Percentages show up everywhere: discounts, tips, taxes, test scores, investment returns, loan interest. Yet many people struggle with them. "What's 15% of $87?" or "How much did the price increase by?" This guide gives you the formulas, shortcuts, and mental math tricks to handle any percentage problem confidently.
Key Takeaways
- 1Three types: find X% of Y, find what % X is of Y, find the whole from a part
- 2Percentage change: ((New - Old) / Old) × 100; positive = increase, negative = decrease
- 3Stacked discounts multiply, not add: 30% off + 20% off = 44% off total, not 50%
- 4To find original after discount, divide by (1 - discount rate), don't add the percentage
- 5Mental math: 10% = move decimal left; 15% = 10% + half; 20% = 10% × 2
Percentage Basics
"Percent" means "per hundred." 25% means 25 out of 100, or 25/100, or 0.25. This simple idea unlocks all percentage calculations.
| Percentage | Fraction | Decimal |
|---|---|---|
| 10% | 1/10 | 0.10 |
| 25% | 1/4 | 0.25 |
| 33.33% | 1/3 | 0.333... |
| 50% | 1/2 | 0.50 |
| 75% | 3/4 | 0.75 |
| 100% | 1/1 | 1.00 |
| 150% | 3/2 | 1.50 |
To convert percent to decimal: divide by 100 (move decimal two places left). 25% → 0.25. To convert decimal to percent: multiply by 100. 0.25 → 25%.
2The Three Types of Percentage Problems
Every percentage question falls into one of three categories. Master these, and you can solve anything.
| Type | Question Format | Formula |
|---|---|---|
| Find percentage of number | What is 20% of 150? | Number × (Percent/100) |
| Find what percent | 30 is what percent of 150? | (Part/Whole) × 100 |
| Find the whole | 30 is 20% of what number? | Part / (Percent/100) |
Example: Type 1: Finding a Percentage of a Number
Scenario
A $85 item is 20% off. How much do you save?
Solution
$85 × 0.20 = $17 savings. Final price: $85 - $17 = $68.
Example: Type 2: Finding What Percent
Scenario
You scored 42 out of 50 on a test. What percentage?
Solution
(42 ÷ 50) × 100 = 84%.
Example: Type 3: Finding the Whole
Scenario
$30 is 15% of the original price. What was the original?
Solution
$30 ÷ 0.15 = $200 original price.
Percentage Change (Increase/Decrease)
Percentage change shows how much something grew or shrank relative to its original value. This is crucial for comparing prices, salaries, investments, and more.
Percentage Change = ((New Value - Old Value) / Old Value) × 100. Positive result = increase. Negative result = decrease.
| Old Value | New Value | Change | Calculation |
|---|---|---|---|
| $100 | $120 | +20% | (120-100)/100 × 100 |
| $100 | $80 | -20% | (80-100)/100 × 100 |
| 50 kg | 55 kg | +10% | (55-50)/50 × 100 |
| $1000 | $1500 | +50% | (1500-1000)/1000 × 100 |
Percentage changes aren't reversible! If a price increases 20% then decreases 20%, you don't get the original. $100 → $120 (+20%) → $96 (-20%). You lost $4.
4Discount Calculations
Sales, coupons, and markdowns all use percentages. Here's how to calculate your actual savings and final price.
| Original Price | Discount | You Save | You Pay |
|---|---|---|---|
| $100 | 10% | $10 | $90 |
| $100 | 25% | $25 | $75 |
| $100 | 33% | $33 | $67 |
| $100 | 50% | $50 | $50 |
| $100 | 70% | $70 | $30 |
Quick mental math: For 10%, move decimal left. For 5%, take half of 10%. For 15%, add 10% + 5%. For 20%, double 10%. Example: 15% of $80 = $8 + $4 = $12.
Example: Stacked Discounts
Scenario
An item is 30% off, then you have an extra 20% coupon. Is that 50% off?
Solution
No! Apply sequentially. $100 × 0.70 = $70 (after 30% off). $70 × 0.80 = $56 (after extra 20%). Total discount: 44%, not 50%.
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Open Discount CalculatorTips, Taxes & Markups
Adding percentages (tips, taxes, markups) is the reverse of discounts. You're increasing a base amount.
| Base Amount | Addition | Total |
|---|---|---|
| $50 meal | 15% tip | $50 × 1.15 = $57.50 |
| $50 meal | 20% tip | $50 × 1.20 = $60 |
| $100 item | 8% sales tax | $100 × 1.08 = $108 |
| $200 item | 18% GST | $200 × 1.18 = $236 |
Example: Tipping on Pre-Tax Amount
Scenario
Meal: $50. Tax: 8%. Tip: 20% on pre-tax. What's the total?
