Meaning of Percentage
Percentage means “per hundred”. The symbol is %.
20% = 20/100 = 1/5
📊 Percentages
Understand “per hundred,” convert between forms, calculate increases and decreases, use fast shortcuts and solve percentage problems step by step.
🎯 1. Learning Objectives
After completing this topic, you should be able to:
- Explain percentage as a fraction with denominator 100.
- Convert fractions and decimals into percentages and back.
- Find a percentage, a part or a whole.
- Calculate percentage increase and decrease using the original value.
- Solve successive-change and reverse-percentage problems.
- Apply percentage reasoning to marks, prices, data and programs.
📖 2. Percentage Fundamentals
Finding a Percentage of a Value
Replace the percentage sign with division by 100, then multiply.
20% of 500 = (20/100) × 500 = 100
🔄 3. Fractions, Decimals and Percentages
| Fraction | Decimal | Percentage | Useful interpretation |
|---|---|---|---|
| 1/2 | 0.5 | 50% | Half |
| 1/4 | 0.25 | 25% | One quarter |
| 3/4 | 0.75 | 75% | Three quarters |
| 1/5 | 0.2 | 20% | One fifth |
| 1/8 | 0.125 | 12.5% | One eighth |
Conversions: decimal × 100 = percentage; percentage ÷ 100 = decimal; fraction × 100% = percentage.
📌 4. Core Formulas
Find the Percentage
Percentage = (Part / Whole) × 100
Find the Part
Part = (Percentage / 100) × Whole
Find the Whole
Whole = Part × 100 / Percentage
Percentage Change
Change % = (Change / Original) × 100
📈 5. Percentage Increase and Decrease
Increase
New value = Original × (1 + p/100)
Percentage increase = Increase ÷ Original × 100.
Decrease
New value = Original × (1 − p/100)
Percentage decrease = Decrease ÷ Original × 100.
Important: the denominator is the original value, not the new value.
🔁 6. Successive Percentage Changes
Two percentage changes act one after the other, so their multipliers are multiplied.
Net change % = a + b + (ab/100)
Use a negative sign for a decrease. For +20% followed by −20%:
20 − 20 + (20 × −20)/100 = −4%
Equal percentage increase and decrease do not cancel because the second change uses a different base.
↩️ 7. Reverse Percentage
When the final value is known, undo the multiplier to recover the original.
After an Increase
Original = Final ÷ (1 + p/100)
After a Decrease
Original = Final ÷ (1 − p/100)
Example: after a 20% increase, a value becomes 600. Original = 600 ÷ 1.20 = 500.
⚡ 8. Fast Percentage Shortcuts
10%
Divide the number by 10.
5%
Find 10%, then divide it by 2.
25%
Divide the number by 4.
50%
Divide the number by 2.
1%
Divide the number by 100.
12.5%
Divide the number by 8.
Interchange shortcut: x% of y = y% of x. For example, 16% of 25 = 25% of 16 = 4.
🧪 9. Percentage Explorer
Select a calculation, enter the values and click Calculate Percentage. Changing an input hides the previous result.
Result
🧾 10. Solved Examples
Example 1: Find a Part
Problem: Find 20% of 500.
Solution: (20/100) × 500 = 100.
Example 2: Find a Percentage
Problem: A student scored 72 out of 90.
Solution: (72/90) × 100 = 80%.
Example 3: Percentage Increase
Problem: A value increases from 200 to 240.
Solution: Increase = 40; (40/200) × 100 = 20%.
Example 4: Reverse Percentage
Problem: A ₹750 price is after a 25% discount.
Solution: Original = 750 ÷ 0.75 = ₹1,000.
💻 11. Percentages in Programming
Progress Indicators
Percentage complete = completed tasks ÷ total tasks × 100.
Data Analysis
Percentages compare categories even when datasets have different sizes.
Performance Changes
Programs report speed-up, memory reduction and error-rate changes as percentages.
UI Scaling
Relative dimensions and responsive layouts often use percentage-based sizes.
⚠️ 12. Common Mistakes
Wrong Denominator
Using the new value instead of the original value for percentage change.
Missing × 100
Part ÷ whole gives a decimal; multiply by 100 to express it as a percentage.
Treating % as a Number
20% means 0.20, not 20, when used as a multiplier.
Cancelling Opposite Changes
+20% and −20% produce −4%, not 0%.
✍️ 13. Practice Problems
Solve each problem first, then use Show Solution to verify your method.
1. Find 25% of 800.
25% of 800 = (25/100) × 800 = 200.
2. Find 15% of 600.
15% of 600 = (15/100) × 600 = 90.
3. A number increases from 400 to 500. Find the percentage increase.
Increase = 100. Percentage increase = (100/400) × 100 = 25%.
4. A price decreases from ₹800 to ₹680. Find the percentage decrease.
Decrease = 120. Percentage decrease = (120/800) × 100 = 15%.
5. 30 is what percentage of 150?
(30/150) × 100 = 20%.
📝 14. Quick Revision
- Percentage means per hundred.
- Percentage = part ÷ whole × 100.
- Part = percentage ÷ 100 × whole.
- Percentage change always uses the original value as the denominator.
- Increase uses 1 + p/100; decrease uses 1 − p/100.
- For successive changes, multiply the change factors or use a + b + ab/100.
- Reverse a percentage by dividing by the final multiplier.
- Common values such as 10%, 25%, 50% and 12.5% support fast mental calculation.
INTERACTIVE PRACTICE
Start 20-Question Test →📊 Percentages Practice
Complete 20 questions from easy to hard. Your score, every option, correct answers and explanations appear only after final submission.