Percentage — Study Notes for TNPSC Group II/IIA
Overview
Percentage is one of the most fundamental and frequently tested topics in the Aptitude section of TNPSC Group II/IIA. The word "percent" literally means "per hundred" — it represents a fraction with denominator 100. Almost every quantitative topic you'll encounter (profit-loss, simple/compound interest, data interpretation, ratio-proportion) builds upon percentage calculations.
Questions typically involve: finding percentage of a quantity, percentage increase/decrease, successive percentage changes, and comparison problems. Mastering quick mental calculation techniques here will save valuable time across the entire Aptitude section.
The key to scoring well is not just knowing formulas but developing the ability to convert between fractions, decimals, and percentages instantly. Students who memorize common fraction-percentage equivalents consistently outperform those who calculate everything from scratch.
Key Concepts
- Basic Definition: x% = x/100. To find x% of a number N, calculate (x × N)/100 or equivalently N × (x/100).
- Fraction-Percentage Conversion: Any fraction a/b can be converted to percentage by multiplying by 100. Thus, a/b = (a/b) × 100 %.
- Percentage Increase: When a quantity increases from A to B, percentage increase = [(B − A)/A] × 100. Always divide by the original value.
- Percentage Decrease: When a quantity decreases from A to B, percentage decrease = [(A − B)/A] × 100.
- Successive Percentage Change: For two successive changes of a% and b%, the net effect = a + b + (ab/100) %. This single formula handles both increases (positive) and decreases (negative).
- Reverse Percentage: If a value after x% increase is V, then original = V × 100/(100 + x). For decrease, original = V × 100/(100 − x).
- Population/Depreciation Formula: After n periods of r% growth, final value = P × (1 + r/100)ⁿ. For depreciation, use (1 − r/100)ⁿ.
- Base Change Awareness: "A is what percent of B" and "B is what percent of A" give different answers. Identify the base (denominator) carefully.
Formulas / Key Facts
| Concept | Formula |
|---|---|
| x% of N | (x × N)/100 |
| Percentage increase | [(New − Old)/Old] × 100 |
| Percentage decrease | [(Old − New)/Old] × 100 |
| Successive change (a% then b%) | Net % = a + b + (ab/100) |
| Original after x% increase | Original = Final × 100/(100 + x) |
| Original after x% decrease | Original = Final × 100/(100 − x) |
| After n periods at r% growth | P × (1 + r/100)ⁿ |
Must-Memorize Fraction-Percentage Equivalents:
- 1/2 = 50%, 1/3 = 33.33%, 1/4 = 25%, 1/5 = 20%
- 1/6 = 16.67%, 1/7 = 14.28%, 1/8 = 12.5%, 1/9 = 11.11%
- 1/10 = 10%, 1/11 = 9.09%, 1/12 = 8.33%
- 2/3 = 66.67%, 3/4 = 75%, 2/5 = 40%, 3/5 = 60%
Worked Examples
Example 1: Basic Percentage Calculation
Question: What is 35% of 840?
Solution:
- 35% of 840 = (35 × 840)/100
- = 35 × 8.4
- = 294
Shortcut: 10% of 840 = 84. So 30% = 252, and 5% = 42. Total = 252 + 42 = 294
Example 2: Percentage Increase/Decrease
Question: A shopkeeper's sales increased from ₹45,000 to ₹54,000. Find the percentage increase.
Solution:
- Increase = 54,000 − 45,000 = ₹9,000
- Percentage increase = (9,000/45,000) × 100
- = (9/45) × 100 = (1/5) × 100
- = 20%
Example 3: Successive Percentage Change
Question: The price of a commodity first increases by 20% and then decreases by 10%. What is the net percentage change?
Solution:
- Using formula: Net % = a + b + (ab/100)
- Here a = +20, b = −10
- Net % = 20 + (−10) + (20 × −10)/100
- = 20 − 10 − 2
- = +8% (Net increase of 8%)
Example 4: Reverse Percentage
Question: After a 15% increase, a person's salary becomes ₹46,000. What was the original salary?
Solution:
- Let original salary = X
- X + 15% of X = 46,000
- X × (115/100) = 46,000
- X = 46,000 × (100/115)
- X = 46,000 × (20/23)
- X = ₹40,000
Common Mistakes
- Wrong base selection → When asked "A is what percent more than B," students often divide by A instead of B. Fix: The phrase "more than B" means B is the base; divide by B.
- Adding successive percentages directly → Students calculate 20% increase followed by 20% decrease as "net zero." Fix: Use the formula a + b + (ab/100). Here: 20 − 20 − 4 = −4%, a net decrease.
- Confusing percentage change with percentage points → If interest rate moves from 8% to 10%, the increase is 2 percentage points, but 25% change. Fix: Read questions carefully for "percentage points" vs "percent change."
- Forgetting to convert back → After finding a fraction like 3/8, students forget to multiply by 100 to express as percentage. Fix: Always check if answer should be in % form.
- Misapplying growth formula → Using simple multiplication instead of compounding for multi-year problems. Fix: For repeated percentage changes over time, always use (1 ± r/100)ⁿ.
Quick Reference
- x% of N = N × x/100 — multiply first, then divide for easier calculation.
- Percentage change = (Difference/Original) × 100 — original is always the base.
- Successive changes: a + b + ab/100 — signs matter; use negative for decrease.
- To reverse an x% increase, multiply by 100/(100+x).
- 1/8 = 12.5%, 1/6 = 16.67%, 1/7 ≈ 14.28% — memorize these for speed.
- If A is x% more than B, then B is [x/(100+x)] × 100 % less than A.