Percentage Math: Every Formula You Need

Percentages appear everywhere: discounts, interest rates, election results, nutrition labels, grade scores. Most percentage problems fall into one of six patterns. Learn these once and you'll never be confused again.

The Six Core Percentage Problems

1. What is X% of Y?

Formula: Result = (X ÷ 100) × Y

Example: What is 15% of 240? - 15 ÷ 100 = 0.15 - 0.15 × 240 = 36

Use case: Calculating a tip, a discount amount, a tax.

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2. X is what percent of Y?

Formula: Percentage = (X ÷ Y) × 100

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

Use case: "I scored 45 out of 180 — what percentage is that?"

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3. X is Y% of what number?

Formula: Whole = X ÷ (Y ÷ 100) = X × (100 ÷ Y)

Example: 36 is 15% of what? - 36 ÷ 0.15 = 240

Use case: You paid $36 in tax (15%) — what was the pre-tax amount?

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4. Percentage increase / decrease

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

Example: Price rose from $80 to $100 - (100 − 80) ÷ 80 = 0.25 - 0.25 × 100 = +25%

Example: Price fell from $100 to $80 - (80 − 100) ÷ 100 = −0.20 - −0.20 × 100 = −20%

⚠️ Note: a 25% increase is NOT reversed by a 25% decrease. Going up 25% then down 25% gives you: - $100 → $125 → $93.75 (not $100)

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5. Adding a percentage (gross-up)

Formula: New value = Old × (1 + X/100)

Example: $200 + 20% VAT - 200 × 1.20 = $240

To find the pre-tax amount from the gross amount: - Pre-tax = Gross ÷ (1 + X/100) - $240 ÷ 1.20 = $200

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6. Compound percentage

When applying multiple percentage changes sequentially, don't add them — multiply:

Example: 10% increase followed by 5% decrease - Multiplier: 1.10 × 0.95 = 1.045 → +4.5% overall (not +5%)

Compound interest formula:

Final = Principal × (1 + rate)^n

Where rate is the per-period rate and n is the number of periods.

$1,000 at 5% annual interest for 3 years: - 1000 × (1.05)³ = 1000 × 1.157625 = $1,157.63

Quick Mental Math Shortcuts

Find 10%: Move the decimal point one place left. 10% of 85 = 8.5

Find 1%: Divide by 100. 1% of 364 = 3.64

Build up from 10% and 1%: - 15% of 80 = 10% + 5% = 8 + 4 = 12 - 25% of any number = divide by 4 - 33% ≈ divide by 3 - 50% = divide by 2 - 75% = ¾ = × 0.75

Common Percentage Traps

"Of" vs. "off"

• 20% of $50 = $10 (you're finding 20% of the amount)
• 20% off $50 = $40 (you're subtracting 20% from the amount)
• These are NOT the same calculation (though the discount amount is the same — $10 in this case)

Percentage points vs. percent change

• "The interest rate rose from 2% to 3%" = 1 percentage point increase, but a 50% increase in the rate itself
• "The approval rating fell 5 percentage points" (from 45% to 40%) ≠ "fell 5%"

Headlines often confuse these deliberately or accidentally.

Averaging percentages

Wrong: Average of 60% and 80% = 70% Only correct if the sample sizes are equal.

Correct approach (weighted average): - 60% on a 10-question test + 80% on a 50-question test - Total correct: (0.6 × 10) + (0.8 × 50) = 6 + 40 = 46 - Total questions: 60 - Average: 46 ÷ 60 = 76.7% (not 70%)

Practical Reference Table

What you want	Formula	Example
X% of Y	X/100 × Y	20% of 150 = 30
X as % of Y	X/Y × 100	30/150 × 100 = 20%
% change	(New−Old)/Old × 100	(180−150)/150 × 100 = 20%
Add X% to Y	Y × (1+X/100)	150 × 1.20 = 180
Remove X% from Y	Y × (1−X/100)	180 × (1−1/6) = 150
Y after X% increase is Z, find Y	Z/(1+X/100)	180/1.20 = 150

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