Class 8 • Maths • Chapter 1
🎭 Fractions in Disguise
Complete study notes on Percentages — from basics to compounding & profit/loss, crafted for NCERT Class 8 students.
📐 Fractions & %
💰 Profit & Loss
🏦 Simple Interest
📈 Compounding
📉 Depreciation
💰 Profit & Loss
🏦 Simple Interest
📈 Compounding
📉 Depreciation
📋 Contents
- What is a Percentage?
- Fractions as Percentages
- Percentage of a Quantity
- The FDP Trio (Fractions, Decimals, Percentages)
- Using Percentages — Comparison, Profit & Loss
- Percentage Increase & Decrease
- Growth & Compounding (Simple vs Compound Interest)
- Decline & Depreciation
- Tricky Percentages
- Chapter Summary
- Exam Practice Questions
1. What is a Percentage?
The word per cent comes from the Latin phrase per centum, meaning “out of hundred”. The symbol % is read as “per cent”.
In Real Life
“Mega Sale — up to 50% off!” means 50 out of every 100 rupees is saved. “Hiya scored 83%” means 83 marks out of every 100.
“Mega Sale — up to 50% off!” means 50 out of every 100 rupees is saved. “Hiya scored 83%” means 83 marks out of every 100.
x% = x / 100 So, 25% = 25/100 = 1/4
📝 Key Idea
Percentages are simply fractions with denominator 100.
Example20% = 20/100 = 1/5
Example33% = 33/100
History of %
The idea of “per hundred” appears in Kautilya’s Arthaśhāstra (4th century BCE) and in ancient Roman tax records (1/20, 1/100). Italian manuscripts from the 15th century used expressions like “xx p cento” (20%), “x p cento” (10%).
The idea of “per hundred” appears in Kautilya’s Arthaśhāstra (4th century BCE) and in ancient Roman tax records (1/20, 1/100). Italian manuscripts from the 15th century used expressions like “xx p cento” (20%), “x p cento” (10%).
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Percentages Around Us!The human body is ~60% water. Ice cream is 30–50% air. About 99.86% of the Solar System’s mass is in the Sun. Over 80% of teenagers globally don’t meet daily exercise recommendations!
2. Fractions as Percentages (and Vice Versa)
➡️ Fraction → Percentage
To convert any fraction to a percentage, multiply the fraction by 100.
Percentage = (Fraction) × 100
Method 1 — Equivalent Fraction
Make denominator = 100
3/4 = (3×25)/(4×25) = 75/100 = 75%
Make denominator = 100
3/4 = (3×25)/(4×25) = 75/100 = 75%
Method 2 — Multiply by 100
x/100 = 3/4
x = 3/4 × 100 = 75
So, 3/4 = 75%
x/100 = 3/4
x = 3/4 × 100 = 75
So, 3/4 = 75%
Quick Rule
To express any fraction as a %, just multiply by 100. Example: 2/5 × 100 = 40%
To express any fraction as a %, just multiply by 100. Example: 2/5 × 100 = 40%
⬅️ Percentage → Fraction
A percentage, z%, is the same as the fraction z/100. Simplify if needed.
z% = z/100
For example: 24% = 24/100 = 12/50 = 6/25
| Fraction | Calculation | Percentage |
|---|---|---|
| 3/5 | 3/5 × 100 | 60% |
| 7/14 | 7/14 × 100 = 1/2 × 100 | 50% |
| 9/20 | 9/20 × 100 | 45% |
| 1/3 | 1/3 × 100 | 33.33% |
Why is 100 special?
Our number system is base 10, so 100 fits neatly with decimals. 31% = 31/100 = 0.31. Converting becomes quick and intuitive. 100 is large enough for detail but simple enough to understand mentally.
Our number system is base 10, so 100 fits neatly with decimals. 31% = 31/100 = 0.31. Converting becomes quick and intuitive. 100 is large enough for detail but simple enough to understand mentally.
3. Percentage of a Quantity
📐 The Core Formula
y% of a value = (y/100) × value
Important!
