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Class 8 · Ganita Prakash · Chapter 1
Fractions in Disguise
Everything about Percentages — from basics to compounding 🚀
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1 · What is Per Cent?
The disguise of every fraction
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Origin of the word
"Per cent" comes from Latin per centum = "out of hundred".
The symbol % is a shorthand for ÷100.
The symbol % is a shorthand for ÷100.
x% =
x100
= x ÷ 100
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Simple way to think about it
25% = 25 out of every 100
→ 25 students out of 100
→ ₹25 out of ₹100
→ 25 marks out of 100 marks
→ 25 students out of 100
→ ₹25 out of ₹100
→ 25 marks out of 100 marks
Visualise: 25% of a whole
25%
0%25%50%75%100%
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Why 100? Why not 10 or 1000?
100 is the sweet spot — large enough to give detail, small enough to understand mentally. Our number system is base-10, so converting to decimals is instant: 31% = 0.31.
🌍 Percentages Around Us
60%
of human body is water (by weight)
30–50%
of ice cream is air by volume!
99.86%
of Solar System's mass is the Sun
45%
of world watched 2022 FIFA World Cup
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2 · Fractions ↔ Percentages
Two ways to convert, always the same answer
⭐
Golden Rule
To convert any fraction → percentage, just multiply by 100.
To convert percentage → fraction, write it over 100 and simplify.
To convert percentage → fraction, write it over 100 and simplify.
📐 Fraction → Percentage (Two Methods)
| Method 1 – Equivalent Fraction | Method 2 – Multiply by 100 |
|---|---|
| Make denominator = 100 3/4 = (3×25)/(4×25) = 75/100 = 75% |
Fraction × 100 = % 3/4 × 100 = 75% |
| 2/5 = 20/50 = 40/100 = 40% | 2/5 × 100 = 40% |
📐 Percentage → Fraction
24% =
24100
=
1250
=
625
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Quick Reference — Common Fractions as %
1/2 = 50% | 1/4 = 25% | 3/4 = 75%
1/5 = 20% | 2/5 = 40% | 1/10 = 10%
1/3 ≈ 33.33% | 2/3 ≈ 66.67%
1/5 = 20% | 2/5 = 40% | 1/10 = 10%
1/3 ≈ 33.33% | 2/3 ≈ 66.67%
3/4 and 75% are the SAME thing
3/4 = 75%
0 (0%)1/4 (25%)1/2 (50%)3/4 (75%)1 (100%)
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Why do we need %?
Comparing 9/34 and 13/45 is hard! But 26.47% vs 28.88% — instantly clear! Percentages put everything on the same scale of 100.
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3 · Percentage of a Quantity
Finding the actual amount from a percentage
y% of some value z =
y100
× z
🔍 Example — Sugar in Biscuits
Madhu eats 120 g biscuits with 25% sugar.
Sugar = 25/100 × 120 = 30 g
Sugar
0g30g60g90g120g
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Important Caution!
Don't compare percentages when the totals are different!
25% of 120g vs 35% of 95g → you must calculate actual amounts first!
25% of 120g vs 35% of 95g → you must calculate actual amounts first!
🧠 Free-Hand Computation Tricks
1
25% = 1/4 → Divide by 4
2
10% → Move decimal one place left
3
20% = 2 × 10% → Double the 10% value
4
5% = 10% ÷ 2 → Halve the 10% value
5
15% = 10% + 5% | 25% = 20% + 5%
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Millet Example (Ratio → %)
Millet : Water = 2 : 7 → Total = 9 parts
Millet fraction = 2/9
Millet % = 2/9 × 100 = 22.22%
Water % = 100 − 22.22 = 77.78%
Millet fraction = 2/9
Millet % = 2/9 × 100 = 22.22%
Water % = 100 − 22.22 = 77.78%
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A ratio can be converted to a fraction, then to a %
Always estimate first! It helps catch calculation mistakes.
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4 · The FDP Trio
Fractions · Decimals · Percentages — they're the same family!
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Three Forms, One Value
50% = 50/100 = 1/2 = 0.5
25% = 25/100 = 1/4 = 0.25
10% = 10/100 = 1/10 = 0.1
25% = 25/100 = 1/4 = 0.25
10% = 10/100 = 1/10 = 0.1
| Percent | Fraction | Decimal |
|---|---|---|
| 100% | 100/100 = 1 | 1.0 |
| 50% | 1/2 | 0.5 |
| 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 |
| 20% | 1/5 | 0.2 |
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Multiply by decimal to find percentage!
