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 "per hundred." When you say 40%, you mean 40 out of every 100 — or equivalently, the fraction 40/100 = 0.40.
The percent symbol % is shorthand for "÷ 100." So 35% simply means 35 ÷ 100 = 0.35. This conversion between percentage, fraction, and decimal is the key to every percentage calculation.
| Percentage | Fraction | Decimal | Meaning |
|---|---|---|---|
| 1% | 1/100 | 0.01 | 1 out of every 100 |
| 10% | 1/10 | 0.10 | 1 out of every 10 |
| 25% | 1/4 | 0.25 | 1 out of every 4 |
| 50% | 1/2 | 0.50 | Half |
| 75% | 3/4 | 0.75 | Three quarters |
| 100% | 1/1 | 1.00 | The whole thing |
| 150% | 3/2 | 1.50 | One and a half times |
Converting a percentage to a decimal is always the same operation: divide by 100 (or move the decimal point two places to the left). 37% → 0.37. 8.5% → 0.085. 120% → 1.20. This decimal form is what you always use in actual calculations.
The 3 Core Percentage Formulas
Every percentage problem — no matter how complex it appears — is a variation of one master equation. This equation has three variables: the Part, the Whole, and the Percentage. Know any two and you can always find the third.
A: 80 × 0.20 = 16
A: (16 ÷ 80) × 100 = 20%
A: 16 ÷ 0.20 = 80
Find the Part — "What is X% of a Number?"
This is the most common percentage calculation. You know the whole amount and the percentage, and you want to find the actual value that percentage represents.
Example: What is 35% of 200? → 200 × 0.35 = 70
Step-by-Step Method
- Identify the whole (the base amount) and the percentage you need to find.
- Convert the percentage to a decimal by dividing by 100. E.g., 35% → 35 ÷ 100 = 0.35.
- Multiply the whole by the decimal. E.g., 200 × 0.35 = 70.
- The result is your Part. In this example, 35% of 200 = 70.
Example 1: What is 15% of $60?
Step 1: 15 ÷ 100 = 0.15
Step 2: $60 × 0.15 = $9.00
Example 2: What is 8% of 3,500?
Step 1: 8 ÷ 100 = 0.08
Step 2: 3,500 × 0.08 = 280
Example 3: What is 6.5% of $12,000? (loan interest)
Step 1: 6.5 ÷ 100 = 0.065
Step 2: $12,000 × 0.065 = $780
Example 4: What is 110% of 400? (result exceeds original)
Step 1: 110 ÷ 100 = 1.10
Step 2: 400 × 1.10 = 440
Use the main Percentage Calculator to find any percentage of any number instantly, with full step-by-step working.
Find the Percentage — "What % is A of B?"
You have two numbers and want to express one as a percentage of the other. Common uses: test scores, what fraction of a budget was spent, how much of a group meets a condition.
Example: 45 is what % of 180? → (45 ÷ 180) × 100 = 25%
Step-by-Step Method
- Identify the Part (the number you're expressing as a percentage) and the Whole (the base/total).
- Divide the Part by the Whole. E.g., 45 ÷ 180 = 0.25.
- Multiply by 100 to convert to a percentage. E.g., 0.25 × 100 = 25%.
Example 1: A student scored 72 out of 90 on a test. What is their percentage?
(72 ÷ 90) × 100 = 0.8 × 100 = 80%
Example 2: A shop sold 340 of its 500 units. What % was sold?
(340 ÷ 500) × 100 = 0.68 × 100 = 68%
Example 3: You spent $37.50 out of a $150 budget. What percentage did you use?
(37.50 ÷ 150) × 100 = 0.25 × 100 = 25%
Example 4: 14 people out of 56 surveyed prefer option A. What % prefer A?
(14 ÷ 56) × 100 = 0.25 × 100 = 25%
Always divide by the base/reference value (the Whole). "30 out of 120" → 30 is the Part, 120 is the Whole. If you accidentally divide the wrong way (120 ÷ 30 = 4 → 400%) you will get a nonsensical answer.
The question phrasing tells you: "A is what % of B?" → B is always the Whole (denominator).
Find the Whole — Reverse Percentage
You know a part and the percentage it represents, but you need to find the original total. This is called a reverse percentage. Common examples: finding the original price before a discount, finding a pre-tax price, or recovering a total from a known slice.
Example: 30 is 25% of what number? → 30 ÷ 0.25 = 120
Step-by-Step Method
- Identify the Part and the Percentage it represents.