Never compare percentages of different quantities directly. Madhu eats biscuits with 25% sugar and Madhav eats biscuits with 35% sugar — but if Madhu eats more biscuits, Madhu could eat more sugar in total!
Never compare percentages of different quantities directly. Madhu eats biscuits with 25% sugar and Madhav eats biscuits with 35% sugar — but if Madhu eats more biscuits, Madhu could eat more sugar in total!
🔢 Worked Example
Madhu eats 120g biscuits (25% sugar). Madhav eats 95g biscuits (35% sugar). Who ate more sugar?
Madhu’s sugar = 25/100 × 120 = 30 gMadhav’s sugar = 35/100 × 95 = 33.25 g
∴ Madhav ate more sugar.
🧮 Free-hand (Mental) Computations
Useful percentage equivalents to remember:
| Percentage | Fraction Equivalent | Quick Trick |
|---|---|---|
| 25% | 1/4 | Divide by 4 |
| 50% | 1/2 | Divide by 2 |
| 10% | 1/10 | Divide by 10 |
| 20% | 1/5 | Divide by 5 |
| 5% | 1/20 | Half of 10% |
| 1% | 1/100 | Divide by 100 |
Useful Relationship
20% of y + 5% of y = 25% of y. Similarly, you can build any % by adding simpler ones!
To find 15%: 10% + 5% = 15%.
20% of y + 5% of y = 25% of y. Similarly, you can build any % by adding simpler ones!
To find 15%: 10% + 5% = 15%.
📖 Percentages Greater than 100%
Percentages can exceed 100! This means the quantity is more than the reference amount.
If a shopkeeper’s daily target is ₹5000 and he earns ₹6000:
% of target = 6000/5000 × 100 = 120% (20% above target)
% of target = 6000/5000 × 100 = 120% (20% above target)
250% means 2.5 times the original value. Example: A farmer harvested 260 kg last year, 650 kg this year:
% of last year’s harvest = 650/260 × 100 = 250%
(This year’s harvest is 2.5 times last year’s)
(This year’s harvest is 2.5 times last year’s)
4. The FDP Trio — Fractions, Decimals & Percentages
Fractions, decimals, and percentages are all just different ways of writing the same thing!
Fraction ↔ Decimal ↔ Percentage
Example: 4/10 = 0.4 = 40%
| Percentage | Fraction | Decimal |
|---|---|---|
| 50% | 1/2 | 0.5 |
| 100% | 1 | 1.0 |
| 25% | 1/4 | 0.25 |
| 75% | 3/4 | 0.75 |
| 10% | 1/10 | 0.1 |
| 1% | 1/100 | 0.01 |
| 5% | 1/20 | 0.05 |
Decimal Shortcut for Finding %
50% of 24 = 0.5 × 24 = 12. You can always replace any percentage with its decimal equivalent for multiplication!
50% of 24 = 0.5 × 24 = 12. You can always replace any percentage with its decimal equivalent for multiplication!
📝 Worked Example
Maximum marks in a test = 75. Zubin needs ≥ 80% to get Grade A. Minimum marks needed?
Method 1 (Fraction): 80/100 × 75 = 4/5 × 75 = 60Method 2 (Decimal): 0.8 × 75 = 60
Method 3 (Proportion): 80 out of 100 → ? out of 75
75 × 80/100 = 60
5. Using Percentages — Comparison & Profit/Loss
🔍 Comparing Proportions
When maximum marks (or totals) are different, convert to % to compare fairly.
Eesha: English 42/50 = 42/50 × 100 = 84% | Science 70/80 = 70/80 × 100 = 87.5%
∴ She did better in Science.
∴ She did better in Science.
🏪 Profit and Loss — Key Terms
Cost Price (CP)The price a shopkeeper pays to buy the item.
Marked Price (MP)The price quoted/labelled on the item (MRP).
Selling Price (SP)The price the customer actually pays after any discount.
