Three modes: percent of a number, share of a total, and change rate.
Percentage formulas
X% of Y = Y × X ÷ 100
X is what % of Y = X ÷ Y × 100%
Change X→Y = (Y − X) ÷ X × 100%
What is a percentage?
A percentage expresses a number as a fraction of 100. The word "percent" comes from Latin "per centum" meaning "by the hundred". So 25% means 25 out of 100, or 0.25 as a decimal. Percentages make it easy to compare proportions across different sizes — a 20% discount on a ¥100 item and a ¥1,000 item both mean the same relative reduction, even though the yuan amounts differ.
Three common percentage calculations
- X% of Y (find a part): What is 15% of 200? = 200 × 15 ÷ 100 = 30.
- X is what % of Y (find the rate): 30 is what % of 200? = 30 ÷ 200 × 100% = 15%.
- Percentage change (find increase/decrease): From 100 to 130 = (130 − 100) ÷ 100 × 100% = +30%.
Practical examples
- Sales tax: A ¥800 item with 13% VAT. Tax = 800 × 13 ÷ 100 = ¥104. Total = ¥904.
- Tip at a restaurant: 15% tip on a ¥368 bill = 368 × 0.15 = ¥55.20. Total = ¥423.20.
- Salary raise: Monthly salary ¥12,000 with a 10% increase = 12,000 × 1.10 = ¥13,200. The raise is ¥1,200.
- Stock gain/loss: A stock bought at ¥50 now worth ¥62. Gain % = (62 − 50) ÷ 50 × 100% = +24%.
- Exam score: 88 correct out of 100 questions = 88 ÷ 100 × 100% = 88%.
The asymmetry trap
A common mistake: a 50% gain followed by a 50% loss does not bring you back to the start. If you invest ¥100 and gain 50%, you have ¥150. A 50% loss on ¥150 leaves you with ¥75 — a net loss of ¥25. This is why percentage gains and losses are not symmetric. To recover a 50% loss, you need a 100% gain.
Tips for working with percentages
- Convert to decimals first. 15% = 0.15. Multiply directly: 200 × 0.15 = 30. This is faster than dividing by 100 each time.
- For percentage increases, use the multiplier. A 20% increase = multiply by 1.20. A 20% decrease = multiply by 0.80.
- Watch the base. The "of" word tells you the base. "20% of 500" uses 500 as the base, not 100.
- Don't average percentages. If a fund returns +50% one year and −50% the next, the average is 0% but the actual return is −25%.
FAQ
How do I calculate percentage change?
Change % = (New − Original) ÷ Original × 100%. Example: 100 → 130 is +30%; 130 → 100 is −23.1% (the base differs, so up and down are not symmetric).
How do I find X% of Y?
Multiply Y by X and divide by 100. Example: 15% of 200 = 200 × 15 ÷ 100 = 30.
Are percentage gains and losses symmetric?
No. A 50% gain then a 50% loss from ¥100 = ¥75, not ¥100. You need a 100% gain to recover a 50% loss.
How do I calculate a percentage increase?
New value = original × (1 + percentage ÷ 100). A 20% increase on ¥12,000 = 12,000 × 1.20 = ¥14,400.
How do I reverse a percentage?
If a price after 20% off is ¥80, the original = 80 ÷ 0.80 = ¥100. Divide by (1 − discount) to find the original.