Australian Curriculum v9 / ACiQ Year 8 Mathematics - Unit 1 - Percentage and financial modelling

Percentage and financial modelling

Model percentage increase and decrease, discounts, profit and loss and simple financial decisions.

Updated 2026-07-26 - 12 min read

Percentage and financial modelling is taught here as a connected set of decisions, not a list of facts. Work through the prerequisite recall, explicit models, carefully faded examples, misconception repairs and transfer task before using the target in Check, Practice, Review or Rapid Revision.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Calculate percentage increase and decrease

Multiply the original amount by 1 + r for an increase or 1 − r for a decrease, where r is the percentage written as a decimal. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by calculate percentage increase and decrease and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: A percentage is relative to the original amount; 15% means 0.15 times that base.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

Increase $240 by 15%.

Step 1 - identify the governing idea: Multiply the original amount by 1 + r for an increase or 1 − r for a decrease, where r is the percentage written as a decimal.

Step 2 - apply it to this evidence: 15% of $240 is $36, so the new amount is $276.

Result: $276

The relationship is visible in the working: 15% of $240 is $36, so the new amount is $276. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • $255 — It does not agree with the required relationship: 15% of $240 is $36, so the new amount is $276.
  • $204 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • $360 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Decrease 680 by 12%.

Step 1 - identify the governing idea: Multiply the original amount by 1 + r for an increase or 1 − r for a decrease, where r is the percentage written as a decimal.

Step 2 - apply it to this evidence: The decrease multiplier is 0.88, and 680 × 0.88 = 598.4.

Result: 598.4

The relationship is visible in the working: The decrease multiplier is 0.88, and 680 × 0.88 = 598.4. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 668 — It does not agree with the required relationship: The decrease multiplier is 0.88, and 680 × 0.88 = 598.4.
  • 761.6 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 81.6 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

A population rises from 800 to 920. What is the percentage increase?

Step 1 - identify the governing idea: Multiply the original amount by 1 + r for an increase or 1 − r for a decrease, where r is the percentage written as a decimal.

Step 2 - apply it to this evidence: The increase is 120; 120 ÷ 800 × 100% = 15%.

Result: 15%

The relationship is visible in the working: The increase is 120; 120 ÷ 800 × 100% = 15%. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 13.04% — It does not agree with the required relationship: The increase is 120; 120 ÷ 800 × 100% = 15%.
  • 120% — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 1.15% — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Model discounts, markups, profit and loss

Apply each percentage to its stated base and distinguish cost price, selling price and the amount of profit or loss. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by model discounts, markups, profit and loss and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: The percentages use different bases, so equal percentage changes in opposite directions do not cancel.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

A $150 jacket is discounted by 30%. What is the sale price?

Step 1 - identify the governing idea: Apply each percentage to its stated base and distinguish cost price, selling price and the amount of profit or loss.

Step 2 - apply it to this evidence: The customer pays 70%: $150 × 0.70 = $105.

Result: $105

The relationship is visible in the working: The customer pays 70%: $150 × 0.70 = $105. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • $120 — It does not agree with the required relationship: The customer pays 70%: $150 × 0.70 = $105.
  • $45 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • $195 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

A shop buys an item for $80 and sells it for $104. What is the profit percentage on cost?

Step 1 - identify the governing idea: Apply each percentage to its stated base and distinguish cost price, selling price and the amount of profit or loss.

Step 2 - apply it to this evidence: Profit is $24, and $24 ÷ $80 × 100% = 30%.

Result: 30%

The relationship is visible in the working: Profit is $24, and $24 ÷ $80 × 100% = 30%. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 23.08% — It does not agree with the required relationship: Profit is $24, and $24 ÷ $80 × 100% = 30%.
  • 24% — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 130% — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

A $200 item is marked up 20% then discounted 20%. What is the final price?

Step 1 - identify the governing idea: Apply each percentage to its stated base and distinguish cost price, selling price and the amount of profit or loss.

Step 2 - apply it to this evidence: The multipliers are 1.20 and 0.80; $200 × 1.20 × 0.80 = $192.

Result: $192

The relationship is visible in the working: The multipliers are 1.20 and 0.80; $200 × 1.20 × 0.80 = $192. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • $200 — It does not agree with the required relationship: The multipliers are 1.20 and 0.80; $200 × 1.20 × 0.80 = $192.
  • $160 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • $240 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Compare financial options

Convert all offers to final dollar amounts using the same quantities and time period before deciding which is better. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by compare financial options and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Percentages may use different bases or conditions, so compare final costs or returns on a common basis.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

A $300 purchase has either 25% off or $70 off. Which offer is cheaper?

Step 1 - identify the governing idea: Convert all offers to final dollar amounts using the same quantities and time period before deciding which is better.

Step 2 - apply it to this evidence: 25% of $300 is $75, giving $225 compared with $230.

Result: 25% off

The relationship is visible in the working: 25% of $300 is $75, giving $225 compared with $230. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • $70 off — It does not agree with the required relationship: 25% of $300 is $75, giving $225 compared with $230.
  • They are equal — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • There is not enough information — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Plan A costs $18 per month. Plan B costs $200 per year. Which is cheaper for 12 months?

Step 1 - identify the governing idea: Convert all offers to final dollar amounts using the same quantities and time period before deciding which is better.

Step 2 - apply it to this evidence: Plan A costs 12 × $18 = $216, so Plan B is $16 cheaper.

Result: Plan B by $16

The relationship is visible in the working: Plan A costs 12 × $18 = $216, so Plan B is $16 cheaper. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Plan A by $16 — It does not agree with the required relationship: Plan A costs 12 × $18 = $216, so Plan B is $16 cheaper.
  • Plan B by $18 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • They cost the same — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

A $500 account earns simple interest of 4% for one year. What is the closing balance?

Step 1 - identify the governing idea: Convert all offers to final dollar amounts using the same quantities and time period before deciding which is better.

Step 2 - apply it to this evidence: Interest is 0.04 × $500 = $20, added to the principal.

Result: $520

The relationship is visible in the working: Interest is 0.04 × $500 = $20, added to the principal. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • $504 — It does not agree with the required relationship: Interest is 0.04 × $500 = $20, added to the principal.
  • $20 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • $700 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. Decrease 680 by 12%.
  2. A shop buys an item for $80 and sells it for $104. What is the profit percentage on cost?
  3. Plan A costs $18 per month. Plan B costs $200 per year. Which is cheaper for 12 months?

Answers

  1. 598.4 — The decrease multiplier is 0.88, and 680 × 0.88 = 598.4.
  2. 30% — Profit is $24, and $24 ÷ $80 × 100% = 30%.
  3. Plan B by $16 — Plan A costs 12 × $18 = $216, so Plan B is $16 cheaper.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources