Percentage — Study Notes for OTET Paper I
Overview
Percentage is one of the most frequently tested topics in OTET Paper I Mathematics. It forms the foundation for many real-life calculations and connects directly to other arithmetic topics like profit-loss, simple interest, and data interpretation. Questions typically test your ability to convert between fractions, decimals, and percentages, calculate percentage increase or decrease, and solve word problems involving discounts, marks, and population changes.
For primary-level teaching, understanding percentage is essential because teachers must help young learners connect the abstract concept of "per hundred" to concrete situations like exam scores, discounts in shops, and recipe adjustments. Expect 2–4 questions on percentage or its direct applications in the exam. Mastery here also speeds up your work in related topics.
Key Concepts
- **Meaning of Percent**: "Percent" means "per hundred" or "out of 100." Writing 25% means 25 out of every 100 units.
- **Conversion Triangle**: Percentage ↔ Fraction ↔ Decimal are interchangeable. Moving fluently between these three forms is the core skill.
- **Base Value Matters**: Percentage is always calculated with respect to some base (original) value. Identifying the correct base is crucial in word problems.
- **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 Changes**: When two percentage changes occur one after another, they cannot simply be added. The second change acts on the result of the first.
- **Percentage Points vs Percentage Change**: A rise from 20% to 25% is an increase of 5 percentage points but a 25% increase in the rate itself. Exam questions sometimes test this distinction.
- **Reverse Percentage Problems**: If a value after increase/decrease is given, finding the original requires working backwards using the percentage formula.
Formulas / Key Facts
**Basic Conversion Formulas**
- Percentage to Fraction: x% = x/100
- Fraction to Percentage: (a/b) × 100%
- Decimal to Percentage: Multiply by 100 (0.45 = 45%)
- Percentage to Decimal: Divide by 100 (45% = 0.45)
**Finding Percentage of a Quantity**
- x% of N = (x/100) × N
**What Percentage is A of B?**
- Percentage = (A/B) × 100%
**Percentage Increase**
- New Value = Original + (Increase% × Original) = Original × (1 + Increase%/100)
- Percentage Increase = [(New − Original)/Original] × 100%
**Percentage Decrease**
- New Value = Original × (1 − Decrease%/100)
- Percentage Decrease = [(Original − New)/Original] × 100%
**Successive Percentage Changes**
- If a% and b% are successive changes: Net effect = a + b + (ab/100)%
- Use + for increase, − for decrease in the formula.
**Commonly Used Fraction-Percent 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%, 1/20 = 5%
Worked Examples
**Example 1: Finding Percentage of a Number**
*Question*: Find 35% of 240.
*Solution*:
- 35% of 240 = (35/100) × 240
- = 35 × 2.4
- = 84
**Answer**: 84
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**Example 2: What Percentage is A of B?**
*Question*: In a class of 50 students, 12 are absent. What percentage of students are absent?
*Solution*:
- Percentage absent = (12/50) × 100%
- = (12 × 100)/50
- = 1200/50
- = 24%
**Answer**: 24%
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**Example 3: Percentage Increase**
*Question*: The price of a book increased from ₹150 to ₹180. Find the percentage increase.
*Solution*:
- Increase = 180 − 150 = ₹30
- Percentage Increase = (30/150) × 100%
- = (30 × 100)/150
- = 3000/150
- = 20%
**Answer**: 20%
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**Example 4: Reverse Percentage (Finding Original)**
*Question*: After a 20% discount, a shirt costs ₹480. What was the original price?
*Solution*:
- After 20% discount, the shirt costs 80% of original price.
- 80% of Original = 480
- Original = 480 × (100/80)
- = 480 × 1.25
- = ₹600
**Answer**: ₹600
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**Example 5: Successive Percentage Changes**
*Question*: A population first increases by 10% and then decreases by 10%. Find the net percentage change.
*Solution*:
- Using formula: Net change = a + b + (ab/100)
- Here a = +10, b = −10
- Net change = 10 + (−10) + (10 × −10)/100
- = 0 + (−100/100)
- = 0 − 1
- = −1%
**Answer**: Net decrease of 1%
Common Mistakes
- **Taking percentage of the wrong base**: When finding percentage decrease, students sometimes divide by the new value instead of the original. → Always use the original value as the base unless specifically asked otherwise.
- **Adding successive percentages directly**: A 20% increase followed by 20% decrease is NOT 0% net change. → Apply each percentage to the resulting value, or use the successive change formula.
- **Confusing "percentage of" with "percentage more than"**: 120% of a value means the whole plus 20% more, not just 20%. → Read carefully: "of" means multiply; "more than" means add to 100%.
- **Forgetting to multiply by 100 when converting to percentage**: Students write 12/50 = 0.24 and stop there. → Always multiply by 100 to express as percentage (0.24 × 100 = 24%).
- **Calculation errors with fractions**: Converting 1/3 to percentage as 30% instead of 33.33%. → Memorise common fraction-percent equivalents to avoid such slips.
Quick Reference
- Percent means "per hundred" — always think "out of 100."
- x% of N = (x × N)/100
- Percentage = (Part/Whole) × 100%
- For reverse problems: Original = Final Value × (100/Remaining%)
- Successive changes: Net = a + b + ab/100 (use signs correctly)
- Memorise: 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%, 1/3 ≈ 33.33%