What Is a Percentage, Really?
A percentage is just a fraction with denominator 100. That’s it. When we write 45%, we mean 45 out of 100, or the fraction 45/100.
The word “percent” comes from the Latin per centum — meaning “out of a hundred.” So every time we see %, we can immediately replace it with ”÷ 100” or “out of 100” and the problem becomes much simpler.
Why does this matter? Because comparison becomes easy when everything is out of the same base (100). Saying “Riya scored 43 out of 50 and Arjun scored 85 out of 100” is hard to compare. But say “Riya scored 86% and Arjun scored 85%” — now it’s instant.
Percentages appear everywhere in Class 7–8 NCERT: profit and loss, discount, taxes, simple interest, data handling. Cracking percentages cleanly means these chapters almost solve themselves.
Key Terms and Definitions
Percentage: A ratio expressed as a fraction of 100. Written with the symbol %.
Base (Whole): The number we calculate the percentage of. In “30% of 200”, the base is 200.
Rate: The percentage value itself (e.g., 30% is the rate).
Part: The actual quantity that corresponds to the percentage of the base. Here, Part = 60.
Percentage Increase/Decrease: How much a quantity has grown or shrunk, expressed as a percentage of the original value.
If the result is positive, it’s an increase. Negative means a decrease.
Always identify the “whole” (base) before setting up any percentage problem. The most common reason students get wrong answers is using the wrong base — especially in percentage change problems.
Core Methods
Method 1: Converting Between Fractions, Decimals, and Percentages
These three forms are interchangeable. Learn the flow:
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 2/5 | 0.4 | 40% |
| 1/8 | 0.125 | 12.5% |
| 1/3 | 0.333… | 33.33% |
Memorise the top 10 fraction-percentage equivalents. They save enormous time in exams.
Method 2: Finding the Percentage of a Number
To find x% of N:
Example: Find 35% of 240.
Method 3: Finding What Percentage One Number Is of Another
The question: “P is what percent of Q?”
Example: 15 is what percent of 60?
Method 4: Finding the Original Number
“If X% of a number is N, find the number.” Work backwards:
Example: 40% of a number is 56. Find the number.
Method 5: Percentage Increase and Decrease
To find the new value after a percentage increase of r%:
To find the new value after a percentage decrease of r%:
Solved Examples
Easy — CBSE Class 7
Q: A class has 40 students. 25 are girls. What percentage of students are girls?
We need: (Girls ÷ Total) × 100
62.5% of the students are girls.
Q: Convert 3/8 to a percentage.
Medium — CBSE Class 8
Q: The price of a bag increased from ₹800 to ₹960. Find the percentage increase.
Increase = 960 − 800 = 160
The price increased by 20%.
A very common error here: students divide by the new price (960) instead of the original price (800). The base for percentage change is ALWAYS the original/old value.
Q: In an election, 75,000 votes were cast. Candidate A got 48% of the votes. How many votes did Candidate B get? (Only two candidates.)
Votes for A = 48% of 75,000 = (48/100) × 75,000 = 36,000
Votes for B = 75,000 − 36,000 = 39,000 votes
Hard — CBSE Class 8 / NTSE Level
Q: A shopkeeper marks his goods 25% above cost price and gives a 10% discount on the marked price. Find his overall profit percentage.
Let the cost price = ₹100 (we pick 100 for easy calculation — this is a standard trick).
Marked Price = 100 + 25% of 100 = ₹125
Discount = 10% of 125 = ₹12.50
Selling Price = 125 − 12.50 = ₹112.50
Profit = 112.50 − 100 = ₹12.50
Overall profit is 12.5%.
Whenever cost price isn’t given, assume CP = ₹100. It removes all the fraction complications and the percentage answer comes out directly.
Q: A number is first increased by 20% and then decreased by 20%. Find the net change.
Let the number be 100.
After 20% increase: 100 × (1 + 20/100) = 100 × 1.2 = 120
After 20% decrease: 120 × (1 − 20/100) = 120 × 0.8 = 96
Net change = 96 − 100 = −4, i.e., a decrease of 4%.
This is a classic trap question. Equal percentage increase and decrease does NOT give zero net change — you always end up with a net loss. The formula for net effect: if the rate is r%,
Here: −(20²)/100 = −400/100 = −4%. Matches perfectly.
Exam-Specific Tips
CBSE Class 7 (Chapter: Comparing Quantities) Marks weightage: 10–12 marks in a 80-mark paper (direct + application). Questions are mostly straightforward conversions and “find the percentage” problems. Always show the formula, then substitution, then answer — 3-step solutions fetch full marks.
CBSE Class 8 (Chapter: Comparing Quantities) Percentage is embedded in profit/loss, discount, and tax problems. A typical 5-mark question will chain two percentage operations (like the shopkeeper problem above). The NCERT exercise questions are directly board-exam-worthy — solve all of them, not just examples.
NTSE / Olympiad Level Successive percentage changes, population growth, and “percentage of percentage” problems appear here. The assumption CP = 100 trick and the net change formula (−r²/100) are must-know shortcuts. Speed is critical — aim to solve medium percentage problems within 45 seconds.
Common Mistakes to Avoid
Mistake 1: Wrong base in percentage change “Price went from 500 to 600, so percentage increase = 100/600 × 100 = 16.67%.” Wrong. The base is always the ORIGINAL value (500), so the answer is 100/500 × 100 = 20%.
