Percentages appear in everyday life constantly, discounts, tax, tips, test scores, interest rates. This guide covers every type of percentage calculation with clear formulas and worked examples you can apply immediately.
What Is a Percentage?
A percentage is a number expressed as a fraction of 100. The word comes from the Latin per centum, meaning "by the hundred." When you say 35%, you mean 35 out of every 100, or equivalently, 35/100 = 0.35 as a decimal.
Percentages make comparisons between different quantities straightforward. Saying "the tax rate is 20%" communicates the same rate regardless of whether the item costs $5 or $50,000.
Type 1: Finding X% of a Number
The most common percentage calculation: "What is 20% of 85?"
Formula: Result = (Percentage ÷ 100) × Number Example: What is 20% of 85? Result = (20 ÷ 100) × 85 = 0.20 × 85 = 17 Example: What is 7.5% of $240? Result = 0.075 × 240 = $18
Mental math shortcut: To find 10%, move the decimal point one place left. To find 5%, halve the 10% value. To find 20%, double the 10% value. To find 15%, add the 10% and 5% values together.
15% of 80: 10% of 80 = 8 5% of 80 = 4 15% = 8 + 4 = 12
Type 2: What Percent Is X of Y?
This finds the percentage that one number represents of another: "42 is what percent of 168?"
Formula: Percentage = (Part ÷ Whole) × 100 Example: What percent is 42 of 168? Percentage = (42 ÷ 168) × 100 = 0.25 × 100 = 25% Example: A student scored 73 out of 80. What percentage is that? Percentage = (73 ÷ 80) × 100 = 0.9125 × 100 = 91.25%
Type 3: Percentage Increase
Use this when a value has grown: a price has risen, a population has grown, or a salary has increased.
Formula: Percentage increase = ((New − Old) / Old) × 100 Example: A shirt cost $40. It now costs $52. What is the percentage increase? Percentage increase = ((52 − 40) / 40) × 100 = (12 / 40) × 100 = 30% Example: A company's revenue grew from $2.4M to $3.1M. What was the growth rate? Percentage increase = ((3.1 − 2.4) / 2.4) × 100 = (0.7 / 2.4) × 100 ≈ 29.2%
Type 4: Percentage Decrease
Use this when a value has fallen: a price has dropped, a weight has reduced, or a rate has declined.
Formula: Percentage decrease = ((Old − New) / Old) × 100 Example: A laptop was $1,200. It is now $900. What is the percentage decrease? Percentage decrease = ((1200 − 900) / 1200) × 100 = (300 / 1200) × 100 = 25%
Percentage increase and decrease use the same formula, the only difference is which value is larger. Both divide by the original (Old) value, not the new one.
Type 5: Finding the Original Value (Reverse Percentage)
This is the trickiest type: you know the final amount after a percentage was applied, and you want to recover the original. A common mistake is to just apply the percentage again, that gives the wrong answer.
After an increase
Formula: Original = Final ÷ (1 + Percentage/100) Example: After a 25% increase, a price is $75. What was the original price? Original = 75 ÷ (1 + 0.25) = 75 ÷ 1.25 = $60 Check: 60 × 1.25 = 75 ✓ Common mistake: 75 − 25% of 75 = 75 − 18.75 = $56.25 ✗ (That calculates 25% of the NEW price, not the original.)
After a decrease
Formula: Original = Final ÷ (1 − Percentage/100) Example: After a 20% discount, an item costs $48. What was the original price? Original = 48 ÷ (1 − 0.20) = 48 ÷ 0.80 = $60 Check: 60 × 0.80 = 48 ✓
Type 6: Applying Multiple Percentage Changes
When consecutive percentage changes are applied, they cannot simply be added together. A 10% increase followed by a 10% decrease does not return to the original value.
Example: $100 increased by 10%, then decreased by 10%. After increase: $100 × 1.10 = $110 After decrease: $110 × 0.90 = $99 (not $100!) The combined effect: multiply the multipliers. 1.10 × 0.90 = 0.99 → net change of −1% Two consecutive 20% increases: 1.20 × 1.20 = 1.44 → net increase of 44% (not 40%)
Percentages in Real Life
VAT and Sales Tax
Price including 20% VAT = original price × 1.20 Price excluding VAT from VAT-inclusive price = VAT price ÷ 1.20 Item listed at $84 including 20% VAT: Pre-VAT price = 84 ÷ 1.20 = $70 VAT amount = $84 − $70 = $14
Discounts and Sale Prices
Sale price = Original price × (1 − Discount%/100) A $120 jacket with 35% off: Sale price = 120 × (1 − 0.35) = 120 × 0.65 = $78 Saving = $120 − $78 = $42
Tip Calculations
15% tip on $63.40: Tip = 0.15 × 63.40 = $9.51 Total = $63.40 + $9.51 = $72.91 Quick mental math: 10% of $63.40 = $6.34 5% = $3.17 15% = $6.34 + $3.17 = $9.51
Profit Margin
Profit margin = ((Revenue − Cost) / Revenue) × 100 Revenue = $800, Cost = $560 Profit = $240 Profit margin = (240 / 800) × 100 = 30%
Quick Reference Card
| Problem type | Formula |
|---|
| X% of Y | (X/100) × Y |
| X is what % of Y? | (X/Y) × 100 |
| % increase from A to B | ((B−A)/A) × 100 |
| % decrease from A to B | ((A−B)/A) × 100 |
| Original before a % increase | Final ÷ (1 + %/100) |
| Original before a % decrease | Final ÷ (1 − %/100) |
Use our free
Percentage Calculator to solve any of these problem types instantly, including reverse percentages, percentage change, and finding the original value after tax or discount.
