075i - Solving Ratio Models

Lesson 75 — Interpreting and Solving Ratio-Based Practical Problems

Strand: Measurement | Descriptor: AC9M7M06 | Duration: 45 minutes

Learning Intentions

• To carry ratio models through the solve and interpret stages of the modelling cycle.
• To recognise when a mathematical answer needs adjusting for the real situation.

Success Criteria

I can:

1. Solve a formulated ratio model accurately.
2. Interpret the mathematical answer in the context.
3. Adjust answers for real-world constraints (whole units, capacities, safety margins).
4. Loop back and revise the model when the answer does not fit.

Warmup

(6 minutes — interpret these answers, pairs)

Each calculation is correct. Interpret each answer in context — what should actually be reported or done?

1. Buses needed: .
2. Paint tins: L needed, sold in L tins → tins.
3. Concentrate for the week: L, sold in L bottles.
4. Each of friends owes: 100 \div 7 = $14.285714\ldots$

Answers: 1. buses; 2. tins ( L spare); 3. bottles ( L, L spare); 4. 14.293$14.28$100.03$; discuss briefly).

The lesson’s theme: the maths produces a number; the interpreter produces a decision.

Activities

Activity 1 — Full-cycle Worked Example (12 min)

The problem. “A community garden is being planted with natives and vegetables in the ratio . The beds hold plants in total. Natives come in trays of , vegetables in trays of . Trays cost 21$18$ (vegetables). What will planting cost?”

Formulate (recap from Lesson 74): plants split ; whole trays must be bought; assume every plant space is filled and no spares are wanted beyond forced leftovers.

Solve:

Interpret: 2282464424 : 642:5$ — is that acceptable? A judgement call to state).

Communicate: report the cost, the tray counts, the leftover, and the assumption that the ratio may flex slightly in practice.

Emphasise the loop: the spare seedlings sent us back to the formulation. That loop is normal, not failure.

Activity 2 — Paired Problems through the Cycle (16 min)

Pairs. Each problem must show all four stages, labelled.

Problem 1 — The sports carnival ribbons. Ribbons for 1st, 2nd and 3rd are ordered in ratio (extra 3rds for participation). The school expects events… ribbons come in rolls that make ribbons each. How many rolls?

Problem 2 — The care packages. A charity packs boxes each containing tinned food, toiletries and stationery by mass in ratio . Each box holds kg. They have kg of tinned food donated. (a) How many complete boxes can they fill? (b) What else must they source, and how much?

Problem 3 — The two-stroke fuel. A mower uses petrol : oil . The tank holds L. Oil comes in mL bottles. How much petrol and oil for a full tank, and how many bottles must be opened?

Socratic scaffolding for Problem 2:

Prompt	Purpose
What does each box need of each item?	One part kg: food kg, toiletries kg, stationery kg.
What limits the number of boxes?	The donated food: boxes.
So the food is the limiting resource?	Yes — like the flour in Lesson 71’s muffins.
What must be sourced?	Toiletries kg; stationery kg.
Interpret: is boxes the final answer?	Only if they can source the rest; otherwise the boxes are limited by the scarcest item.
Communicate	Report boxes, the shopping list, and the dependency.

Answers:

1. Events : ribbons ; rolls rolls (assume every event runs; spares absorbed).
2. (a) boxes (b) kg toiletries and kg stationery.
3. Parts : petrol L; oil L mL bottle exactly. (A satisfying exact fit — point out that real products are often designed around common mix ratios.)

Activity 3 — Inquiry: when the Answer Breaks the Model (6 min)

Whole class.

A cafe’s iced chocolate uses milk : ice cream : syrup by volume, in a mL glass. A customer asks for a mL “super size”.