ProfitWhen SP > CP. Profit = SP − CP
LossWhen SP < CP. Loss = CP − SP
DiscountReduction given on the Marked Price. Discount = MP − SP
📊 Profit & Loss Formulas
Profit% = (Profit / CP) × 100
Loss% = (Loss / CP) × 100
SP = CP × (1 + Profit%/100) OR SP = CP × (1 − Loss%/100)
🧾 Worked Examples
Example: Profit
Kishanlal buys sweater at ₹300, sells at ₹430.
Profit = 430 − 300 = ₹130
Profit% = 130/300 × 100 = 43.3%Example: Loss
Raghu bought rice at ₹35/kg. Sells 10kg for ₹300.
CP of 10kg = ₹350. Loss = 350 − 300 = ₹50
Loss% = 50/350 × 100 = 14.28%Example: Selling at a loss %
Shyamala buys vase at ₹2650. Sells at 18% loss.
SP = 82% of 2650 = 0.82 × 2650 = ₹2173
Kishanlal buys sweater at ₹300, sells at ₹430.
Profit = 430 − 300 = ₹130
Profit% = 130/300 × 100 = 43.3%Example: Loss
Raghu bought rice at ₹35/kg. Sells 10kg for ₹300.
CP of 10kg = ₹350. Loss = 350 − 300 = ₹50
Loss% = 50/350 × 100 = 14.28%Example: Selling at a loss %
Shyamala buys vase at ₹2650. Sells at 18% loss.
SP = 82% of 2650 = 0.82 × 2650 = ₹2173
💸 Discount
30% discount means the price is reduced by 30% of the Marked Price.
SP = MP × (1 − Discount%/100)
Example: Cooker MRP = ₹1800, discount = 35%
SP = 1800 × (1 − 35/100) = 1800 × 0.65 = ₹1170
💼 Gross Profit vs Net Profit
Gross ProfitRevenue − Cost of goods sold (before deducting expenses)
Net ProfitGross Profit − Other expenses (rent, salary, bills, etc.)
🧾 Taxes (GST)
GST (Goods and Services Tax) is a tax added on top of the price. The rate is given as a %.
Final price = Original price × (1 + GST%/100)
Example: Phone ₹8250, GST 18%: Final = 8250 × 1.18 = ₹9735
6. Percentage Increase & Decrease
⬆️ Percentage Increase
% Increase = (Amount of Increase / Original Value) × 100
Example: Tomato price was ₹30, now ₹42. Increase = ₹12.
% Increase = 12/30 × 100 = 40%
⬇️ Percentage Decrease
% Decrease = (Amount of Decrease / Original Value) × 100
Example: Theater footfall was 160, now 100. Decrease = 60.
% Decrease = 60/160 × 100 = 37.5%
Equivalent Statements
“Population in 1991 is 165% of 1961 population” means the SAME as “Population increased by 65% from 1961 to 1991.”
Both mean: new = 1.65 × old
“Population in 1991 is 165% of 1961 population” means the SAME as “Population increased by 65% from 1961 to 1991.”
Both mean: new = 1.65 × old
7. Growth & Compounding (Simple vs Compound Interest)
📚 Key Terms
- Principal (P): The amount of money deposited/invested.
- Rate of Interest (r): The percentage interest per year (p.a. = per annum).
- Time (t): Number of years the money is invested.
- Amount: Principal + Interest earned.
📊 Without Compounding (Simple Interest)
The interest is the same every year and is paid out. The principal stays the same.
Total Amount = P(1 + rt)
where r is the rate as a decimal (e.g., 10% → r = 0.10)
where r is the rate as a decimal (e.g., 10% → r = 0.10)
P = ₹6000, r = 10% = 0.10, t = 3 yearsInterest per year = 6000 × 0.10 = ₹600
Total Interest (3 years) = 600 × 3 = ₹1800
Total Amount = 6000 + 1800 = ₹7800
= 6000 × (1 + 0.10 × 3) = 6000 × 1.3
Total Interest (3 years) = 600 × 3 = ₹1800
Total Amount = 6000 + 1800 = ₹7800
= 6000 × (1 + 0.10 × 3) = 6000 × 1.3
📈 With Compounding (Compound Interest)
The interest is added back to the principal every year, so interest grows on interest!
