Percentage

Percentage — Study Notes for Bihar TET Paper I

Overview

Percentage is one of the most scoring and frequently tested topics in Bihar TET Paper I Mathematics. It forms the foundation for understanding profit-loss, discount, simple interest, and data interpretation — all of which appear in the exam. A primary-school teacher must be able to explain percentage concepts to young learners and solve standard calculation problems quickly.

The topic tests two core skills: converting between fractions, decimals, and percentages, and applying percentage concepts to real-life situations like increase, decrease, and comparison. Mastering this topic also helps in the pedagogy section when discussing how to teach commercial mathematics to children.

For exam success, focus on memorising common fraction-percentage equivalents, understanding the concept of "base value," and practising word problems involving percentage change.

Key Concepts

• Meaning of Percent: "Percent" means "per hundred." Writing 25% means 25 out of every 100, or 25/100.
• Conversion Rule: To convert percentage to fraction, divide by 100. To convert fraction to percentage, multiply by 100. For example, 3/4 = (3/4) × 100 = 75%.
• Percentage of a Quantity: To find x% of a number N, calculate (x/100) × N. For example, 20% of 150 = (20/100) × 150 = 30.
• What Percent One Number is of Another: If A is what percent of B, then answer = (A/B) × 100. For example, 45 is what percent of 180? Answer = (45/180) × 100 = 25%.
• Percentage Increase: When a value increases from old to new, percentage increase = [(New − Old)/Old] × 100.
• Percentage Decrease: When a value decreases from old to new, percentage decrease = [(Old − New)/Old] × 100.
• Base Value Matters: Always identify which quantity is the "base" (denominator). A common exam trap is confusing the base in increase vs decrease problems.
• Successive Percentage Change: When two percentage changes occur one after another, the net effect is NOT the simple sum. Use the formula for combined effect.

Formulas / Key Facts

Must-Memorise Equivalents:

• 1/2 = 50%
• 1/3 = 33.33% (or 33⅓%)
• 1/4 = 25%
• 1/5 = 20%
• 1/8 = 12.5%
• 1/10 = 10%
• 2/5 = 40%
• 3/4 = 75%
• 1/6 = 16.67% (or 16⅔%)

Worked Examples

Example 1: Finding Percentage of a Quantity

Question: A school has 450 students. If 36% are girls, how many girls are there?

Solution:

• Girls = 36% of 450
• = (36/100) × 450
• = 36 × 4.5
• = 162

Answer: 162 girls

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Example 2: What Percent One Number is of Another

Question: In a class of 80 students, 64 passed the exam. What percentage of students passed?

Solution:

• Percentage passed = (Number passed / Total students) × 100
• = (64/80) × 100
• = 0.8 × 100
• = 80%

Answer: 80% passed

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Example 3: Percentage Increase

Question: The price of rice increased from ₹40 per kg to ₹50 per kg. Find the percentage increase.

Solution:

• Increase = New price − Old price = 50 − 40 = ₹10
• Percentage increase = (Increase/Old price) × 100
• = (10/40) × 100
• = 25%

Answer: 25% increase

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Example 4: Successive Percentage Change

Question: A shopkeeper increases the price of an item by 20% and then gives a 10% discount. What is the net percentage change?

Solution:

• Using formula: Net change = a + b + (ab)/100
• Here a = +20, b = −10
• Net change = 20 + (−10) + (20 × −10)/100
• = 20 − 10 − 2
• = 8%

Answer: 8% net increase

Common Mistakes

• Wrong base in percentage change → Students calculate (50−40)/50 × 100 = 20% instead of using old value 40 as base. Fix: For increase/decrease, always use the ORIGINAL value as the denominator.
• Adding successive percentages directly → Thinking 20% increase followed by 10% decrease equals 10% increase. Fix: Use the successive change formula or calculate step by step on actual values.
• Confusing "of" and "is" → In "A is what percent of B," students sometimes divide B by A. Fix: Remember the word "of" comes after the base. A/B × 100.
• Forgetting to multiply by 100 → Writing the answer as 0.25 instead of 25%. Fix: Percentage is always expressed as a number out of 100, so multiply the decimal by 100.
• Misreading the question → Finding percentage increase when percentage of total is asked, or vice versa. Fix: Underline key words — "increase," "decrease," "of total," "more than," "less than."

Quick Reference

• Percent means "out of 100" — always divide by 100 to convert.
• x% of N = (x × N) ÷ 100
• For percentage change, base is always the ORIGINAL value.
• Memorise: 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%
• Successive changes: a% then b% gives net = a + b + ab/100
• When population or price increases by x%, new value = old × (1 + x/100)