Percentage Formulas: Complete Reference with Examples
Percentages appear in almost every numerical context — discounts, taxes, exam scores, investment returns, tips, and statistics. This reference covers every common percentage formula with worked examples. Use the percentage calculator to apply any of these instantly without doing the arithmetic manually.
Core Percentage Formulas
| Calculation | Formula | Example | Result |
|---|---|---|---|
| Find X% of Y | Result = (X ÷ 100) × Y | 15% of 240 | 36 |
| X is what % of Y | Result = (X ÷ Y) × 100 | 36 is what % of 240? | 15% |
| Percentage change | ((New − Old) ÷ |Old|) × 100 | 80 to 100 | +25% |
| Increase Y by X% | Result = Y × (1 + X/100) | Increase 400 by 15% | 460 |
| Decrease Y by X% | Result = Y × (1 − X/100) | Decrease 400 by 15% | 340 |
Reverse Percentage (Finding the Original Value)
Reverse percentage finds the original value before a percentage was applied. Common use: a sale price with tax included, and you want the pre-tax amount.
- Formula: Original = Final ÷ (1 + X/100) for an increase
- Formula: Original = Final ÷ (1 − X/100) for a decrease
Example: A product costs $108 including 8% tax. Pre-tax price = 108 ÷ 1.08 = $100.
Example: A coat costs $85 after a 15% discount. Original price = 85 ÷ 0.85 = $100.
Practical Percentage Calculations
Sales tax
To add tax: Total = Price × (1 + tax rate/100). $45 + 8% tax = 45 × 1.08 = $48.60. To check tax included: Tax amount = Total − (Total ÷ 1.08) = 48.60 − 45 = $3.60.
Discount
Sale price = Original × (1 − discount/100). $120 at 25% off = 120 × 0.75 = $90. Saving = $30, which is 25% of $120.
Restaurant tip
Tip = Bill × (tip%/100). 20% tip on $65 bill = 65 × 0.20 = $13. Total = $78. Per person (4 people) = $78 ÷ 4 = $19.50.
Exam score
Score % = (Marks obtained ÷ Total marks) × 100. 68 out of 80 = (68 ÷ 80) × 100 = 85%. To find marks needed for 70%: 0.70 × 80 = 56 marks.
Compound percentage change
When a percentage change is applied multiple times (e.g., annual interest or inflation): Final = Initial × (1 + r)ⁿ, where r is the rate per period and n is the number of periods. $1,000 invested at 5% annual return for 10 years = 1000 × (1.05)¹⁰ = $1,628.89. Note: this is not 1000 + 50% (linear) — it is compounded, which gives a higher result.
Common Percentage Mistakes
Percentages of percentages
"A 5% increase followed by a 5% decrease" does NOT return to the original. 100 × 1.05 × 0.95 = 99.75 — a 0.25% net loss. This asymmetry matters in investing, where recovering from a loss requires a larger gain than the loss itself.
Percentage points vs percentages
An interest rate rising from 2% to 3% is a 1 percentage point increase but a 50% percentage change ((3−2)/2 × 100). These are very different statements and are frequently confused in financial news.
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