1. Scale the recipe. Does anything break?
2. The cafe’s scoop dispenses ice cream only in mL scoops. Now what?

Socratic scaffolding:

Prompt	Purpose
Scale the recipe for mL.	One part mL: milk mL, ice cream mL, syrup mL.
Where does reality intrude?	Ice cream comes in mL scoops — mL is scoops.
The options?	scoops ( mL) and slightly less milk, or scoops and a milkier drink.
Which preserves the ratio better?	Neither exactly. scoops changes the ratio to — close.
Looking back	The model guided the decision but the final call was practical. Interpret, adjust, and say what you did.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Trail mix is packed in ratio into g bags. Find the mass of each ingredient per bag.
2. A model says tables are needed for a function. Interpret this.
3. Fuel is mixed . How much oil for L of the mixture?
4. A charity can fill boxes needing kg of rice each and has kg of rice. How many complete boxes, and how much rice is left?
5. Reasoning. Give an example of a solve-stage answer that must be rounded down when interpreted, and one that must be rounded up.

Answers: 1. One part g: g, g, g; 2. tables — you cannot seat people at of a table; 3. Parts : oil L; 4. boxes; kg left; 5. Down: complete boxes/bags from a fixed stock. Up: buses, tins, rolls — anything where a shortfall fails the task.

Common Misconceptions

Misconception	How to pre-empt it
Reporting the raw solve-stage number as the answer.	The warmup makes interpretation a separate, visible step.
Rounding the wrong direction for the context.	Revisit Lesson 61’s rule: shortfall-fails → up; fixed-stock → down.
Treating leftover material as an error.	The garden example frames leftovers as information to communicate.
Ignoring the limiting resource in multi-ingredient problems.	Problem 2’s scaffolding names it explicitly.
Forcing an exact ratio when reality quantises (scoops, trays).	The iced-chocolate inquiry legitimises stated approximation.
Skipping the loop back to formulation.	The garden example models the loop as routine practice.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Fuel is mixed petrol : oil . How much oil is needed with L of petrol?

Answer

L mL.

E2 (AMC Junior style). Punch is mixed juice : soda : cordial in a L bowl. Cordial comes in mL bottles. What fraction of a bottle is used?

Answer

Cordial L mL of a bottle.

E3 (Challenge). A bakery packs gift boxes with croissants : muffins . They have croissants and muffins. What is the largest number of complete boxes of items () they can pack, and what limits it?

Answer

From croissants: boxes. From muffins: boxes. The muffins limit it: boxes, leaving croissants over.

E4 (Challenge). Concrete is mixed (cement : sand : gravel). A job needs of concrete. Cement bags are ; sand and gravel are sold by the scoop. Write the shopping list.

Answer

One part : cement ( bags exactly), sand ( scoops, spare), gravel ( scoops, spare).

Homework

1. A smoothie mixes banana : berries : yoghurt by mass into g servings. Find each ingredient’s mass per serving.
2. A model gives marquees for a fair. Interpret, and justify your rounding.
3. Fuel is mixed . (a) How much oil with L of petrol? (b) Oil comes in mL bottles — how many bottles?
4. A florist makes bouquets with roses : lilies . She has roses and lilies. (a) How many complete bouquets of stems? (b) What limits her, and what is left over?
5. A cordial station needs L of drink mixed . (a) How much concentrate? (b) Concentrate comes in L bottles — how many? (c) Interpret any leftover.
6. Care packages need rice : lentils : flour by mass, kg per box. The charity has kg of rice, kg of lentils, kg of flour. How many complete boxes, and which ingredient limits them?
7. Reasoning. Explain, with an example, what it means for a resource to be “limiting”.
8. Challenge. An orchardist plants apples : pears : plums across tree spaces. Trees come in bundles: apples , pears , plums . Write the full order and note all spares.

Answers: Q1 — one part g: g, g, g. Q2 — marquees; running short leaves stalls uncovered. Q3 — (a) L (b) bottles. Q4 — rose-limited: ; lily check: ; so bouquets, leaving rose and lilies. Q5 — (a) L (b) bottles exactly (c) none. Q6 — per box: rice kg, lentils kg, flour kg; limits: rice , lentils , flour ; lentils limit at boxes. Q7 — the ingredient that runs out first sets the maximum output, e.g. Lesson 71’s flour. Q8 — apples ( bundles exactly), pears ( bundles , spare), plums ( bundles exactly).