AC9M7M06 • Year 7 Maths • Measurement • Learn
use mathematical modelling to solve practical problems involving ratios; formulate problems, interpret and communicate solutions in terms of the situation, justifying choices made about the representation.
We are learning to formulate, solve, interpret and justify practical ratio models using appropriate representations.
A ratio model represents a multiplicative comparison in a stated order. Begin by naming the quantities, recording assumptions and constraints, and deciding whether the ratio compares parts with parts or a part with the total.
Choose a tape diagram, double number line, ratio table or digital spreadsheet that keeps corresponding quantities aligned. Generate equivalent ratios by multiplying or dividing every term by the same non-zero scale factor; do not add the same amount to each term.
Interpret the calculated values in the practical context, including whole-number or minimum constraints. Communicate the representation, strategy and conclusion, then justify the model by simplifying the final ratio, checking a common scale factor or testing the boundary case.
Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.
Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.
Example 1
Final answer: Use 6 L of concentrate and 15 L of water.
Check: The quantities total 21 L and 6:15 is equivalent to 2:5.
Example 2
Final answer: At least 7 adults are required for 67 students.
Check: Six adults cover at most 60 students, whereas 7 adults cover up to 70 students.
Example 3
Problem: A map scale is 1:25 000. Two locations are 7.6 cm apart on the map. Find the actual distance in kilometres and explain the unit conversion.
Final answer: 7.6×25 000 = 190 000 cm = 1 900 m = 1.9 km.
Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Example 4
Problem: Relief packs must keep water:food in the ratio 5:3, with at least 70 packs in total. Find the smallest exact-ratio order and justify it with a ratio table.
Final answer: Order 45 water packs and 27 food packs, 72 packs total; scale factor 9 is the smallest whole factor whose 8-part group reaches at least 70.
Check: Each exact-ratio group contains 5+3=8 packs; scale factor 8 gives only 64, while factor 9 gives 45:27 and 72 total, so 72 is the smallest valid order.
Common mistake: Adding the same number to both terms creates an equivalent ratio.
Correction: Equivalent ratios are created by multiplying or dividing every term by the same non-zero scale factor.
Common mistake: The order of a ratio does not matter.
Correction: Ratio order names which quantity each term represents, so 2:5 and 5:2 describe different comparisons.
Common mistake: A decimal answer is always acceptable in a practical ratio problem.
Correction: Interpret the context and constraints; whole people or packs may require rounding up and then checking the requirement.
Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.
1. Simplify the ratio 12:18.
Answer: 2:3.
Hint: Divide both terms by their highest common factor, 6.
Why: Dividing 12 and 18 by 6 gives the equivalent ratio 2:3.
2. Scale the ratio 4:7 by a factor of 3.
Answer: 12:21.
Hint: Multiply both terms by 3.
Why: Multiplying both terms by the same scale factor gives 4×3:7×3=12:21.
3. A basket has red and blue balls in the ratio 3:5. How many equal parts are in the model?
Answer: 8 equal parts.
Hint: Add the two ratio terms to count all parts.
Why: The part-to-part ratio 3:5 contains 3+5=8 equal parts altogether.
4. Cordial concentrate and water are mixed in the ratio 1:4. Model a 15-cup batch.
Answer: Use 3 cups of concentrate and 12 cups of water.
Hint: Divide 15 by the 5 total parts, then scale both ratio terms.
Why: Each of the 5 parts is 3 cups, so the model gives 1×3=3 cups of concentrate and 4×3=12 cups of water.
5. Red and white paint are mixed in the ratio 2:3. How much of each is needed for 10 litres?
Answer: Use 4 L of red paint and 6 L of white paint.
Hint: Find the size of one of the 5 equal parts.
Why: Ten litres divided into 5 equal parts gives 2 L per part, so the mixture needs 4 L red and 6 L white.
6. A school provides computers to students in the ratio 1:3. How many computers model a group of 27 students?
Answer: 9 computers.
Hint: Match 3 students to one computer, then find the scale factor.
