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.
In CG TET, expect 2–4 questions directly or indirectly involving percentage. 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.
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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%.
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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%
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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
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### 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
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### 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%
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### 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
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Common Mistakes
1. **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%
2. **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.
3. **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
4. **Forgetting to convert percentage to decimal/fraction before multiplying**
- Wrong: 15% of 200 = 15 × 200 = 3000
- Correct: 15% of 200 = (15/100) × 200 = 30
5. **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%
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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