Total Amount = P × (1 + r)t
P = ₹6000, r = 0.10, t = 3 yearsYear 1: 6000 × 1.10 = ₹6600
Year 2: 6600 × 1.10 = ₹7260
Year 3: 7260 × 1.10 = ₹7986= 6000 × (1.1)³ = 6000 × 1.331 = ₹7986
Year 2: 6600 × 1.10 = ₹7260
Year 3: 7260 × 1.10 = ₹7986= 6000 × (1.1)³ = 6000 × 1.331 = ₹7986
Without Compounding₹7800 returned (30% gain over 3 years)
With Compounding₹7986 returned (33.1% gain over 3 years) — More! 🎉
Why Compounding is Powerful
Without compounding = Linear growth (same interest each year). With compounding = Exponential growth (interest grows on interest). The longer you invest, the bigger the difference!
Without compounding = Linear growth (same interest each year). With compounding = Exponential growth (interest grows on interest). The longer you invest, the bigger the difference!
🔢 General Formulas
| Without Compounding | With Compounding | |
|---|---|---|
| Amount after t years | P(1 + rt) | P(1 + r)t |
| Total Interest | P × r × t | P(1+r)t − P |
| Growth type | Linear | Exponential |
Note on ‘r’
In formulas, r must be in decimal form. So 10% → r = 0.10, 5% → r = 0.05. The formula P(1+r)ᵗ already accounts for this.
In formulas, r must be in decimal form. So 10% → r = 0.10, 5% → r = 0.05. The formula P(1+r)ᵗ already accounts for this.
8. Decline & Depreciation
Depreciation is the reduction in value of an item due to age and use.
Value after depreciation = Original Value × (1 − r)t
where r = depreciation rate as a decimal
where r = depreciation rate as a decimal
📝 Worked Examples
TV Depreciation
TV bought at ₹21,000. Depreciates 5% after 1 year.
New value = 95% of 21,000 = 0.95 × 21,000 = ₹19,950Population Decline
Village population: 1250. Reduces 10% per decade.
After 3 decades:
1250 × (0.9)³ = 1250 × 0.729 = ≈ 911
TV bought at ₹21,000. Depreciates 5% after 1 year.
New value = 95% of 21,000 = 0.95 × 21,000 = ₹19,950Population Decline
Village population: 1250. Reduces 10% per decade.
After 3 decades:
1250 × (0.9)³ = 1250 × 0.729 = ≈ 911
Pattern
Growth: multiply by (1 + r) each period. Decline: multiply by (1 − r) each period. Both are compound effects!
Growth: multiply by (1 + r) each period. Decline: multiply by (1 − r) each period. Both are compound effects!
9. Tricky Percentages — Watch Out!
⚠️ 30% + 20% ≠ 50% in Discounts!
When shops say “30% + 20% off”, they apply discounts sequentially (on the reduced price), not together. This is compounding of discounts.
Cake worth ₹200. Shop offers 30% + 20% off.After 30%: 200 − 60 = ₹140
After 20% on ₹140: 140 − 28 = ₹112vs. a 50% off shop: 200 × 0.5 = ₹100 (cheaper!)
After 20% on ₹140: 140 − 28 = ₹112vs. a 50% off shop: 200 × 0.5 = ₹100 (cheaper!)
The Mishap Problem
A shopkeeper adds 50% profit margin, then gives 50% discount. Does she break even? NO! She makes a 25% LOSS because 50% of (1.5x) = 0.75x, which is less than the original cost x.
A shopkeeper adds 50% profit margin, then gives 50% discount. Does she break even? NO! She makes a 25% LOSS because 50% of (1.5x) = 0.75x, which is less than the original cost x.
🔢 Interesting Property
x% of y = y% of x (always!)