50% of 24 = 0.5 × 24 = 12 (same as dividing by 2!)
80% of 75 = 0.8 × 75 = 60 → multiply by fraction form
80% of 75 = 0.8 × 75 = 60 → multiply by fraction form
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5 · Percentages Greater Than 100
Yes! You can have more than the whole
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What does 120% mean?
If target = ₹5000 and you earned ₹6000:
6000/5000 × 100 = 120% = 1.2 times the target
→ You earned 20% more than the target!
6000/5000 × 100 = 120% = 1.2 times the target
→ You earned 20% more than the target!
📊 % vs Fraction vs Decimal (including >100%)
120%
1.2×
200%
2.0×
250%
2.5×
🌾 Example — Farmer's Harvest
Last year: 260 kg | This year: 650 kg
This year's harvest = 650/260 × 100 = 250% of last year
250% = 2.5 times the original value
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Key Conversions
100% = 1 (exactly the whole)
200% = 2 (double / twice)
50% = 0.5 (half)
150% = 1.5 (one and a half times)
200% = 2 (double / twice)
50% = 0.5 (half)
150% = 1.5 (one and a half times)
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6 · Comparing Proportions
When totals differ, use % to compare fairly
🏆 Example — Test Scores
Eesha: 42/50 in English → 42/50 × 100 = 84%
Eesha: 70/80 in Science → 70/80 × 100 = 87.5%
Science score is higher → Eesha did better in Science!
✅
Why convert to %?
Percentages put everything on the same scale of 100, making comparisons fair and easy — even when the original totals are different.
🛒 Know Your Contents (KYC) — Food Labels
🍫
DEF Badam Mix (150g total)
Sugar 99g → 99/150 × 100 = 66% 😮
Milk Solids 30g → 20% | Badam 12g → 8% | Food Chemicals 9g → 6%
Milk Solids 30g → 20% | Badam 12g → 8% | Food Chemicals 9g → 6%
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Always do KYC — Know Your Contents!
Converting ingredient weights to percentages helps compare different-sized packs instantly. The percentages of all ingredients must add up to 100%.
📉📈
7 · Percentage Increase & Decrease
Express changes as a rate relative to the original
% Increase =
Amount of IncreaseOriginal Amount (Base)
× 100
% Decrease =
Amount of DecreaseOriginal Amount (Base)
× 100
📈 Increase
Tomato: ₹30 → ₹42
Increase = 12
% increase = 12/30 × 100 = 40%
📉 Decrease
Theatre: 160 → 100 visitors
Decrease = 60
% decrease = 60/160 × 100 = 37.5%
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Two Statements — Same Meaning
"Population in 1991 is 165% of 1961" = "Population increased by 65% from 1961 to 1991"
Both mean: New = Old × 1.65
Both mean: New = Old × 1.65
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8 · Profit, Loss & Discount
The language of buying and selling
Cost Price (CP) Price shopkeeper pays
Marked Price (MP) Price quoted / MRP
Selling Price (SP) Price customer pays
| Concept | Formula | Remember |
|---|---|---|
| Profit | SP − CP | SP > CP |
| Loss | CP − SP | CP > SP |
| % Profit | (Profit/CP) × 100 | Always on CP |
| % Loss | (Loss/CP) × 100 | Always on CP |
| Discount | MP − SP | % discount on MP |
| SP (with discount) | MP × (1 − d/100) | d = discount % |
| SP (with profit) | CP × (1 + p/100) | p = profit % |
🧥 Example — Sweater
CP = ₹300 | SP = ₹430
Profit = 430 − 300 = ₹130
% Profit = 130/300 × 100 = 43.3%
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Gross Profit vs Net Profit
Gross Profit = Revenue − Cost of goods sold
Net Profit = Gross Profit − Other expenses (rent, salary, electricity…)
In this chapter, "profit" = gross profit.
Net Profit = Gross Profit − Other expenses (rent, salary, electricity…)
In this chapter, "profit" = gross profit.
🧮
GST — Tax is a Percentage!
GST (Goods & Services Tax) is added to the price as a percentage.
