Year 7 Mathematics · AC9M7M06

Practical Ratio Modelling

use mathematical modelling to solve practical problems involving ratios; formulate problems, interpret and communicate solutions in terms of the situation, justifying…

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Learning goalsSay it simply

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.

Success criteria

  • I can formulate a practical problem as an ordered ratio with clear assumptions and constraints.
  • I can use an appropriate ratio representation or digital tool and scale equivalent ratios accurately.
  • I can interpret, communicate and justify a ratio-model solution in its practical context.
Key conceptTeach from the board

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

Formulate a ratio model for a drink mix

  1. Define the concentrate-to-water ratio as 2:5 and represent it with 7 equal parts.
  2. For 21 litres in total, calculate the scale factor 21÷7=3.
  3. Scale both terms by 3 to obtain 6 litres of concentrate and 15 litres of water.
  4. Interpret the result in context and check that 6:15 simplifies to 2:5.

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

Interpret a staffing ratio with a practical constraint

  1. Represent the requirement as at least 1 adult for every 10 students.
  2. For 67 students, calculate 67÷10=6.7 groups of students.
  3. Round up to 7 adults because a fraction of an adult cannot supervise the remaining students.
  4. Communicate the recommendation and justify it against the minimum-ratio constraint.

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

Application problem 1

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.

  1. Plan: Represent the information first, then calculate, interpret and independently check the result.
  2. Work: 7.6×25 000 = 190 000 cm = 1 900 m = 1.9 km.
  3. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

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

Application problem 2

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.

  1. Plan: Use totals of 8, 16, 24 and so on, then choose the first total at least 70.
  2. Work: 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.
  3. Interpret: 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.

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.

Clean visual examplesOne-page board

Clean one-page examples

AC9M7M06 - Practical Ratio Modelling
Example 1

Example 1 Formulate a ratio model for a drink mix Define the concentrate-to-water ratio as 2:5 and represent it with 7 equal parts. For 21 litres in total, calculate the scale factor 21÷7=3. Scale both terms by 3 to obtain 6 litres of concentrate and 15 litres of water. Interpret the result in context and check that 6:15 simplifies to 2:5. 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

Example 2 Interpret a staffing ratio with a practical constraint Represent the requirement as at least 1 adult for every 10 students. For 67 students, calculate 67÷10=6.7 groups of students. Round up to 7 adults because a fraction of an adult cannot supervise the remaining students. Communicate the recommendation and justify it against the minimum-ratio constraint. 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

Example 3 Application problem 1 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. Plan: Represent the information first, then calculate, interpret and independently check the result. Work: 7.6×25 000 = 190 000 cm = 1 900 m = 1.9 km. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills. 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

Example 4 Application problem 2 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. Plan: Use totals of 8, 16, 24 and so on, then choose the first total at least 70. Work: 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. Interpret: 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. 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.

Curriculum examplesCopied content
  • Content description: 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.
  • E1: use fractions to model ratio problems, including part–part and part–whole relations.
  • E2: model practical ratios of length, capacity or mass in contexts such as construction, design, food or textile production.
  • E3: use manipulatives, diagrams and mathematical discussion, including colour-mixing models.
  • E4: investigate commercialised substances founded on First Nations Australians’ knowledges and understand how ratios are used in their development.
Questions and answersWith answers

Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.

  1. 1. Simplify the ratio 12:18.

    Check answer

    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. 2. Scale the ratio 4:7 by a factor of 3.

    Check answer

    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. 3. A basket has red and blue balls in the ratio 3:5. How many equal parts are in the model?

    Check answer

    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. 4. Cordial concentrate and water are mixed in the ratio 1:4. Model a 15-cup batch.

    Check answer

    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. 5. Red and white paint are mixed in the ratio 2:3. How much of each is needed for 10 litres?

    Check answer

    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. 6. A school provides computers to students in the ratio 1:3. How many computers model a group of 27 students?

    Check answer

    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. 7. A student claims 3:5 and 5:7 are equivalent because 2 was added to both terms. Evaluate the claim.

    Check answer

    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. 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.

    Check answer

    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. 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.

    Check answer

    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. 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.

    Check answer

    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.

Practice and reviewReady for practice

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.

Curriculum alignmentStart here

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.

Success criteria

  • I can formulate a practical problem as an ordered ratio with clear assumptions and constraints.
  • I can use an appropriate ratio representation or digital tool and scale equivalent ratios accurately.
  • I can interpret, communicate and justify a ratio-model solution in its practical context.
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