Chapter 1: Fractions in Disguise
Mastering Percentages: FDP Trio, Commercial Arithmetic, Simple & Compound Growth, and Percentage Traps
💯 1. Fractions as Percentages
Per Centum ConceptDerived from Latin per centum, meaning "out of 100". A percentage is merely a fraction with a fixed denominator of 100.
x% = x / 100 | Fraction to % = Fraction × 100
• 3/4 = (3/4) × 100% = 75%
• 2/5 = (2/5) × 100% = 40%
• Simplifies direct comparison across unequal totals!
🔄 2. The FDP Trio Equivalence
Fraction • Decimal • %Every rational amount seamlessly transitions between three interconnected disguises:
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.50 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/10 | 0.10 | 10% |
| 1/1 | 1.00 | 100% |
📜 3. Ancient Historical Roots
Kautilya to RenaissanceKautilya's Arthashastra (4th Century BCE): First recorded use of interest computed "per month per cent" across trade, forests, and maritime commerce.
15th-century Italian merchants wrote per cento (p cento), which evolved into the modern % symbol.
🧠 4. Free-Hand Mental Math
Benchmark ChunkingBreak complex percentages into intuitive building blocks (10%, 5%, 1%):
- 10% of x: Shift decimal 1 place left (x / 10).
- 5% of x: Exactly half of 10% value.
- 20% of x: Double the 10% value.
- 15% of x: (10% value) + (5% value).
- Commutative Law: x% of y = y% of x (e.g., 16% of 250 = 250% of 16 = 2.5 × 16 = 40)!
📈 5. Percentages > 100%
Scale & MultipliersPercentages exceeding 100% represent values larger than the original reference base:
• 100% of Base: Equal to Base (1.0 × Base)
• 150% of Base: 1.5 × Base (50% increase)
• 200% of Base: Exactly 2.0 × Base (Doubled)
• 250% of Base: 2.5 × Base (e.g., 650 kg harvest vs 260 kg target = 250%)
🏷️ 6. Commercial Profit & Loss
CP • SP • MP • GSTProfit / Loss % is ALWAYS calculated on Cost Price (CP):
- Profit %: [(SP − CP) / CP] × 100
- Loss %: [(CP − SP) / CP] × 100
- Discount %: Calculated on Marked Price (MP) → SP = MP − Discount.
- GST / Taxes: Added onto final Selling Price → Total = SP × (1 + GST%).
📊 Growth Models: Simple Growth vs Compounding vs Depreciation
Mathematical models governing financial interest, demographic growth, bacteria count, and asset decline:
| Financial / Physical Model | Core Mechanism | Mathematical Formula | 3-Year Example (P = ₹6,000, r = 10% p.a.) |
|---|---|---|---|
| Simple Growth (No Compounding) | Interest is paid out regularly; Principal remains fixed throughout. | Amount = P(1 + rt) Interest = P × r × t |
Amount = 6000 × (1 + 0.10 × 3) = ₹7,800 (30% total gain) |
| Compounded Growth | Interest is added back to principal at each term (Exponential growth). | Amount = P(1 + r)t | Amount = 6000 × (1.10)3 = ₹7,986 (33.1% total gain) |
| Depreciation / Decay | Value decreases at a constant percentage rate each period due to use/wear. | Value = P(1 − r)t | TV bought at ₹21,000 (r = 5% decay) → 21000 × 0.95 = ₹19,950 |
⚠️ 7. Tricky Percentages & Retail Traps
Successive Discounts Trap: "30% + 20% OFF" is NOT 50% OFF!
- Cakely (30% + 20% on ₹200 cake): 1st discount → ₹140. 2nd discount (20% of ₹140 = ₹28) → Final Price = ₹112 (Effective discount = 44%).
- Cakify (Single 50% on ₹200 cake): Final Price = ₹100 (Cheaper!).
- The Equal Gain/Loss Fallacy: Setting a +50% margin and then applying a −50% discount results in a 25% net loss (1.50 × 0.50 = 0.75)!
♟️ 8. Puzzle Time: Peaceful Knights
The Challenge: Place 8 knights on an 8 × 8 chessboard such that no knight attacks any other knight.
Mathematical Principle: A knight always moves between opposite colors (from white to red, or red to white).
Solution Strategy: Placing all 8 knights exclusively on squares of the same color (e.g., all on white squares, such as an entire row/file) ensures they can never attack one another!
⚠️ Common Pitfalls & Exam Misconceptions
Comparing Raw Percentages
A larger percentage does not guarantee a larger amount unless the base is equal! 25% of 120 g (30 g) is greater than 35% of 80 g (28 g).
Base Switching in Profit/Loss
Profit % is calculated over Cost Price (CP), not Selling Price (SP). If a ₹500 item is sold for ₹750, the profit is 50% on cost (250/500), but 33.3% on revenue (250/750).
Relative Growth Illusion
A small club growing from 1 to 2 members grew by 100%, whereas a club growing from 500 to 900 grew by 80%. High percentage growth can hide small absolute size!