Year 7 Mathematics · AC9M7N09

Modelling Rational and Percentage Problems

use mathematical modelling to solve practical problems, involving rational numbers and percentages, including financial contexts; formulate problems, choosing…

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Learning goalsSay it simply

We are learning to use mathematical modelling to solve practical problems involving rational numbers and percentages, including financial contexts; choose representations, efficient strategies and digital tools; interpret and communicate justified solutions.

A mathematical model represents the quantities, relationships and constraints relevant to a practical problem. Financial models must include the correct percentage base as well as fixed fees, thresholds and timing.

A complete comparison calculates final costs under the same assumptions. For a $480 item, 15% off costs $408, while $90 off plus a $25 fee costs $415.

Sensitivity analysis varies an input to test whether a recommendation changes. A break-even point occurs when two modelled costs are equal, and conclusions should state assumptions and limits rather than claim universal validity.

Success criteria

  • I can identify quantities, constraints and assumptions in a practical model.
  • I can represent and solve rational-number or percentage comparisons.
  • I can test a break-even point and communicate a justified recommendation.
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

Compare discount offers

  1. Plan A gives 15% off $480: 480×0.85=$408.
  2. Plan B gives $90 off then adds $25: 480−90+25=$415.
  3. Compare complete costs: $408<$415.
  4. Recommend A, cheaper by $7, assuming identical products and no other fees.

Final answer: Plan A costs $408 and is $7 cheaper.

Check: Recalculate each offer including every fee.

Example 2

Find a break-even price

  1. Offer A is 20% off price p, so cost is 0.8p.
  2. Offer B is $60 off plus $12 fee, so cost is p−48.
  3. Set 0.8p=p−48.
  4. Solve 0.2p=48, giving p=$240.

Final answer: The offers break even at $240.

Check: Both offers cost $192 when p=$240.

Example 3

Application problem 1

Problem: Plan A gives 25% off a $240 item plus a $12 fee. Plan B gives 18% off with no fee. Which is cheaper, by how much, and what assumptions are you making?

  1. Plan: Represent the information first, then calculate, interpret and independently check the result.
  2. Work: Plan A costs $240×0.75+$12 = $192. Plan B costs $240×0.82 = $196.80. Plan A is cheaper by $4.80, assuming the discounts apply to the same original price and there are no other fees.
  3. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Final answer: Plan A costs $240×0.75+$12 = $192. Plan B costs $240×0.82 = $196.80. Plan A is cheaper by $4.80, assuming the discounts apply to the same original price and there are no other fees.

Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Application problem 2

Problem: A store offers 25% off then charges 10% tax on the discounted price. Find the final cost of $240 and compare with subtracting 15% once.

  1. Plan: Apply each multiplier in order.
  2. Work: Sequential cost is 240×0.75×1.10=$198; 15% off gives $204, so sequential is $6 cheaper.
  3. Interpret: Successive percentages use changing bases.

Final answer: Sequential cost is 240×0.75×1.10=$198; 15% off gives $204, so sequential is $6 cheaper.

Check: Successive percentages use changing bases.

Clean visual examplesOne-page board

Clean one-page examples

AC9M7N09 - Modelling Rational and Percentage Problems
Example 1

Example 1 Compare discount offers Plan A gives 15% off $480: 480×0.85=$408. Plan B gives $90 off then adds $25: 480−90+25=$415. Compare complete costs: $408&lt;$415. Recommend A, cheaper by $7, assuming identical products and no other fees. Final answer: Plan A costs $408 and is $7 cheaper. Check: Recalculate each offer including every fee.

Example 2

Example 2 Find a break-even price Offer A is 20% off price p, so cost is 0.8p. Offer B is $60 off plus $12 fee, so cost is p−48. Set 0.8p=p−48. Solve 0.2p=48, giving p=$240. Final answer: The offers break even at $240. Check: Both offers cost $192 when p=$240.

Example 3

Example 3 Application problem 1 Problem: Plan A gives 25% off a $240 item plus a $12 fee. Plan B gives 18% off with no fee. Which is cheaper, by how much, and what assumptions are you making? Plan: Represent the information first, then calculate, interpret and independently check the result. Work: Plan A costs $240×0.75+$12 = $192. Plan B costs $240×0.82 = $196.80. Plan A is cheaper by $4.80, assuming the discounts apply to the same original price and there are no other fees. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills. Final answer: Plan A costs $240×0.75+$12 = $192. Plan B costs $240×0.82 = $196.80. Plan A is cheaper by $4.80, assuming the discounts apply to the same original price and there are no other fees. Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Example 4 Application problem 2 Problem: A store offers 25% off then charges 10% tax on the discounted price. Find the final cost of $240 and compare with subtracting 15% once. Plan: Apply each multiplier in order. Work: Sequential cost is 240×0.75×1.10=$198; 15% off gives $204, so sequential is $6 cheaper. Interpret: Successive percentages use changing bases. Final answer: Sequential cost is 240×0.75×1.10=$198; 15% off gives $204, so sequential is $6 cheaper. Check: Successive percentages use changing bases.

