Percentage — Study Notes for TNPSC Group IV
Overview
Percentage is one of the most fundamental topics in the Aptitude section of TNPSC Group IV. It forms the foundation for several other important topics like Profit & Loss, Simple & Compound Interest, and Data Interpretation. You cannot afford to be slow or inaccurate here.
The exam typically tests your ability to convert between percentages, fractions, and decimals, calculate percentage increase or decrease, and apply percentage concepts to real-world scenarios like population, price changes, and examination results. Mastering percentage calculations with speed is essential because these questions appear directly and also as sub-steps in other problems.
Questions are usually straightforward but designed to trap careless students. Focus on understanding the "of what" part—percentage is always calculated with respect to some base value, and identifying that base correctly is half the battle.
Key Concepts
- Percentage means "per hundred": 25% simply means 25 out of 100, written as 25/100 or 0.25.
- Base value identification: When we say "A is 20% more than B," the base is B. When we say "A is 20% of B," the base is still B. Always ask: "percentage of what?"
- Percentage increase formula: If a value increases from Original to New, then Percentage Increase = [(New − Original) / Original] × 100.
- Percentage decrease formula: Percentage Decrease = [(Original − New) / Original] × 100.
- Successive percentage changes don't add directly: A 10% increase followed by a 10% decrease does NOT give you the original value. The net effect formula handles this.
- Percentage points vs percentage change: Going from 40% to 50% is a 10 percentage point increase, but a 25% increase in the percentage itself (10/40 × 100).
- Multiplying factors: A 20% increase means multiplying by 1.20; a 20% decrease means multiplying by 0.80. This shortcut speeds up calculations enormously.
Formulas / Key Facts
| Concept | Formula |
|---|---|
| Percentage of a number | x% of N = (x/100) × N |
| What percent is A of B | (A/B) × 100 |
| Percentage increase | [(New − Old) / Old] × 100 |
| Percentage decrease | [(Old − New) / Old] × 100 |
| Net effect of successive changes a% and b% | a + b + (ab/100) % |
| If A is x% more than B | A = B × (1 + x/100) |
| If A is x% less than B | A = B × (1 − x/100) |
| To reverse a% increase | Decrease by [a/(100+a)] × 100 % |
| To reverse a% decrease | Increase by [a/(100−a)] × 100 % |
Must-memorize fraction-percentage conversions:
- 1/2 = 50%, 1/3 = 33.33%, 1/4 = 25%, 1/5 = 20%
- 1/6 = 16.67%, 1/8 = 12.5%, 1/10 = 10%, 1/12 = 8.33%
- 2/3 = 66.67%, 3/4 = 75%, 2/5 = 40%, 3/5 = 60%
Worked Examples
Example 1: Basic Percentage Calculation
A student scored 72 marks out of 90. What is the percentage?
Percentage = (Marks obtained / Total marks) × 100 = (72/90) × 100 = (4/5) × 100 = 80%
Answer: 80%
Example 2: Percentage Increase
The price of rice increased from ₹40 per kg to ₹46 per kg. Find the percentage increase.
Percentage Increase = [(New − Old) / Old] × 100 = [(46 − 40) / 40] × 100 = (6/40) × 100 = 15%
Answer: 15%
Example 3: Successive Percentage Changes
The population of a town increases by 10% in the first year and decreases by 10% in the second year. If the original population was 10,000, what is the population after two years?
Method 1 (Step-by-step): After Year 1: 10,000 × 1.10 = 11,000 After Year 2: 11,000 × 0.90 = 9,900
Method 2 (Net effect formula): Net change = 10 + (−10) + (10 × −10)/100 = 10 − 10 − 1 = −1%
Final population = 10,000 × 0.99 = 9,900
Answer: 9,900
Example 4: Reverse Percentage
If A's salary is 25% more than B's salary, then B's salary is what percent less than A's salary?
Let B's salary = 100 A's salary = 125
Difference = 25 B is less than A by = (25/125) × 100 = 20%
Shortcut: Required % = [x/(100+x)] × 100 = [25/125] × 100 = 20%
Answer: 20%
Example 5: Expenditure Problem
The price of sugar increases by 25%. By what percent should a family reduce consumption so that expenditure remains the same?
Using formula: Reduction = [x/(100+x)] × 100 = [25/125] × 100 = 20%
Answer: 20%
Common Mistakes
- Confusing the base value: "A is 30% more than B" and "B is 30% less than A" are NOT the same. In the first, base is B; in the second, base is A. If A = 130 and B = 100, then B is not 30% less than A—it's about 23% less.
- Adding successive percentages directly: A 20% increase followed by a 30% increase is NOT a 50% increase. Use the net effect formula: 20 + 30 + (20×30)/100 = 56%.
- Forgetting that percentage decrease then increase doesn't restore original: 50% decrease followed by 50% increase gives net = 50 − 50 + (−2500/100) = −25%, not zero.
- Calculation errors with decimals: When finding 37.5% of 240, convert to fraction first: 37.5% = 3/8, so answer = 240 × 3/8 = 90. Faster and error-free.
- Misreading "percentage points": If pass percentage rises from 60% to 75%, the increase is 15 percentage points, but the percentage increase is (15/60) × 100 = 25%.
Quick Reference
- Percentage = (Part/Whole) × 100 — always identify the "whole" correctly.
- For increase by x%, multiply by (100+x)/100; for decrease, multiply by (100−x)/100.
- Successive changes: Net % = a + b + ab/100.
- To reverse x% increase, decrease by 100x/(100+x) percent.
- Memorize fraction-percentage pairs: 1/4 = 25%, 1/8 = 12.5%, 1/6 ≈ 16.67%.
- When in doubt, assume base = 100 and work out the actual numbers.