Why: Because 27 students are 9 groups of 3, scaling 1:3 by 9 gives 9 computers for 27 students.
7. A student claims 3:5 and 5:7 are equivalent because 2 was added to both terms. Evaluate the claim.
Answer: The claim is false; changing 3 to 5 uses a factor of 5/3, but changing 5 to 7 uses a different factor of 7/5.
Hint: Check whether one common scale factor transforms both terms.
Why: Adding the same amount does not preserve a ratio; there is no single scale factor that maps both 3 to 5 and 5 to 7.
8. A camp requires at least 1 adult for every 10 students. Formulate and solve a ratio model for 67 students, then justify the practical decision.
Answer: Model adult:student as at least 1:10; 67÷10=6.7, so 7 whole adults are required because 6 cover only 60 students.
Hint: Divide the student count by 10, then interpret the decimal using the whole-person constraint.
Why: The ratio constraint gives 6.7 adult groups, which must be rounded up; 7 adults cover up to 70 students, while 6 do not meet the requirement for 67.
9. Relief packs must keep water:food in the ratio 5:3, with at least 70 packs in total. Find the smallest exact-ratio order and justify it with a ratio table.
Answer: Order 45 water packs and 27 food packs, 72 packs total; scale factor 9 is the smallest whole factor whose 8-part group reaches at least 70.
Hint: Use totals of 8, 16, 24 and so on, then choose the first total at least 70.
Why: Each exact-ratio group contains 5+3=8 packs; scale factor 8 gives only 64, while factor 9 gives 45:27 and 72 total, so 72 is the smallest valid order.
10. A map scale is 1:25 000. Two locations are 7.6 cm apart on the map. Find the actual distance in kilometres and explain the unit conversion.
Answer: 7.6×25 000 = 190 000 cm = 1 900 m = 1.9 km.
Hint: Represent the information first, then calculate, interpret and independently check the result.
Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
These are closest-topic connections rather than exact code equivalents.
| Region | Closest connection |
|---|---|
| Australia | Australian Curriculum v9.0 — AC9M7M06 mathematical modelling with ratios. |
| Victoria | Level 7 ratio, proportion and mathematical modelling connections within Number and applied problem solving. |
| NSW | Stage 4 proportional reasoning, fractions and ratio problem-solving connections. |
| United States | Common Core Grade 7 Ratios and Proportional Relationships, especially proportional relationships and multi-step real-world problems. |
| England | Key Stage 3 ratio, proportion and rates of change. |
| New Zealand | Level 4–5 proportional reasoning and multiplicative thinking connections. |
| India | Class 7 ratio/proportion and applied arithmetic connections. |
The SkillrHub lesson remains the primary learning resource. This optional video reinforces the explanation; you can complete the lesson and practice without watching.
Before you watch:
Math Antics — Represent a comparison between quantities using a ratio and apply it to a practical model.
As you watch: Why does the order of the quantities in a ratio matter?
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Try it: A drink uses cordial and water in the ratio 1:5. Model enough for 18 equal measures and explain the quantities you choose.
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Mapped skill: use mathematical modelling to solve practical problems involving ratios; formulate problems, interpret and communicate solutions in terms of the situation, justifying choices made about the representation
These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.
| Region | Curriculum framework | Closest level or code |
|---|---|---|
| Australia | Australian Curriculum v9.0 | AC9M7M06 · Year 7 |
| Victoria | Victorian Curriculum F–10 Version 2.0 — Mathematics | VC2M7M06 · Level 7 |
| New South Wales | NSW Mathematics K–10 Syllabus (2022) | MA4-RAT-C-01 · Stage 4 |
| United States (USA) | Common Core State Standards for Mathematics | Grade 7 |
| Canada (Ontario) | Ontario Curriculum — Mathematics | Grade 7 |
| United Kingdom (England) | National Curriculum in England — Mathematics | Year 8, Key Stage 3 |
| India | NCERT / CBSE — Mathematics | Class 7 |
Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.
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Topic reference: AC9M7M06 — Practical Ratio Modelling
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