Percentage is one of the most practical and frequently tested topics in JTET Paper I Mathematics. The word "percent" comes from the Latin "per centum," meaning "per hundred." A percentage is simply a way of expressing a number as a fraction of 100, making it easier to compare quantities of different sizes.
This topic forms the foundation for many real-life calculations—discounts in shops, marks obtained in exams, population growth, and bank interest rates. For JTET, you must be comfortable converting between fractions, decimals, and percentages, and solving word problems involving increase, decrease, and comparison. Questions typically test both computational speed and conceptual clarity, so mastering the underlying logic is essential.
Percentage also connects directly to other Paper I topics like Profit-Loss-Discount and Simple Interest. A strong grip here will make those topics significantly easier.
Key Concepts
**Percentage means "out of 100"**: 25% means 25 out of every 100, written as 25/100 or 0.25 in decimal form.
**Conversion trio**: Any percentage can be written as a fraction (divide by 100) or decimal (move decimal point two places left), and vice versa. Example: 40% = 40/100 = 2/5 = 0.40.
**Finding percentage of a quantity**: To find x% of a number N, calculate (x/100) × N. Example: 15% of 200 = (15/100) × 200 = 30.
**What percentage is A of B?**: Use the formula (A/B) × 100. Example: 45 is what percent of 180? Answer: (45/180) × 100 = 25%.
**Percentage increase**: When a value rises from old to new, increase% = [(New − Old)/Old] × 100.
**Percentage decrease**: When a value falls, decrease% = [(Old − New)/Old] × 100.
**Successive percentage change**: If two successive changes of a% and b% occur, net effect = a + b + (ab/100). This can be positive or negative depending on increase/decrease.
**Reverse percentage (finding original)**: If a number after x% increase becomes N, the original = N × (100/(100 + x)).
Formulas / Key Facts
| Formula / Fact | Context | |----------------|---------| | x% of N = (x × N)/100 | Finding a percentage of any quantity | | (A/B) × 100 = percentage | Finding what percent A is of B | | Increase% = [(New − Old)/Old] × 100 | Calculating percentage increase | | Decrease% = [(Old − New)/Old] × 100 | Calculating percentage decrease | | Net change = a + b + (ab/100) | Successive changes of a% and b% | | Original = Final × [100/(100 ± change%)] | Finding original after increase (+) or decrease (−) | | 1/2 = 50%, 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5% | Common fraction-percentage equivalents | | 1/3 = 33.33%, 2/3 = 66.67%, 1/6 = 16.67% | Recurring decimal percentages | | If A is x% more than B, then B is [x/(100+x)] × 100% less than A | Reverse comparison between two quantities |
Worked Examples
**Example 1: Basic percentage calculation**
*A school has 450 students. If 36% are girls, how many boys are in the school?*
Step 1: Number of girls = 36% of 450 = (36/100) × 450 = 162
Step 2: Number of boys = 450 − 162 = 288
**Answer: 288 boys**
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**Example 2: Percentage increase**
*The price of rice increased from ₹40 per kg to ₹50 per kg. Find the percentage increase.*
Step 1: Increase = 50 − 40 = ₹10
Step 2: Percentage increase = (10/40) × 100 = 25%
**Answer: 25% increase**
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**Example 3: Successive percentage change**
*A shopkeeper increases the price of an item by 20% and then offers a 10% discount. What is the net percentage change in price?*
Step 1: Use formula: Net change = a + b + (ab/100)
*After a 25% increase, the population of a village became 7500. What was the original population?*
Step 1: Let original = x
Step 2: x + 25% of x = 7500 x × (125/100) = 7500
Step 3: x = 7500 × (100/125) = 6000
**Answer: Original population was 6000**
Common Mistakes
**Confusing base in increase/decrease**: Students calculate percentage change using the new value instead of the old (original) value. Always use the original value as the base for calculating percentage change.
**Adding successive percentages directly**: Assuming 20% increase followed by 10% decrease equals 10% increase. This is wrong because the second change applies to the changed value, not the original. Use the successive change formula instead.
**Forgetting to convert percentage to fraction/decimal**: Writing 25% × 400 = 10000 instead of (25/100) × 400 = 100. Always convert percentage to its fractional or decimal form before multiplying.
**Reversing the comparison incorrectly**: If A is 25% more than B, students assume B is 25% less than A. The correct calculation: B is (25/125) × 100 = 20% less than A.
**Misreading "of" vs "is"**: "What is 20% of 50" means find the value (answer: 10), while "20 is what percent of 50" asks for the percentage (answer: 40%). Read questions carefully.
Quick Reference
**Percent to fraction**: Divide by 100 (e.g., 75% = 75/100 = 3/4)
**Fraction to percent**: Multiply by 100 (e.g., 3/5 × 100 = 60%)
**x% of N = xN/100** — the most-used formula
**Percentage change always uses original as base**
**Successive changes: a + b + ab/100** — memorize this shortcut