5% of 40 = 2
40% of 5 = 2 ← Same!25% of 12 = 3
12% of 25 = 3 ← Same!Algebraic proof: x/100 × y = y/100 × x ✓
40% of 5 = 2 ← Same!25% of 12 = 3
12% of 25 = 3 ← Same!Algebraic proof: x/100 × y = y/100 × x ✓
🎲 Comparing % vs Absolute Values
Key Principle
A higher % doesn’t always mean more in absolute terms. And a higher absolute amount doesn’t mean higher %. Always be clear about WHAT the percentage is of (the base/whole).
A higher % doesn’t always mean more in absolute terms. And a higher absolute amount doesn’t mean higher %. Always be clear about WHAT the percentage is of (the base/whole).
10. Chapter Summary
% Definitionx% = x/100. Fractions with denominator 100.
Fraction → %Multiply fraction by 100.
% of a quantity(y/100) × value
FDP ConnectionFraction ↔ Decimal ↔ %. All interchangeable!
% Increase/Decrease(Change / Original) × 100
Profit% / Loss%Always calculated on Cost Price (CP).
Simple AmountP(1 + rt) — linear growth
Compound AmountP(1 + r)ᵗ — exponential growth
DepreciationValue × (1 − r)ᵗ
DiscountSP = MP × (1 − d%) on Marked Price
GSTFinal Price = Price × (1 + GST rate)
Key Trickx% of y = y% of x always!
Most Important Formula Box:
Without Compounding: A = P(1 + rt)
With Compounding: A = P(1 + r)t
Depreciation: Value = P(1 − r)t
Profit%: = (SP − CP) / CP × 100
Loss%: = (CP − SP) / CP × 100
Without Compounding: A = P(1 + rt)
With Compounding: A = P(1 + r)t
Depreciation: Value = P(1 − r)t
Profit%: = (SP − CP) / CP × 100
Loss%: = (CP − SP) / CP × 100
11. Exam Practice Questions
Q1. Express 3/5 as a percentage.
Ans: 3/5 × 100 = 60%
Q2. Nandini has 25 marbles, of which 15 are white. What percentage are white?
Ans: 15/25 × 100 = 60% (Option iv)
Q3. Find 25% of 160.
Ans: 25/100 × 160 = 1/4 × 160 = 40
Q4. A shopkeeper buys a geometry box for ₹75 and sells it for ₹110. What is his profit margin?
Ans: Profit = 110 − 75 = ₹35. Profit% = 35/75 × 100 = 46.67%
Q5. A cyclist completes 40% of a journey and has covered 92 km. How much is left?
Ans: Total distance = 92 × 100/40 = 230 km. Remaining = 230 − 92 = 138 km
Q6. Deposit ₹20,000 at 10% p.a. for 2 years. Find amount with and without compounding.
Ans: Without compounding: 20000 × (1 + 0.10 × 2) = ₹24,000. With compounding: 20000 × (1.1)² = 20000 × 1.21 = ₹24,200
Q7. A TV bought at ₹21,000 depreciates 5% per year. Value after 1 year?
Ans: 21000 × 0.95 = ₹19,950
Q8. The petrol price was ₹60 in 2015 and ₹100 in 2025. What is the % increase?
Ans: Increase = 40. % increase = 40/60 × 100 = 66.66% (Option iv)
Q9. A number increased by 20% becomes 90. What is the number?
Ans: Let the number = x. x × 1.20 = 90 → x = 90/1.20 = 75
Q10. Is 10% of a day longer than 1% of a week?
Ans: 10% of 24h = 2.4 hours = 144 min. 1% of 7 days = 0.07 days = 100.8 min. 10% of a day is longer!
Exam Strategy
Always estimate before calculating. Draw bar models to visualise the problem. Remember: % Increase/Decrease is always calculated on the ORIGINAL (base) value, not the new value!
Always estimate before calculating. Draw bar models to visualise the problem. Remember: % Increase/Decrease is always calculated on the ORIGINAL (base) value, not the new value!
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