Solution
Tax: $50 × 0.08 = $4. Tip: $50 × 0.20 = $10. Total: $50 + $4 + $10 = $64.
Tip on pre-tax or post-tax? Either is acceptable. Pre-tax is more common. For 20% of post-tax total: $54 × 0.20 = $10.80 tip instead of $10.
6Reverse Percentage Calculations
Sometimes you know the final amount and need to find the original. These "reverse" calculations trip up many people.
| Scenario | Given | Find | Formula |
|---|---|---|---|
| Original price after discount | $80 (after 20% off) | Original | $80 / 0.80 = $100 |
| Pre-tax price | $108 (incl. 8% tax) | Pre-tax | $108 / 1.08 = $100 |
| Original before increase | $120 (after 20% raise) | Original | $120 / 1.20 = $100 |
Common mistake: To find original price after 20% discount, DON'T add 20% to the sale price. $80 + 20% = $96, not $100. Instead, divide by 0.80.
Example: Finding Pre-Tax Price
Scenario
Receipt shows $118 total with 18% GST included. What was the base price?
Solution
Base price = $118 / 1.18 = $100. GST amount = $118 - $100 = $18.
Mental Math Shortcuts
You don't always need a calculator. These tricks let you estimate percentages quickly.
| Percentage | Mental Trick |
|---|---|
| 1% | Divide by 100 (move decimal 2 left) |
| 10% | Divide by 10 (move decimal 1 left) |
| 5% | Half of 10% |
| 15% | 10% + 5% (10% + half of 10%) |
| 20% | 10% × 2 |
| 25% | Divide by 4 |
| 33% | Divide by 3 |
| 50% | Divide by 2 |
| 75% | 50% + 25% |
Example: Quick Tip Calculation
Scenario
Calculate 18% tip on $73 bill
Solution
10% = $7.30. 8% = $7.30 × 0.8 ≈ $5.84. Total: $7.30 + $5.84 ≈ $13.14. Or simpler: 20% = $14.60, so 18% is slightly less ≈ $13.
For 15% tip: Find 10% and add half. $60 bill → 10% is $6, half is $3, so 15% = $9. Takes 3 seconds.
8Common Percentage Mistakes
These errors are surprisingly common. Knowing them helps you avoid costly mistakes.
| Mistake | Example | Correct Approach |
|---|---|---|
| Adding successive percentages | 30% off + 20% off = 50% off | Apply sequentially: 0.70 × 0.80 = 0.56 (44% off) |
| Reversing by adding | Original from $80 (20% off): $80 + 20% | Divide: $80 / 0.80 = $100 |
| Percentage point vs percent | 5% to 10% is "5% increase" | It's 5 percentage points, or 100% increase |
| Base confusion | "50% more" vs "50% of" | "50% more" = 150% of original |
| Order matters with tax/discount | Tax first then discount | Usually discount first, then tax on discounted price |
Percentage points vs percent: Interest rate rising from 2% to 4% is a 2 percentage point increase, but a 100% increase (it doubled). Media often confuses these.
Frequently Asked Questions
What is the formula for calculating percentage?
There are three main formulas: 1) Percentage of a number: Number × (Percent/100). Example: 20% of 150 = 150 × 0.20 = 30. 2) What percent is X of Y: (X/Y) × 100. Example: 30 is what % of 150? (30/150) × 100 = 20%. 3) Find the whole: Part / (Percent/100). Example: 30 is 20% of what? 30 / 0.20 = 150.
How do I calculate percentage increase or decrease?
Use the formula: ((New Value - Old Value) / Old Value) × 100. If the result is positive, it's an increase; negative means decrease. Example: Price went from $80 to $100. Change = (100-80)/80 × 100 = 25% increase.
How do I find the original price before a discount?
Divide the sale price by (1 - discount rate). For a 20% discount: Original = Sale Price / 0.80. Example: Item costs $64 after 20% off. Original = $64 / 0.80 = $80. Don't make the mistake of adding 20% to the sale price.
How do I calculate a tip quickly in my head?
For 10%: Move decimal one place left. For 15%: Find 10%, add half. For 20%: Find 10%, double it. Example: $73 bill. 10% = $7.30, so 20% = $14.60, and 15% = $7.30 + $3.65 ≈ $11.
What's the difference between "percent" and "percentage points"?
Percent describes relative change; percentage points describe absolute change. If interest rises from 2% to 4%, that's 2 percentage points increase but a 100% relative increase (it doubled). This distinction matters in finance, statistics, and news reporting.