Percentage — Study Notes for Assam TET Paper I
Overview
Percentage is one of the most frequently tested topics in the Mathematics section of Assam TET Paper I. It forms the foundation for understanding profit-loss, simple interest, and data interpretation problems that appear in the exam. The word "percent" comes from Latin "per centum" meaning "per hundred," so percentage is simply a way of expressing a number as a fraction of 100.
For primary-level teaching, you must not only solve percentage problems yourself but also understand how to make this concept accessible to young learners. Questions typically test your ability to convert between fractions, decimals, and percentages, calculate percentage increase or decrease, and apply percentages to real-life situations like discounts, marks obtained, or population changes.
Mastering percentage requires comfort with basic arithmetic operations and a clear understanding of the relationship between part, whole, and rate. Most exam questions can be solved quickly using standard formulas or mental math shortcuts once the core concepts are internalised.
Key Concepts
- **Percentage means "out of 100"**: When we say 25%, we mean 25 out of every 100, or 25/100, or 0.25 as a decimal.
- **The three quantities in any percentage problem**: Part (the portion we're finding), Whole (the total or base), and Rate (the percentage). If you know any two, you can find the third.
- **Percentage of a number**: To find x% of a number N, calculate (x/100) × N or equivalently (x × N)/100.
- **Converting fractions to percentage**: Multiply the fraction by 100. For example, 3/5 = (3/5) × 100 = 60%.
- **Percentage change always uses original value as base**: Whether increase or decrease, the change is calculated with respect to the original (initial) value, not the new value.
- **Successive percentage changes do not add directly**: A 10% increase followed by a 10% decrease does not return to the original — there is always a net change.
- **Percentage points vs percentage**: An increase from 20% to 25% is a rise of 5 percentage points, but a 25% increase in the rate itself.
Formulas / Key Facts
| Formula/Fact | Context | |--------------|---------| | Percentage = (Part/Whole) × 100 | Finding what percentage one number is of another | | Part = (Percentage × Whole)/100 | Finding a given percentage of a number | | Whole = (Part × 100)/Percentage | Finding the total when part and percentage are known | | Percentage Increase = [(New − Original)/Original] × 100 | When value goes up | | Percentage Decrease = [(Original − New)/Original] × 100 | When value goes down | | New Value after x% increase = Original × (100 + x)/100 | Quick calculation for increased value | | New Value after x% decrease = Original × (100 − x)/100 | Quick calculation for decreased value | | 1/2 = 50%, 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%, 1/10 = 10% | Must-memorise fraction-to-percentage conversions |
Worked Examples
**Example 1: Finding percentage of a number**
*A student scored 72 marks out of 80. What is the percentage of marks obtained?*
Step 1: Identify Part = 72, Whole = 80 Step 2: Apply formula: Percentage = (72/80) × 100 Step 3: Simplify: 72/80 = 9/10 = 0.9 Step 4: Calculate: 0.9 × 100 = 90%
**Answer: 90%**
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**Example 2: Finding a given percentage of a number**
*What is 35% of 240?*
Step 1: Convert percentage to fraction: 35% = 35/100 Step 2: Multiply: (35/100) × 240 = (35 × 240)/100 Step 3: Calculate: 8400/100 = 84
**Answer: 84**
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**Example 3: Percentage increase problem**
*The population of a village increased from 25,000 to 28,000. Find the percentage increase.*
Step 1: Find the increase: 28,000 − 25,000 = 3,000 Step 2: Use original value as base: Percentage increase = (3,000/25,000) × 100 Step 3: Simplify: 3000/25000 = 3/25 = 0.12 Step 4: Calculate: 0.12 × 100 = 12%
**Answer: 12% increase**
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**Example 4: Reverse percentage (finding original value)**
*After a 20% discount, a shirt costs ₹480. What was the original price?*
Step 1: After 20% discount, the price is 80% of original (100% − 20% = 80%) Step 2: Let original price = x Step 3: 80% of x = 480, so (80/100) × x = 480 Step 4: x = 480 × (100/80) = 480 × 1.25 = 600
**Answer: ₹600**
Common Mistakes
- **Using new value instead of original for percentage change**: Students often calculate percentage increase as (Change/New Value) × 100. This is wrong. Always divide by the original value. Fix: Ask yourself "what was the starting point?" and use that as the denominator.
- **Adding successive percentages directly**: A 20% increase followed by a 10% decrease is not a net 10% increase. The correct approach is to apply each percentage sequentially. A ₹100 item after 20% increase becomes ₹120, then after 10% decrease becomes ₹108 (not ₹110). Fix: Calculate step by step, not by adding/subtracting percentages.
- **Forgetting to multiply by 100 when converting to percentage**: Students sometimes write 72/80 = 0.9 as the final answer instead of 90%. Fix: Remember that percentage must be expressed "per hundred" — always multiply the decimal by 100.
- **Confusing "percentage of" with "percentage more than"**: 25% of 80 is 20, but 25% more than 80 is 80 + 20 = 100. Fix: Read the question carefully — "of" means direct multiplication, "more than" means add the percentage to the original.
- **Calculating discount on selling price instead of marked price**: Discount is always calculated on the marked/original price, not on the reduced price. Fix: Identify which price is given and ensure you're using the correct base.
Quick Reference
- **x% of N = (x × N)/100** — the most-used formula
- **To convert fraction to percentage, multiply by 100**
- **Percentage change = (Change/Original) × 100 — always use original as base**
- **After x% increase, multiply original by (100 + x)/100**
- **After x% decrease, multiply original by (100 − x)/100**
- **Memorise: 1/4 = 25%, 1/5 = 20%, 1/3 = 33.33%, 1/8 = 12.5%**