The word percent comes from the Latin per centum, which means per hundred. The symbol % is just a shorthand way of writing "÷ 100."
Definition: A percentage is a number or ratio expressed as a fraction of 100. When you write 45%, you are saying "45 out of every 100" or the fraction 45/100.
This makes percentages incredibly useful because they put everything on a common scale. Whether you are comparing test scores, sales figures, or nutrition labels, percentages let you make apples-to-apples comparisons instantly.
Every percentage has two equivalent forms:
As a fraction: 45% = 45/100 (which simplifies to 9/20)
As a decimal: 45% = 0.45
Being able to switch between all three forms — fraction, decimal, percentage — is the core skill this lesson builds.
The Conversion Triangle: Fractions ↔ Decimals ↔ Percentages
Think of these three forms as three languages that say the same thing. You need to be fluent in all three and able to translate instantly.
Fraction → Decimal: Divide top by bottom
Divide the numerator by the denominator using long division (or a calculator).
Fraction → Percentage: Divide, then multiply by 100
First convert the fraction to a decimal (Step 1), then convert the decimal to a percentage (Step 2).
Full example: What is 3/8 as a percentage?Step 1: 3 ÷ 8 = 0.375
Step 2: 0.375 × 100 = 37.5%
Answer: 3/8 = 37.5%
Finding a Percentage of a Number
This is the most common percentage task in daily life: "What is 20% of $85?" or "What is 35% of 240 students?" The method is always the same.
Percentage of a Number = (Percent ÷ 100) × Whole number
Or equivalently: convert the percent to a decimal first, then multiply.
Example 1 — What is 20% of 85?Step 1: Convert percent to decimal
20% → 20 ÷ 100 = 0.20
Step 2: Multiply by the whole number
0.20 × 85 = 17
Answer: 20% of 85 = 17
Example 2 — What is 15% tip on a $64 meal?Step 1: 15% → 0.15
Step 2: 0.15 × 64 = 9.60
Answer: Leave a $9.60 tip
Real-world shortcut: 10% of $64 = $6.40
5% is half of that = $3.20
10% + 5% = $9.60 ✓
Example 3 — A shirt is 30% off its $45 original price. What do you save?Step 1: 30% → 0.30
Step 2: 0.30 × 45 = $13.50 savings
Step 3: Sale price = $45.00 − $13.50 = $31.50
Percent Increase and Percent Decrease
Percent change tells you how much something has grown or shrunk, expressed relative to where it started. You see this everywhere: population growth, price changes, exam score improvements.
Percent Change = ((New Value − Old Value) ÷ Old Value) × 100
If the answer is positive, it is a percent increase. If it is negative, it is a percent decrease.
Example 1 — Percent increase: a salary rises from $40,000 to $46,000Step 1: Find the change: $46,000 − $40,000 = $6,000
Step 2: Divide by the OLD value: $6,000 ÷ $40,000 = 0.15
Step 3: Multiply by 100: 0.15 × 100 = 15%
Answer: 15% salary increase
Example 2 — Percent decrease: a stock drops from $80 to $68Step 1: Change: $68 − $80 = −$12 (negative = decrease)
Step 2: Divide by OLD: −$12 ÷ $80 = −0.15
Step 3: × 100: −15%
Answer: 15% decrease in value
Example 3 — Test score improved from 60 to 75. By what percentage?Change: 75 − 60 = 15
Old value: 60
Percent change: (15 ÷ 60) × 100 = 25%
Answer: 25% improvement
Working Backwards: Finding the Original Value
Sometimes you know the final value and the percent change, and you need the original. Rearrange the formula:
Original = Final Value ÷ (1 + percent change as decimal)
Example — A price including 8% tax is $162. What was the pre-tax price?Tax rate = 8% = 0.08
Formula: Original = Final ÷ (1 + 0.08) = 162 ÷ 1.08 = $150
Answer: The pre-tax price was $150
Check: $150 × 1.08 = $162 ✓
⚠ 5 Most Common Percentage Mistakes
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Not dividing by the OLD value for percent change. Always use the starting value as the denominator, not the new value. Dividing by the new value gives a different (wrong) percentage.
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Confusing percent change with percentage points. If a test pass rate goes from 70% to 80%, that is 10 percentage points but a 14.3% relative increase. These mean different things.
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Forgetting to convert percent to decimal before multiplying. 30% of 50 is NOT 30 × 50. It is 0.30 × 50 = 15.
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Applying a percent increase then the same percent decrease and expecting to get back to the original. 100 + 50% = 150. Then 150 − 50% = 75, not 100. Percent changes are relative, not symmetric.
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Misplacing the decimal when converting. 5% is 0.05, not 0.5 (which is 50%). Always verify: 5% should be a small fraction of the whole.
Practice Problems
Try each problem, then reveal the worked solution.
After a 12% price increase, a laptop costs $896. What was the original price?
Formula: Original = Final ÷ (1 + rate)
Original = $896 ÷ (1 + 0.12)
Original = $896 ÷ 1.12
Original = $800
Check: $800 × 1.12 = $896 ✓
Answer: The original price was $800
What is the difference between a percent change and percentage points?
A percent change is relative. A percentage point change is absolute. If interest rates go from 2% to 3%, that is 1 percentage point but a 50% relative increase.
How do you convert a decimal to a percent?
Multiply the decimal by 100 and add the % symbol. For example, 0.75 × 100 = 75%.
Ready for the Next Challenge?
Now that you can work confidently with percentages, tackle algebraic thinking — where letters stand in for unknown numbers.