Year 8 Mathematics · Number and Algebra · 10m read

Percentages — Increase, Decrease and Reverse

Ever noticed how a 20% off sale price doesn't feel like you'd expect, or wondered what the *original* price actually was? Percentage changes—increases, decreases, and working backwards to find what something cost before—are the math behind sales, interest on savings, and tax. Master these skills and you'll spot deals, understand your pay slip, and ace the financial maths that shows up constantly in VCE and real life.

Learning objective: Calculate percentage increase/decrease, find original values and apply to financial problems.

By the end of this lesson you can:

• I calculate percentage increase and decrease.
• I find the original value from a percentage change.
• I apply percentages to profit/loss, GST and simple interest.

Percentages are price tags that change. A 20% discount shrinks the tag; GST grows it by 10%. Multipliers change the tag in one step.

Percentage applications

PERCENTAGE INCREASE:
New value = original × (1 + rate/100)
OR: find the increase, add to original.
Original = $80, increase 25%: $80 × 1.25 = $100.

PERCENTAGE DECREASE:
New value = original × (1 − rate/100)
Original = $80, decrease 15%: $80 × 0.85 = $68.

MULTIPLIER METHOD (efficient):
15% increase → multiply by 1.15
30% decrease → multiply by 0.70 (100% − 30% = 70%)

REVERSE PERCENTAGE (finding original):
You know the FINAL value after a change. Work backwards.
If $68 is 80% (after a 20% discount), what was the original?
$68 ÷ 0.80 = $85.

GST (Goods and Services Tax) in Australia = 10%:
Price + GST: multiply by 1.10.
If price includes GST, find GST amount: price ÷ 11 = GST component.
$110 includes GST. GST = $110/11 = $10. Pre-GST = $100.

PROFIT AND LOSS:
Profit/Loss = Selling price − Cost price.
% Profit = Profit / Cost price × 100.

SIMPLE INTEREST: I = P × r × t
P = principal, r = annual interest rate (as decimal), t = time in years.
I = 2000 × 0.05 × 3 = $300.

Percentage rules

• Multiplier is most efficient: Instead of finding 15% then adding: just × 1.15. Fast, one-step, fewer errors.
• Reverse = divide by the multiplier: If something is 75% of the original (25% off), divide the sale price by 0.75 to find the original.
• GST: ÷ 11 rule: In Australia, to find the GST component in a GST-inclusive price, divide by 11. This only works for 10% GST.

Percentage moves

• Increase: A \times (1 + r)
• Decrease: A \times (1 - r)
• Reverse: \text{Original} = \dfrac{\text{New}}{1 \pm r}

Quick reference

| Change | Multiplier | Example ($80) |
|---|---|---|
| Increase 10% | × 1.10 | $88 |
| Decrease 25% | × 0.75 | $60 |
| Increase 50% | × 1.5 | $120 |
| Reverse: after +10% is $88 | ÷ 1.10 | $80 original |
| % change | change ÷ original × 100 | |

Worked example (Medium)

A jacket is on sale for $119. It's been reduced by 15%. Find the original price.

1. Identify multiplier. 15% decrease → multiplier 0.85. So $119 = 85% of original.
2. Reverse calculation. Original = $119 ÷ 0.85 = $140.
3. Check. $140 × 0.85 = $119 ✓

Answer: Original price = $140

Worked example (Hard)

Apply advanced understanding of percentages — increase, decrease and reverse to a challenging real-world scenario. Demonstrate systematic analysis using key concepts from this topic.

1. Analyse the context. Identify the key principles of percentages — increase, decrease and reverse that are relevant. Consider what specific knowledge is needed to address this scenario.
2. Apply specific knowledge. Use precise mathematical terminology to analyse the situation. Show how concepts interact and build upon each other.
3. Evaluate implications. Consider the broader significance. How does this analysis connect to other aspects of maths?
4. Form a conclusion. Synthesise your analysis into a well-reasoned conclusion supported by specific evidence and terminology.

Answer: Analysis demonstrates deep understanding of percentages — increase, decrease and reverse with specific terminology, systematic reasoning, and real-world application.

Common mistakes

• Providing surface-level responses without depth Students state facts about percentages — increase, decrease and reverse without explaining underlying mechanisms, reasoning, or significance. Fix: Always explain WHY and HOW, not just WHAT. In mathematics assessments, depth of explanation distinguishes high-achieving responses from basic recall.
• Not using precise mathematical terminology Students use everyday language when specific terms would demonstrate stronger understanding. Fix: Build and use a vocabulary of key terms for this topic. Precise terminology shows understanding and earns higher marks in mathematics assessments.

Percentages — Increase, Decrease and Reverse — connecting to the bigger picture

Understanding percentages — increase, decrease and reverse extends beyond this single topic — it connects to broader themes across maths and other disciplines. Recognising these connections transforms isolated knowledge into an integrated framework that supports deeper analysis and creative problem-solving.

In professional practice, experts use conceptual frameworks — organised structures of related ideas — to approach new problems efficiently. They don't recall every fact; they activate connections between concepts that help them identify what's relevant. This is why the most effective study approach is not memorising more facts, but building richer connections between the facts you already know.

Active learning strategies that build these connections include: relating new concepts to personal experience, comparing ideas across different topics, applying knowledge to unfamiliar scenarios, and teaching concepts to others. Each of these forces you to process information more deeply than passive reading or highlighting. The goal is to develop the kind of flexible understanding that allows you to analyse any question about percentages — increase, decrease and reverse — even ones you haven't seen before — by drawing on a network of connected ideas.

Practice questions

1. A phone costs $320 after a 20% price increase. What was the original price?

   1. $256
   2. $264
   3. $280
   4. $400

   (1 marks)

2. Find the simple interest on $5000 at 4% per year for 2.5 years.

   (2 marks)

Summary

• Calculate percentage increase and decrease.
• Find the original value from a percentage change.
• Apply percentages to profit/loss, GST and simple interest.