Percentage — Study Notes for HP TET
Overview
Percentage is one of the most frequently tested topics in the HP TET Mathematics section because it connects arithmetic to real-world applications that teachers must explain to students. The word "percent" comes from Latin *per centum*, meaning "out of hundred." This single concept forms the foundation for profit-loss, discount, simple/compound interest, and data interpretation questions.
For HP TET, you need two things: speed in converting between fractions, decimals, and percentages, and clarity in setting up word problems correctly. Most questions are straightforward if your conceptual base is solid, but careless errors in identifying the "base" value cost many candidates easy marks.
Mastering percentage also helps you teach Classes 5–8 effectively, where NCF emphasizes connecting mathematics to daily-life contexts like shopping discounts, election results, and nutritional labels.
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Key Concepts
- **Definition**: Percentage means "per hundred." Writing 25% is the same as writing 25/100 or 0.25.
- **Conversion Triangle**: Fraction → Percentage: multiply by 100. Percentage → Fraction: divide by 100. Decimal → Percentage: shift decimal two places right.
- **Base Value Principle**: Percentage is always calculated *of* something. Identifying the correct base is the single most important step in any percentage problem.
- **Percentage Change**: When a quantity increases or decreases, the change is expressed as a percentage of the *original* value, not the new value.
- **Successive Percentage Change**: When two percentage changes happen one after another, they do not simply add up. A 10% increase followed by a 10% decrease does *not* return to the original.
- **Reverse Percentage**: If a value *after* increase/decrease is given, you work backward to find the original. The base shifts, so the formula changes.
- **Percentage Points vs Percentage**: A rise from 40% to 50% is a 10 *percentage point* increase but a 25% *percentage increase* (10 is 25% of 40).
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Formulas / Key Facts
| Concept | Formula | |---------|---------| | Percentage of a number | (Percentage × Number) / 100 | | What percent is A of B? | (A / B) × 100 | | Percentage Increase | [(New − Original) / Original] × 100 | | Percentage Decrease | [(Original − New) / Original] × 100 | | New value after x% increase | Original × (1 + x/100) | | New value after x% decrease | Original × (1 − x/100) | | Original value (given new after x% increase) | New / (1 + x/100) | | Successive changes of a% and b% | Net effect = a + b + (ab/100) % |
**Must-Remember Fraction-Percentage Equivalents**
- 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%, 4/5 = 80%
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Worked Examples
### Example 1: Basic Percentage Calculation **Problem**: What is 35% of 240?
**Solution**: Percentage of a number = (Percentage × Number) / 100 = (35 × 240) / 100 = 8400 / 100 = **84**
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### Example 2: Finding Percentage **Problem**: In a class of 60 students, 45 passed an exam. What percentage of students passed?
**Solution**: Percentage = (Part / Whole) × 100 = (45 / 60) × 100 = 0.75 × 100 = **75%**
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### Example 3: Percentage Increase **Problem**: The price of a book increased from ₹400 to ₹460. Find the percentage increase.
**Solution**: Increase = 460 − 400 = ₹60 Percentage increase = (Increase / Original) × 100 = (60 / 400) × 100 = 15%
**Answer**: The price increased by **15%**.
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### Example 4: Successive Percentage Changes **Problem**: A shopkeeper increases the price of an item by 20% and then offers a 20% discount. What is the net percentage change?
**Solution**: Using the formula: Net effect = a + b + (ab/100) Here a = +20 (increase), b = −20 (decrease) Net effect = 20 + (−20) + (20 × −20)/100 = 0 + (−400/100) = −4%
**Answer**: There is a net **4% decrease** from the original price.
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### Example 5: Reverse Percentage **Problem**: After a 25% increase, the salary of an employee became ₹50,000. What was the original salary?
**Solution**: Let original salary = x After 25% increase: x × (1 + 25/100) = 50,000 x × 1.25 = 50,000 x = 50,000 / 1.25 x = **₹40,000**
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Common Mistakes
1. **Using the wrong base in percentage change**
- *Wrong thinking*: Price dropped from ₹100 to ₹80, then rose back to ₹100 — so 20% down and 20% up.
- *Correct fix*: The second change is from ₹80, so rising to ₹100 is a (20/80) × 100 = 25% increase, not 20%.
2. **Adding successive percentages directly**
- *Wrong thinking*: 10% increase + 10% increase = 20% increase.
- *Correct fix*: Use the successive formula: 10 + 10 + (10×10)/100 = 21% increase.
3. **Confusing "percentage of" with "percentage more than"**
- *Wrong thinking*: "A is 25% more than B" means A = 25% of B.
- *Correct fix*: A = B + 25% of B = 1.25B.
4. **Ignoring decimal precision in conversions**
- *Wrong thinking*: 1/3 = 33% exactly.
- *Correct fix*: 1/3 = 33.33...% (recurring). In MCQs, use 33.33% or the fraction form for accuracy.
5. **Reversing the fraction in "what percent" questions**
- *Wrong thinking*: What percent of 25 is 5? Answer: 25/5 = 5 = 500%.
- *Correct fix*: (5/25) × 100 = 20%. The smaller number goes in the numerator when finding what percent one number is of another.
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Quick Reference
- **Percent means per hundred**: x% = x/100
- **Percentage change formula**: [(Change) / Original] × 100
- **Successive changes**: a + b + ab/100 (use signs correctly)
- **Reverse calculation**: Divide by (1 ± rate) to find original
- **1/8 = 12.5%** and **1/6 ≈ 16.67%** — memorise for speed
- **Base always matters**: Always identify *of what* before calculating