Percentage Increase and Decrease

1Summary 重點摘要

Learn how to calculate percentage increase and decrease, know actual price after markup and markdown.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Percentage change formulaThe percentage change formula: percentage change = (new − original) ÷ original × 100%. A positive result is an increase, a negative one a decrease. Example: last month's takings $8000, this month $10000 — percentage change = (10000 − 8000) ÷ 8000 × 100% = 25%, an increase of 25%. This formula is the foundation of the whole topic; memorise it. A common error is dividing by the new value instead of the original; remember, the comparison is always against the original, so the denominator is the original.
2. Step 2: How to find new valueWith the original and the percentage change known, find the new value: for an increase of x%, new = original × (1 + x%); for a decrease of x%, new = original × (1 − x%). Example: a shirt originally $100 rises 20%, so the new price = 100 × 1.2 = $120. Converting x% to a decimal first speeds the work, e.g. 20% = 0.2. A common error is forgetting the 1 when increasing, computing 100 × 0.2 = 20; remember, the 1 in (1 + x%) stands for the original value itself.
3. Step 3: How to find original valueWith the new value and the percentage change known, find the original: original = new ÷ (1 ± x%). Example: a shirt sells at $80 after a 20% cut, so the original = 80 ÷ (1 − 20%) = 80 ÷ 0.8 = $100. Finding the original uses division, because original-to-new is multiplication, and multiplication reverses into division. A common error is computing 80 × 1.2 = 96 as the original; the check: 100 less 20% gives 80, back to the new value, so 100 is the true original. Whenever the original is sought, ask first: 'original times what equals the new value?'
4. Step 4: Common pitfallsThe classic trap: up 20% then down 20% does not return to the start. Example: $100 up 20% becomes $120; then down 20%, 120 × 0.8 = $96 — actually $4 cheaper. The reason is different bases: the first step uses 100 as the base, the second uses 120, so the same 20% stands for $20 first and $24 later. Exams love this trap; in answers, write clearly that the bases differ so the price does not return to the start, and show both steps of working as proof.
5. Step 5: Discount applicationsDiscounts are percentage decrease at work. Example: an item marked $200 at 20% off — paying 80% — sells for 200 × 0.8 = $160, saving $40; at half price, paying 50%, it sells for $100. Note that '20% off' and 'paying 80%' are two wordings of one event, and exams use both. A common error is reading 20% off as paying 20%, computing 200 × 0.2 = $40 — the price and the saving have swapped places; remember, the discount figure is the fraction paid, and the reduction is 100% minus that.
6. Step 6: Growth rate exampleGrowth rate combines the techniques of the whole lesson. Example: a town's population grows from 50000 to 60000 — growth rate = (60000 − 50000) ÷ 50000 × 100% = 10000 ÷ 50000 × 100% = 20%, an increase of two tenths. The reverse: if a population grows 20% to reach 60000, the original = 60000 ÷ 1.2 = 50000. A common error is dividing by the new value in the growth formula, giving 10000 ÷ 60000 ≈ 16.7%; the check is to multiply the original by (1 + 20%) and see whether it returns to the new value.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Percentage change = (New - Old) ÷ Old × 100%
• Increase x%: new = original × (1 + x%)
• Decrease x%: new = original × (1 - x%)
• After increase then decrease won't be back to original
• Discount = markdown