Curriculum examplesCopied content

Content description: use mathematical modelling to solve practical problems, involving rational numbers and percentages, including financial contexts; formulate problems, choosing representations and efficient calculation strategies, using digital tools as appropriate; interpret and communicate solutions in terms of the situation, justifying choices made about the representation.

Questions and answersWith answers

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

  1. 1. Find 15% of $200.

    Check answer

    Answer: $30.

    Hint: Convert 15% to 0.15.

    Why: The percentage amount is 0.15×200.

  2. 2. A $90 item has $12 postage. Find total cost.

    Check answer

    Answer: $102.

    Hint: Add item price and postage.

    Why: A model includes all relevant charges.

  3. 3. State one assumption needed when comparing two phone plans.

    Check answer

    Answer: For example, assume the same usage and comparison period.

    Hint: Identify what must be equal across plans.

    Why: A defensible mathematical model states a constraint or assumption that keeps the phone-plan comparison fair.

  4. 4. A $320 bike is discounted 12.5%. Find its sale price.

    Check answer

    Answer: 12.5% is 1/8; discount $40, sale price $280.

    Hint: Find one eighth of 320.

    Why: Subtract the discount from the original price.

  5. 5. Compare 10% off $500 with $65 off plus a $20 fee.

    Check answer

    Answer: 10% off costs $450; the second costs $455, so 10% off is $5 cheaper.

    Hint: Model each final price separately.

    Why: Complete costs include the fee.

  6. 6. A budget is $600 with a 5% contingency. Find the planned total.

    Check answer

    Answer: $600×1.05=$630.

    Hint: Add 5% of 600.

    Why: A contingency increases the base amount.

  7. 7. Why can a spreadsheet help compare changing prices?

    Check answer

    Answer: It can apply transparent formulas repeatedly and reveal where the cheaper option changes.

    Hint: State the formula and varied input.

    Why: Digital tools support sensitivity analysis after formulas are defined.

  8. 8. Plan A costs $18+$0.08 per message; Plan B costs $10+$0.12 per message. Find the break-even usage.

    Check answer

    Answer: 18+0.08m=10+0.12m, so 8=0.04m and m=200 messages.

    Hint: Set the two complete cost formulas equal.

    Why: Equal costs define the break-even point.

  9. 9. A store offers 25% off then charges 10% tax on the discounted price. Find the final cost of $240 and compare with subtracting 15% once.

    Check answer

    Answer: Sequential cost is 240×0.75×1.10=$198; 15% off gives $204, so sequential is $6 cheaper.

    Hint: Apply each multiplier in order.

    Why: Successive percentages use changing bases.

  10. 10. Plan A gives 25% off a $240 item plus a $12 fee. Plan B gives 18% off with no fee. Which is cheaper, by how much, and what assumptions are you making?

    Check answer

    Answer: Plan A costs $240×0.75+$12 = $192. Plan B costs $240×0.82 = $196.80. Plan A is cheaper by $4.80, assuming the discounts apply to the same original price and there are no other fees.

    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: Discount percentages added to dollar fees.

Correction: Calculate each type with its correct base and unit.

Common mistake: Cheapest-looking headline chosen.

Correction: Include all conditions and costs.

Common mistake: Spreadsheet output lacks formula explanation.

Correction: Show the mathematical relationship.

Curriculum alignmentStart here

We are learning to use mathematical modelling to solve practical problems involving rational numbers and percentages, including financial contexts; choose representations, efficient strategies and digital tools; interpret and communicate justified solutions.

A mathematical model represents the quantities, relationships and constraints relevant to a practical problem. Financial models must include the correct percentage base as well as fixed fees, thresholds and timing.

A complete comparison calculates final costs under the same assumptions. For a $480 item, 15% off costs $408, while $90 off plus a $25 fee costs $415.

Sensitivity analysis varies an input to test whether a recommendation changes. A break-even point occurs when two modelled costs are equal, and conclusions should state assumptions and limits rather than claim universal validity.

Success criteria

  • I can identify quantities, constraints and assumptions in a practical model.
  • I can represent and solve rational-number or percentage comparisons.
  • I can test a break-even point and communicate a justified recommendation.
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