- Convert the percentage to a decimal. E.g., 25% → 0.25.
- Divide the Part by the decimal. E.g., 30 ÷ 0.25 = 120.
- Verify: 25% of 120 = 120 × 0.25 = 30 ✓
Example 1: A shirt costs $42 after a 30% discount. What was the original price?
After 30% off, you paid 70% of the original → $42 is 70% of original
Original = $42 ÷ 0.70 = $60.00
Example 2: VAT (20%) inclusive price is £96. What is the pre-VAT price?
£96 includes 100% + 20% = 120% → £96 is 120% of the pre-VAT price
Pre-VAT = £96 ÷ 1.20 = £80.00
Example 3: 15 students got an A, which is 12% of the class. How many students are in the class?
Total = 15 ÷ 0.12 = 125 students
Example 4: You saved $35, which is 7% of your monthly income. What is your monthly income?
Income = $35 ÷ 0.07 = $500
For reverse percentage calculations — including finding original prices before discounts — use the Discount Calculator (Find Original Price tab) or the main Percentage Calculator.
How to Calculate Percentage Increase
A percentage increase tells you how much a value has grown relative to its original size. It is not the same as the absolute increase — a $10 rise means very different things on a $20 item vs. a $10,000 item.
The result is always relative to the OLD (original) value — never the new value.
Step-by-Step Method
- Subtract Old from New to find the increase amount. New − Old = Increase.
- Divide by the Old value. Increase ÷ Old.
- Multiply by 100. Result is the % increase.
Example 1: A price rose from $40 to $52. What is the % increase?
Increase = $52 − $40 = $12
% Increase = ($12 ÷ $40) × 100 = 30% increase
Example 2: A salary went from $48,000 to $54,000. What % raise is that?
Increase = $6,000
% Increase = ($6,000 ÷ $48,000) × 100 = 12.5% raise
Example 3: Website traffic went from 8,200 to 11,070 visits. What % increase?
Increase = 2,870
% Increase = (2,870 ÷ 8,200) × 100 = 35% increase
Finding the New Value After a % Increase
Example: $200 increased by 15% → $200 × 1.15 = $230
Use the Percentage Increase Calculator for instant results with step-by-step working.
How to Calculate Percentage Decrease
A percentage decrease works exactly the same way as percentage increase, but the new value is smaller than the original. The formula is symmetric — the only change is subtracting New from Old instead.
Always divide by the OLD value. The result is always positive (0 to 100%).
Example 1: A product's price dropped from $80 to $60. What is the % decrease?
Decrease = $80 − $60 = $20
% Decrease = ($20 ÷ $80) × 100 = 25% decrease
Example 2: A company's headcount went from 450 to 378. What % reduction?
Decrease = 72
% Decrease = (72 ÷ 450) × 100 = 16% decrease
Example 3: Your phone bill was $95/month, now it's $76/month. What % did you save?
Decrease = $19
% Decrease = ($19 ÷ $95) × 100 = 20% decrease
Finding the New Value After a % Decrease
Example: $150 decreased by 20% → $150 × (1 − 0.20) = $150 × 0.80 = $120
Percentage Change Formula
Percentage change is the single unified formula that covers both increases and decreases. A positive result means an increase; a negative result means a decrease. It is the standard formula used in finance, science, and data analysis for comparing before-and-after values.
Positive = increase. Negative = decrease. Always divide by the OLD (original) value.
Example 1 (increase): Revenue: $120,000 → $156,000
% Change = ((156,000 − 120,000) ÷ 120,000) × 100 = (36,000 ÷ 120,000) × 100 = +30%
Example 2 (decrease): Errors per day: 24 → 18
% Change = ((18 − 24) ÷ 24) × 100 = (−6 ÷ 24) × 100 = −25%
Example 3 (no change): Value: 500 → 500
% Change = ((500 − 500) ÷ 500) × 100 = 0%
The most common mistake is dividing by the new value instead of the old. Always use the starting value (old, original, base) as the denominator. The formula measures "how much did it change relative to where it started?"
Also: percentage change is not reversible. A 50% increase followed by a 50% decrease does NOT return to the original. ($100 × 1.50 = $150; $150 × 0.50 = $75 — not $100.)
Use the Percentage Change Calculator for any before/after values with a visual result showing direction and magnitude.