Mistake 2: Adding percentages of different bases “Discount is 20% and tax is 10%, so the net effect is 10%.” You can only add/subtract percentages when they apply to the same base. Here, discount is on Marked Price, tax is on Selling Price — they have different bases, so you can’t simply subtract.
Mistake 3: Successive percentage confusion “20% increase then 20% decrease = 0% net change.” We showed above this equals a 4% net LOSS. Equal up and down rates never cancel out — the second percentage operates on the changed value, not the original.
Mistake 4: Converting percentages above 100% Students write 150% = 1.5 correctly, but then use it as if it means 1.5 out of 1 (which is fine), yet get confused in context: “150% of 80” — some students write 1.5 × 80 = 120 but lose marks because they didn’t verify the setup. Always re-read: “150% of 80” = (150/100) × 80 = 120. ✓
Mistake 5: Reversing “of” problems “A is 25% of B” means A = (25/100) × B = B/4. “B is 25% more than A” means B = A + 25% of A = 1.25A. These are completely different. Read the language of the problem very carefully before writing the equation.
Practice Questions
Q1. Find 15% of 320.
Q2. Express 7/20 as a percentage.
Q3. 60 is what percent of 240?
Q4. 35% of a number is 91. Find the number.
Q5. In a school of 800 students, 45% are boys. How many girls are there?
Boys = 45% of 800 = 360
Girls = 800 − 360 = 440 girls
Q6. A shirt costs ₹450. Its price is reduced by 12%. Find the new price.
Reduction = 12% of 450 = (12/100) × 450 = ₹54
New Price = 450 − 54 = ₹396
Or directly: New Price = 450 × (1 − 12/100) = 450 × 0.88 = ₹396 ✓
Q7. The population of a town was 60,000 in 2022. It increased by 5% in 2023 and by 8% in 2024. What is the population in 2024?
After 2023: 60,000 × 1.05 = 63,000
After 2024: 63,000 × 1.08 = 68,040
Population in 2024 = 68,040
Q8. A number is first decreased by 30% and then increased by 30%. Find the net percentage change.
Let the number = 100.
After 30% decrease: 100 × 0.70 = 70
After 30% increase: 70 × 1.30 = 91
Net change = 91 − 100 = −9, so net decrease of 9%.
Using formula: net change = −r²/100 = −(30)²/100 = −900/100 = −9% ✓
Q9. Amit scored 378 out of 450 in his exams. What is his percentage score? Is he eligible for a scholarship that requires at least 80%?
Since 84% > 80%, yes, Amit is eligible for the scholarship.
Q10. A dealer buys a cycle for ₹1,200 and sells it at a gain of 25%. A customer pays with a note and gets ₹50 as change. What denomination note did the customer use?
Selling price = 1200 × (1 + 25/100) = 1200 × 1.25 = ₹1,500
Customer got ₹50 change, so he paid 1500 + 50 = ₹1,550? — Wait.
Re-reading: customer paid with a note and received ₹50 change, meaning the note > ₹1,500.
The nearest denomination above ₹1,500 that fits is ₹2,000.
Change = 2000 − 1500 = ₹500. But the problem says ₹50 change.
So the note was ₹1,550? Notes aren’t in that denomination.
The standard answer: note = ₹2,000, change = ₹500. Likely a ₹50 typo in this version — the logic is: SP = ₹1,500, note = ₹2,000 note (standard denomination), change = ₹500.
Frequently Asked Questions
What is the formula to calculate percentage?
The “whole” is always the base — what you’re calculating the percentage of.
How do I convert a fraction to a percentage?
Multiply the fraction by 100 and add the % sign. For example, 3/5 = (3/5) × 100 = 60%.
Why does an equal percentage increase and decrease not give the original number back?
Because the second operation works on the already-changed value, not the original. A 20% increase on 100 gives 120, and a 20% decrease on 120 gives 96, not 100. The net effect is always a slight loss, equal to −r²/100 percent.
How do I find the original number if I’m given the percentage value?
If 40% of a number = 72, then original = (72 × 100)/40 = 180.
What is percentage change and how is it different from percentage?
Percentage tells you part out of whole. Percentage change tells you how much a value has increased or decreased relative to its original value. Percentage change = [(New − Old)/Old] × 100. The key difference is the base: in percentage, the base is the whole; in percentage change, the base is the old/original value.
Is 100% possible? What does 200% mean?
Yes, 100% means the entire quantity. 200% means twice the original quantity. Percentages can exceed 100 when the “part” is larger than the “whole” — common in profit-and-loss problems (e.g., a 150% profit means you earned 1.5 times your original cost).
How are percentages used in Class 8 profit and loss?
Profit % = (Profit/CP) × 100. Loss % = (Loss/CP) × 100. The base is always the cost price (CP), not the selling price. This connects directly to the percentage formula — just apply it with CP as the “whole.”
What are the most scoring percentage topics in CBSE exams?
Direct percentage calculations and simple percentage increase/decrease (3-mark questions) are the most reliable. Chained problems — like “a price increases by 10%, then decreases by 5%, find net change” — appear as 5-mark questions in Class 8 boards and are worth practising specifically from NCERT Examples 8–12 and Exercise 8.3.