Percentage Calculations: The Complete Guide with Formulas

What Is a Percentage?

A percentage is a way of expressing a number as a fraction of 100. The word comes from the Latin "per centum" meaning "by the hundred." The symbol % means "out of 100."

So 45% means 45 out of every 100, which equals 0.45 as a decimal or 9/20 as a fraction.

Basic Percentage Formula

The fundamental percentage relationship is:

Part = Percentage × Whole / 100

This one formula answers three types of questions:

1. What is X% of a number? What is 30% of 250?

Part = 30 × 250 / 100 = 75

2. What percentage is one number of another? 35 is what percent of 140?

Percentage = 35 / 140 × 100 = 25%

3. Finding the whole from a percentage and part: 60 is 40% of what number?

Whole = 60 / 40 × 100 = 150

Percentage Change (Increase/Decrease)

Percentage change measures how much a value has changed relative to its original value:

Percentage change = (New Value − Old Value) / Old Value × 100

Example — Percentage increase: A salary went from $50,000 to $57,500.

Change = (57,500 − 50,000) / 50,000 × 100 = 15% increase

Example — Percentage decrease: A stock dropped from $80 to $64.

Change = (64 − 80) / 80 × 100 = -20% (20% decrease)

Important: Percentage increases and decreases are not symmetric. A 50% increase followed by a 50% decrease does NOT return to the original value:

• Start: $100
• +50%: $150
• -50%: $75 (not $100)

Discounts

Finding the sale price:

Sale price = Original price × (1 - discount%)

$200 item with 25% off:

Sale price = $200 × (1 - 0.25) = $200 × 0.75 = $150

Finding the discount amount:

Discount = Original price × discount% Discount = $200 × 0.25 = $50

Stacking discounts: Two discounts of 20% and 10% applied sequentially are NOT the same as 30% off:

• $100 − 20% = $80
• $80 − 10% = $72

The effective combined discount is 28%, not 30%.

Tax and Tips

Adding tax:

Price with tax = Price × (1 + tax rate)

$85 item with 8.5% tax:

$85 × 1.085 = $92.23

Calculating tip on a restaurant bill:

Tip = Bill × tip rate

$73 bill with 18% tip:

Tip = $73 × 0.18 = $13.14 Total = $73 + $13.14 = $86.14

Quick tip mental math:

• 10% tip: move decimal left one place ($73 → $7.30)
• 20% tip: double the 10% ($7.30 × 2 = $14.60)
• 15% tip: 10% + half of 10% ($7.30 + $3.65 = $10.95)

Working Backwards from a Percentage

If a price including 20% VAT is $120, what's the pre-tax price?

Pre-tax price = Price with tax / (1 + tax rate) Pre-tax = $120 / 1.20 = $100

Common mistake: subtracting 20% of $120 = $24, giving $96. Wrong — $96 + 20% = $115.20, not $120.

Grades and Academic Percentages

If you scored 87 points out of 120 total:

Grade = 87 / 120 × 100 = 72.5%

To find the minimum score needed for a 90% grade on a 120-point test:

Score = 90% × 120 = 108 points

Percent vs Percentage Point

These terms are often confused:

• If a tax rate changes from 20% to 25%, it increased by 5 percentage points (a simple arithmetic difference)
• But it increased by 25% as a relative change: (25-20)/20 × 100 = 25%

"Percentage point" refers to the arithmetic difference between two percentages. "Percent change" refers to the relative change. Politicians and marketers often exploit this ambiguity — always ask which one is meant.

Interest Rate Example

A central bank raises interest rates from 4% to 5%:

• Increase in percentage points: 1 percentage point
• Percentage increase in the rate: (5-4)/4 × 100 = 25%

The first is more accurate for describing rate changes. The second sounds more dramatic.

Use our free calculators: Percentage Calculator, Percentage Change Calculator, and Discount Calculator.