Mental Math Shortcuts for Percentages
You don't always need a calculator. These shortcuts let you calculate the most common percentages in your head within seconds. The core trick is building from 10% (move the decimal left one place) as your anchor.
10% of 450 = 45
10% of $8.50 = $0.85
1% of 3,400 = 34
1% of $0.90 = $0.009
5% of 240 → 24 ÷ 2 = 12
5% of $150 → $15 ÷ 2 = $7.50
20% of 85 → 8.5 × 2 = 17
20% of $200 → $20 × 2 = $40
15% of $80 → $8 + $4 = $12
15% of 200 → 20 + 10 = 30
25% of 400 = 100
25% of $88 = $22
50% of 730 = 365
50% of $13.50 = $6.75
75% of 80 → 40 + 20 = 60
75% of $24 → $12 + $6 = $18
30% of 500 → 50 × 3 = 150
30% of $45 → $4.50 × 3 = $13.50
33% of 120 = 40
33% of $60 = $20
The "Building Block" Strategy
For any percentage not in the list above, build it from components using 1%, 5%, 10%, 25%, and 50% as building blocks. This avoids all long multiplication:
- 35% = 25% + 10%
- 45% = 50% − 5%
- 18% = 20% − 2% (where 2% = 1% × 2)
- 12% = 10% + 1% + 1%
- 7% = 5% + 1% + 1%
- 60% = 50% + 10%
- 90% = 100% − 10%
- 95% = 100% − 5%
Method 1 (build up): 25% of $240 = $60 · 10% of $240 = $24 · Total: $60 + $24 = $84
Method 2 (pay %, not discount %): 35% → you want 35/100 → 240 × 0.35 = 84. Or think: 240 × 35 ÷ 100 = 8400 ÷ 100 = $84
Real-World Percentage Applications
Percentages appear in almost every area of daily life. Here are the most common practical applications with the exact formula each uses:
Common Percentage Mistakes to Avoid
Mistake 1: Dividing by the Wrong Value
For percentage change, always divide by the original (old) value. Students often accidentally divide by the new value. If something went from 50 to 75, the % increase is (25 ÷ 50) × 100 = 50% — not (25 ÷ 75) × 100 = 33%.
Mistake 2: Adding Percentage Increases
Two successive percentage changes do not simply add. A 20% increase followed by a 20% decrease is NOT zero change. $100 × 1.20 = $120 → $120 × 0.80 = $96. You end up 4% below the start. Changes compound multiplicatively, not additively.
A 50% increase followed by a 50% decrease = −25% net, not 0%. ($100 → $150 → $75). A 100% increase followed by a 50% decrease = 0% net only by coincidence. Never add or subtract percentage changes directly.
Mistake 3: Confusing "X% more than" with "X% of"
"A is 20% more than B" means A = B × 1.20. It does not mean A = B × 0.20. Similarly, "A is 20% less than B" means A = B × 0.80 — not B − 20.
Mistake 4: Percentage Points vs. Percentage Change
If an interest rate rises from 2% to 5%, that is a 3 percentage point increase — but it is a 150% increase in the rate itself ((5−2)÷2×100). These are completely different statements. "Percentage points" measure absolute differences between percentages; "percentage change" measures relative change.
Mistake 5: Forgetting to Convert % to Decimal
A very common arithmetic error: multiplying by 15 instead of 0.15. Always convert your percentage to its decimal form first by dividing by 100. 15% → 0.15. 7.5% → 0.075. 120% → 1.20.
| Mistake | Wrong ❌ | Correct ✅ |
|---|---|---|
| % of a number | 200 × 15 = 3,000 | 200 × 0.15 = 30 |
| % change base | (Δ ÷ New) × 100 | (Δ ÷ Old) × 100 |
| Adding % changes | +20% then −20% = 0% | +20% then −20% = −4% |
| "20% more than 50" | 50 × 0.20 = 10 | 50 × 1.20 = 60 |
| Reverse % | Divide by (1 + %) | Divide by (% as decimal) |
🔗 Use the Right Calculator for Each Problem Type
- Finding X% of a number or what % A is of B → Percentage Calculator
- Price went up → Percentage Increase Calculator
- Price went down → Percentage Decrease Calculator
- Before-and-after comparison → Percentage Change Calculator
- Shopping discount & sale price → Discount Calculator
- Tips, tax, salary → Money Percentage Calculator
- Fraction to % conversion → Fraction to Percentage Calculator