Percentage — Study Notes for CG TET Paper I
Overview
Percentage is one of the most practical and frequently tested topics in CG TET Paper I Mathematics. It forms the foundation for understanding profit-loss, simple interest, and data interpretation — all of which appear in the syllabus. For a primary school teacher, mastery of percentage is essential because it connects classroom mathematics to real-life situations like discounts, marks calculation, and population data.
Questions typically test basic conversion (fraction to percentage), finding percentage of a quantity, percentage increase/decrease, and simple word problems. The pedagogy section may also ask how to teach percentage using real-life examples from the Chhattisgarh context — local market prices, agricultural yield data, or school attendance figures.
Your goal: be fluent in quick mental calculations, understand the concept deeply enough to explain it to Class 4–5 students, and avoid common calculation traps.
Key Concepts
- Definition: Percent means "per hundred." Writing 25% means 25 out of 100, or 25/100 = 0.25.
- Fraction-Percentage-Decimal Triangle: Any fraction can be written as a percentage by multiplying by 100. Any percentage can be written as a decimal by dividing by 100. Example: 3/4 = 75% = 0.75.
- Base Value Concept: When we say "20% of 150," the number 150 is the base. Always identify the base clearly in word problems.
- Percentage Change: Increase or decrease is always calculated on the original (initial) value, not the new value.
- Successive Percentage: When two percentages apply one after another, they don't simply add. You must apply them step by step.
- Reversibility: If a value increases by x%, to restore it you don't decrease by x%. The reverse percentage is different because the base changes.
- Percentage Points vs Percentage: If pass percentage rises from 60% to 75%, the increase is 15 percentage points, but the percentage increase is (15/60) × 100 = 25%.
Formulas / Key Facts
| Concept | Formula |
|---|---|
| Percentage of a number | x% of N = (x/100) × N |
| Fraction to percentage | (a/b) × 100 % |
| Percentage to fraction | x% = x/100 (simplify) |
| Percentage increase | [(New − Old)/Old] × 100 |
| Percentage decrease | [(Old − New)/Old] × 100 |
| New value after increase | Old × (1 + x/100) |
| New value after decrease | Old × (1 − x/100) |
| Successive change (a% then b%) | Net effect = a + b + (ab/100) % |
| Reverse of x% increase | Decrease by [x/(100+x)] × 100 % |
| Reverse of x% decrease | Increase by [x/(100−x)] × 100 % |
Must-remember fraction-percentage equivalents:
- 1/2 = 50%, 1/4 = 25%, 3/4 = 75%
- 1/5 = 20%, 2/5 = 40%, 3/5 = 60%, 4/5 = 80%
- 1/8 = 12.5%, 1/3 = 33.33%, 2/3 = 66.67%
- 1/6 = 16.67%, 1/10 = 10%, 1/20 = 5%
Worked Examples
Example 1: Finding percentage of a quantity
Problem: A farmer in Raipur harvested 450 quintals of paddy. If 18% was damaged due to rain, how much paddy was damaged?
Solution:
- Damaged paddy = 18% of 450
- = (18/100) × 450
- = 18 × 4.5
- = 81 quintals
Answer: 81 quintals
Example 2: Percentage increase
Problem: The population of a village was 12,000 in 2020. It increased to 13,800 in 2023. Find the percentage increase.
Solution:
- Increase = 13,800 − 12,000 = 1,800
- Percentage increase = (Increase/Original) × 100
- = (1800/12000) × 100
- = (1800 × 100)/12000
- = 15%
Answer: 15% increase
Example 3: Successive percentage change
Problem: The price of rice increased by 20% in the first month, then decreased by 10% in the second month. What is the net percentage change?
Solution: Using the formula: Net = a + b + (ab/100)
- Here a = +20, b = −10
- Net = 20 + (−10) + (20 × −10)/100
- = 20 − 10 − 2
- = 8%
Verification: Let original price = ₹100
- After 20% increase = ₹120
- After 10% decrease on 120 = 120 − 12 = ₹108
- Net change = ₹8 on ₹100 = 8%
Answer: Net increase of 8%
Example 4: Reverse percentage
Problem: After a 25% increase, the salary of a teacher became ₹31,250. What was the original salary?
Solution:
- Let original salary = x
- After 25% increase: x × (1 + 25/100) = 31,250
- x × 1.25 = 31,250
- x = 31,250 ÷ 1.25
- x = 25,000
Answer: ₹25,000
Common Mistakes
- Calculating percentage change on the wrong base
- Wrong: "Price went from ₹80 to ₹100, so increase = 20/100 = 20%"
- Correct: Base is the original value (₹80), so increase = 20/80 × 100 = 25%
- Adding successive percentages directly
- Wrong: "20% increase then 20% decrease = no change"
- Correct: Apply the formula — net effect = 20 − 20 + (20 × −20)/100 = −4%. There is a 4% decrease.
- Confusing "percentage of" with "percentage more than"
- "A is 25% of B" means A = 0.25B
- "A is 25% more than B" means A = 1.25B
- Forgetting to convert percentage to decimal/fraction before multiplying
- Wrong: 15% of 200 = 15 × 200 = 3000
- Correct: 15% of 200 = (15/100) × 200 = 30
- Using the new value as base for reverse calculation
- Wrong: "If price increased by 20%, to get original, decrease new price by 20%"
- Correct: To reverse a 20% increase, decrease by (20/120) × 100 = 16.67%
Quick Reference
- Percent = Per Hundred: Always think "out of 100"
- Conversion shortcut: To find x% of N, move decimal two places left in x, then multiply by N
- Percentage change base: Always use the ORIGINAL value as denominator
- Successive change formula: a + b + (ab/100) — memorize this for quick calculation
- Common trap: 50% increase followed by 50% decrease ≠ 0% change (actually −25%)
- Teaching tip for CG TET pedagogy: Use local examples — percentage of tribal population, percentage of forest cover in Bastar, discount at local haat bazaar