelling: Solving Ratio and Rate Models

Strand: Measurement | Descriptor: AC9M8M07 | Duration: 45 minutes

Learning Intentions

• To solve formulated ratio and rate models accurately, particularly in financial contexts.
• To select and apply efficient calculation strategies (unitary method, proportion, scaling) to reach a numerical solution.

Success Criteria

I can:

1. Take a formulated ratio or rate equation and solve it for the unknown.
2. Use the unitary method or direct proportion to solve best-buy, currency, and mixing problems.
3. Solve multi-step financial problems involving rates such as simple interest, wages, and discounts.
4. Check a solution’s numerical accuracy by substituting back into the original relationship.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

Solve each proportion for the unknown as fast as you can.

Answers: 1. ; 2. ; 3. ; 4. .

Teacher note: These are last lesson’s formulated equations, stripped of context. Today we reconnect the solving process to the financial scenarios they came from.

Activities

Activity 1 — Explicit Instruction: Solving Strategies for Rate and Ratio Equations (10 min)

Remind students of the modelling cycle: Formulate → Solve → Interpret → Review. Today’s focus is Solve — carrying out accurate calculations once a model is set up.

I do: “A savings account pays simple interest at per annum. Zac invests 12003$ years. How much interest does he earn?”

Formulated model (recap, do not re-derive): , where , , .

Zac earns 144$1443$12004%$120048 \times 3 = 144$ ✓.

We do: Together solve the formulated model from last lesson: “A currency exchange booth converts AUD to NZD at AUD NZD. Convert 250\text{NZD} = 250 \times 1.08$.

You do: Solve each formulated model, showing full working:

(a) = fuel required (L):

(b) Paint “Sunset” mixed red:yellow:white , total L. Find each colour.

(c) Recipe for people scaled to people: g rice becomes g, where .

Activity 2 — Guided Practice: Multi-step Financial Rate Problems (10 min)

I do: “A casual worker is paid 27.50145$ hours on Saturday. Find her total pay.”

We do: Together solve: “A 8520%10%30%$ discount.)*

You do:

(d) A phone plan charges a 25$0.04600$ MB used.

(e) An investment of 20003.5%18$ months. (Careful with the time unit.)

(f) A shop pays staff 228$-hour public holiday shift.

Activity 3 — Inquiry Task: Solving the Fundraiser Model (14 min)

Pairs. Return to the fundraiser scenario formulated previously:

The Year 8 fundraiser sells raffle tickets at for 52:3:42.4$7401= 0.66$ USD.

Using the equations you formulated in Lesson 92, solve for:

• The cost of buying raffle tickets.
• The mass of butter, sugar and flour needed for the kg tray.
• The AUD value of the 740$ USD in pledges.
• The total funds raised, combining raffle ticket sales (assume all tickets bought by different families, each buying one bundle of for 5$1.50100$ g), and the converted pledge amount.

Socratic scaffolding for the combined total:

Prompt	Purpose
Understand: what final quantity is being asked for?	A single dollar total combining three separately-solved amounts.
What have you already solved?	List each solved sub-amount before combining — don’t recombine unsolved expressions.
Devise a plan: how do the parts fit together?	Addition of independent revenue streams; check no quantity is double-counted.
Carry it out	Compute each part, then sum.
Looking back	Does the total feel like a plausible school fundraiser amount? Which single revenue stream dominates?

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Solve for : .
2. A savings account pays simple interest at p.a. Find the interest earned on 6002$ years.
3. A jumper is discounted from 605%$ loyalty discount is applied. Find the final price.
4. Reasoning. Explain why applying two successive percentage discounts of and does not give the same result as a single discount.

Answers: 1. ; 2. 6060 \times 0.85 \times 0.95 = $48.4548.4560 \times 0.80 = $48.00$, since the second percentage is taken from a smaller base.

Common Misconceptions

Misconception	How to pre-empt it
Adding successive percentage discounts (e.g. off).	Compute both ways side by side on the board and show they differ; the compounding base changes each time.
Using the wrong time unit in simple interest (e.g. months treated as years).	Insist is always converted to years before substituting into .
Forgetting to add the flat fee in two-part rate plans.	Underline the “flat fee rate quantity” structure before solving.
Solving the proportion by cross-multiplying incorrectly (multiplying the wrong pair).	Re-derive explicitly before applying it.
Reporting an unrounded or oddly-rounded currency answer (e.g. 61.1999…$).	Require all money answers rounded to the nearest cent, with a reminder of standard rounding.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). An investment of 50006%$1500$?

Answer

6 full years are needed.

E2 (Kangaroo style). A shirt’s price is increased by , then later decreased by . Is the final price equal to, greater than, or less than the original? Find the overall percentage change.

Answer

Less than the original — a overall decrease. Increase and decrease by the same percentage never cancel out, since the decrease applies to a larger base.

E3 (Challenge). Two currency exchange booths convert AUD to EUR. Booth A charges no fee but uses a rate of AUD EUR. Booth B uses the “true” rate of AUD EUR but charges a flat 5$ AUD fee. For what amount of AUD converted are the two booths equivalent?

Answer

Let = AUD converted.

At $105 AUD, both booths give the same EUR amount ( EUR). Below this, Booth A is better; above it, Booth B is better.

Homework

1. Solve for the unknown: (a) (b) (c) .
2. A worker earns 24.80124$ Sunday hours.
3. Find the simple interest earned on 34004.5%2.5$ years.
4. A 12010%8%$ discount applies at checkout. Find the final price.
5. A recipe uses flour, butter and sugar in the ratio to make g of dough. Solve for the mass of each ingredient needed to make kg of dough.
6. Reasoning. A phone plan charges 0.03$10$0.015$ per MB. Solve for the number of MB at which both plans cost the same, and explain what happens to the comparison for larger data usage.
7. Challenge. An amount of money is invested at p.a. simple interest. After how many years will the total interest earned equal the original amount invested?

Answers: 1(a) (b) (c) . 2. ; Sunday rate , pay ; total 446.40I=3400\times0.045\times2.5=$382.50120\times0.90\times0.92=$99.36=101200/500=2.4=5\times2.4\times… =\frac{5}{10}\times1200=600=\frac{2}{10}\times1200=240=\frac{3}{10}\times1200=3600.03m = 10+0.015m \Rightarrow 0.015m=10 \Rightarrow m=666.7667I=P \Rightarrow P\times0.05\times t=P \Rightarrow t=20$ years.