Percentage Formulas Cheat Sheet: Every Formula You Need

This cheat sheet covers every percentage formula you are likely to need, organized by category with clear examples. Bookmark this page for quick reference whenever you encounter a percentage problem.

The Five Basic Percentage Calculations

Every percentage problem falls into one of five categories. Master these and you can solve any percentage question.

1. Finding a Percentage of a Number

Formula: Result = (Percentage ÷ 100) × Number

This answers "What is X% of Y?"

Example: What is 35% of 800? Result = (35 ÷ 100) × 800 = 0.35 × 800 = 280

When to use: Calculating tips, discounts, tax amounts, commissions, portions of a total.

2. Finding What Percentage One Number Is of Another

Formula: Percentage = (Part ÷ Whole) × 100

This answers "X is what percent of Y?"

Example: 45 is what percent of 180? Percentage = (45 ÷ 180) × 100 = 0.25 × 100 = 25%

When to use: Test scores, budget tracking, completion rates, market share.

3. Percentage Change (Increase or Decrease)

Formula: % Change = ((New − Old) ÷ |Old|) × 100

This answers "What is the percentage change from X to Y?"

Example: Change from 200 to 250? % Change = ((250 − 200) ÷ 200) × 100 = (50 ÷ 200) × 100 = +25%

When to use: Price changes, salary raises, population growth, performance tracking.

4. Reverse Percentage (Finding the Whole)

Formula: Whole = Part ÷ (Percentage ÷ 100)

This answers "X is Y% of what?"

Example: 60 is 40% of what? Whole = 60 ÷ (40 ÷ 100) = 60 ÷ 0.40 = 150

When to use: Finding original prices from sale prices, pre-tax amounts, total from a known portion.

5. Percentage Difference

Formula: % Difference = (|A − B| ÷ ((A + B) ÷ 2)) × 100

This answers "What is the percentage difference between X and Y?"

Example: Difference between 80 and 120? % Difference = (40 ÷ 100) × 100 = 40%

When to use: Comparing two values without a clear before/after relationship.

Percentage Increase and Decrease Formulas

Apply a Percentage Increase

Formula: New Value = Original × (1 + Percentage ÷ 100)

Example: Increase 500 by 20%: New Value = 500 × 1.20 = 600

Apply a Percentage Decrease

Formula: New Value = Original × (1 − Percentage ÷ 100)

Example: Decrease 500 by 20%: New Value = 500 × 0.80 = 400

Find the Original After a Percentage Increase

Formula: Original = New Value ÷ (1 + Percentage ÷ 100)

Example: After a 25% increase, the value is 750. What was the original? Original = 750 ÷ 1.25 = 600

Find the Original After a Percentage Decrease

Formula: Original = New Value ÷ (1 − Percentage ÷ 100)

Example: After a 30% decrease, the value is 350. What was the original? Original = 350 ÷ 0.70 = 500

Compound Percentage Formulas

Compound Growth

Formula: Final = Initial × (1 + Rate)^Periods

Example: $1,000 growing at 8% annually for 10 years: Final = $1,000 × (1.08)^10 = $2,158.92

Compound Interest with Compounding Frequency

Formula: A = P × (1 + r/n)^(n×t)

Where P is principal, r is annual rate, n is compounding periods per year, t is years.

Example: $5,000 at 6% compounded monthly for 3 years: A = $5,000 × (1 + 0.06/12)^(12×3) = $5,000 × (1.005)^36 = $5,983.40

Continuous Compounding

Formula: A = P × e^(r×t)

Example: $5,000 at 6% continuously compounded for 3 years: A = $5,000 × e^(0.06×3) = $5,000 × e^0.18 = $5,986.09

Rule of 72

Formula: Years to Double ≈ 72 ÷ Rate

Example: At 9% growth, how long to double? 72 ÷ 9 = 8 years (approximately)

Sequential Percentage Changes

Stacked Percentage Changes

Formula: Final = Original × (1 ± p1) × (1 ± p2) × ... × (1 ± pn)

Example: $200 with 30% off then 10% off: Final = $200 × 0.70 × 0.90 = $126 (total discount: 37%, not 40%)

Overall Percentage Change from Sequential Changes

Formula: Overall % Change = ((1 ± p1) × (1 ± p2) × ... − 1) × 100

Example: A stock rises 20% then falls 15%: Overall = ((1.20) × (0.85) − 1) × 100 = (1.02 − 1) × 100 = +2%

Ratio and Proportion Formulas

Percentage to Fraction

Formula: Fraction = Percentage ÷ 100

Example: 37.5% = 37.5 ÷ 100 = 3/8

Fraction to Percentage

Formula: Percentage = Fraction × 100

Example: 5/8 = 0.625 × 100 = 62.5%

Common Percentage-Fraction Equivalents

Weighted Percentage Formula

Formula: Weighted Average = Σ(Value × Weight) ÷ Σ(Weights)

Example: Course grade with homework (30%), midterm (30%), final (40%): Scores: 92, 78, 85 Weighted = (92 × 0.30) + (78 × 0.30) + (85 × 0.40) = 27.6 + 23.4 + 34.0 = 85.0

Markup and Margin Formulas

Markup Percentage

Formula: Markup = ((Selling Price − Cost) ÷ Cost) × 100

Margin Percentage

Formula: Margin = ((Selling Price − Cost) ÷ Selling Price) × 100

Selling Price from Desired Margin

Formula: Selling Price = Cost ÷ (1 − Margin)

Example: Cost $40, target 40% margin: Selling Price = $40 ÷ 0.60 = $66.67

Converting Between Markup and Margin

Markup to Margin: Margin = Markup ÷ (1 + Markup) Margin to Markup: Markup = Margin ÷ (1 − Margin)

Percentage Point vs. Percentage

This is a critical distinction. When a rate changes from 8% to 10%:

• The increase is 2 percentage points (absolute)
• The increase is 25% (relative: 2 ÷ 8 × 100)

"Percentage points" measure the absolute difference between two percentages. "Percent" measures the relative change. News and reports often confuse these, leading to significant misinterpretation. Always clarify which one is being discussed.

Mental Math Shortcuts

• 1% trick: Find 1% (move decimal two places left), then multiply. 7% of 350: 1% = 3.5, so 7% = 24.5.
• Reversibility: X% of Y = Y% of X. So 8% of 50 = 50% of 8 = 4.
• Halving and doubling: 15% = 10% + 5%. Find 10%, halve it for 5%, add them.
• Subtract for discounts: 35% off means you pay 65%. Multiply by 0.65 directly.

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