Item ₹8,250 with 18% GST → Final price = 8250 × 1.18 = ₹9,735
Item ₹8,250 with 18% GST → Final price = 8250 × 1.18 = ₹9,735
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9 · Simple vs Compound Interest
Growth and compounding — the magic of reinvestment
Principal (P) Initial deposit
Rate (r) Interest % per year
Time (t) Number of years
| Simple Interest (No Compounding) | Compound Interest |
|---|---|
| Interest same every year Principal stays fixed |
Interest added to principal Principal grows each year |
| Total = P(1 + rt) | Total = P(1 + r)ᵗ |
| ₹6000, 10%, 3 yrs = 6000 × (1 + 0.10 × 3) = 6000 × 1.3 = ₹7800 |
₹6000, 10%, 3 yrs = 6000 × 1.1 × 1.1 × 1.1 = 6000 × 1.331 = ₹7986 |
| Gain = 30% | Gain = 33.1% ✨ More! |
Simple (3yr)
₹7800
Compound (3yr)
₹7986
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Key Insight — Compounding is Exponential Growth!
Each year the principal is larger, so the interest earned is also larger.
Simple interest = linear growth
Compound interest = exponential growth
Simple interest = linear growth
Compound interest = exponential growth
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Amount after 1 year
Principal P at rate r → becomes P × (1 + r) after 1 year
e.g. ₹6000 at 10% → 6000 × 1.1 = ₹6600
e.g. ₹6000 at 10% → 6000 × 1.1 = ₹6600
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10 · Depreciation & Decline
When value decreases over time by a fixed %
📌
Depreciation
Reduction in value of an item due to use and age.
If depreciation rate = d% per year:
Value after 1 year = Current Value × (1 − d/100)
If depreciation rate = d% per year:
Value after 1 year = Current Value × (1 − d/100)
Value after t years = P × (1 − r)ᵗ
📺 TV Depreciation
Cost = ₹21,000
Depreciation = 5%/yr
After 1 yr = 21000 × 0.95
= ₹19,950
🏘️ Village Population
Current = 1250 | −10%/decade
After 3 decades:
= 1250 × 0.9 × 0.9 × 0.9
≈ 911
🧠
Growth vs Decline — Same Formula!
Growth: P × (1 + r)ᵗ → multiply by more than 1
Decline: P × (1 − r)ᵗ → multiply by less than 1
Decline: P × (1 − r)ᵗ → multiply by less than 1
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11 · Tricky Percentage Traps
Don't get fooled — common mistakes to avoid
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Trap 1 — 30% + 20% ≠ 50% in shopping!
When applied successively, discounts compound:
30% off ₹200 → ₹140 | then 20% off ₹140 → ₹112
But 50% off ₹200 = ₹100 (Cheaper!)
Always calculate step by step.
30% off ₹200 → ₹140 | then 20% off ₹140 → ₹112
But 50% off ₹200 = ₹100 (Cheaper!)
Always calculate step by step.
❌
Trap 2 — 50% margin then 50% discount = LOSS!
Buy at x → Mark up 50% → SP = 1.5x
Give 50% discount → Sell at 0.75x
0.75x < x → That's a 25% loss!
Give 50% discount → Sell at 0.75x
0.75x < x → That's a 25% loss!
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Trap 3 — Comparing % without considering the base
"80% growth" does NOT mean more members than "100% growth" club!
A club of 1000 growing 80% = +800 members
A club of 10 growing 100% = only +10 members
Always check what the base (whole) is!
A club of 1000 growing 80% = +800 members
A club of 10 growing 100% = only +10 members
Always check what the base (whole) is!
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Symmetry Property — Cool Trick!
x% of y = y% of x (Always!)
e.g. 5% of 40 = 2 = 40% of 5 ✅
Use this to make hard % easy: 48% of 50 = 50% of 48 = 24
e.g. 5% of 40 = 2 = 40% of 5 ✅
Use this to make hard % easy: 48% of 50 = 50% of 48 = 24
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12 · Quick Summary
Everything at a glance
✓
Percentage = fraction with denominator 100. x% = x/100
✓
Fraction → % : multiply by 100. % → fraction: divide by 100.
✓
y% of z = (y/100) × z. Base is always what "of" refers to.
✓
FDP Trio: Fractions, Decimals, Percentages are interchangeable forms.
✓
% can be >100 when the value exceeds the original whole/target.
✓
% Increase/Decrease: always calculated on the original (base) value.
✓
Profit % and Loss % are always calculated on Cost Price.
✓
Simple Interest: Total = P(1+rt) | Compound: Total = P(1+r)ᵗ
✓
Depreciation: Value after t years = P × (1−r)ᵗ
✓
Successive discounts compound — always calculate step by step!
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Always ESTIMATE before you calculate!
Drawing a rough bar model or diagram helps understand the problem and catch silly mistakes. Estimation builds number sense — a skill that lasts a lifetime!
🔢 🎭 💯
Fractions in Disguise
Chapter 1 · Ganita Prakash · Class 8
x% = x/100
P(1+r)ᵗ
FDP Trio
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