Percentage — Study Notes for JKTET Paper I

Overview

Percentage is one of the most frequently tested topics in the Mathematics section of JKTET Paper I. It forms the foundation for understanding profit and loss, simple interest, discount, and data interpretation — all of which appear in the syllabus. Examiners expect candidates to convert between fractions, decimals, and percentages fluently, and to apply percentage concepts to real-world situations that primary teachers encounter.

For JKTET, you must master three core skills: converting between different forms (fraction ↔ decimal ↔ percentage), calculating percentage of a quantity, and solving word problems involving percentage increase, decrease, and comparison. Questions typically involve contexts familiar to J&K students — population data, agricultural yields, school attendance, or simple commercial transactions.

The good news: percentage problems follow predictable patterns. Once you internalise the base formula and a handful of shortcuts, you can solve most questions in under a minute. This topic rewards practice more than memorisation.

Key Concepts

• **Percentage means "per hundred"**: The symbol % represents a fraction with denominator 100. So 25% = 25/100 = 0.25.

• **The universal percentage formula**: Percentage = (Part / Whole) × 100. This single formula handles almost every percentage question.

• **Converting fraction to percentage**: Multiply the fraction by 100. Example: 3/5 = (3/5) × 100 = 60%.

• **Converting percentage to fraction**: Write percentage over 100 and simplify. Example: 45% = 45/100 = 9/20.

• **Finding percentage of a number**: X% of Y = (X/100) × Y. Example: 20% of 150 = (20/100) × 150 = 30.

• **Percentage increase**: New Value = Original + (Increase% × Original). Increase% = [(New − Original) / Original] × 100.

• **Percentage decrease**: New Value = Original − (Decrease% × Original). Decrease% = [(Original − New) / Original] × 100.

• **Base value matters**: Percentage change is always calculated on the original (base) value, not the new value.

Formulas / Key Facts

| Concept | Formula | |---------|---------| | Percentage of a quantity | X% of Y = (X × Y) / 100 | | What percentage is A of B? | (A / B) × 100 | | Percentage increase | [(New − Old) / Old] × 100 | | Percentage decrease | [(Old − New) / Old] × 100 | | Value after P% increase | Original × (1 + P/100) | | Value after P% decrease | Original × (1 − P/100) | | Successive % changes: first a%, then b% | Net effect on original = 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%, 1/20 = 5%
• 2/3 ≈ 66.67%, 3/4 = 75%, 4/5 = 80%

Worked Examples

**Example 1: Finding percentage of a number**

*Question*: A school in Srinagar has 450 students. If 36% are girls, how many girls are in the school?

*Solution*:

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

**Answer: 162 girls**

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**Example 2: Finding what percentage one number is of another**

*Question*: In a village in Jammu, out of 800 households, 120 have access to piped water. What percentage of households have piped water?

*Solution*:

• Percentage = (Part / Whole) × 100
• = (120 / 800) × 100
• = 0.15 × 100
• = 15%

**Answer: 15%**

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**Example 3: Percentage increase**

*Question*: The price of rice increased from ₹40 per kg to ₹46 per kg. Find the percentage increase.

*Solution*:

• Increase = 46 − 40 = ₹6
• Percentage increase = (Increase / Original) × 100
• = (6 / 40) × 100
• = 15%

**Answer: 15% increase**

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**Example 4: Successive percentage changes**

*Question*: A shopkeeper increases prices by 20% and then offers a 10% discount. What is the net percentage change from the original price?

*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% increase from original price**

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**Example 5: Reverse percentage (finding the original)**

*Question*: After a 25% increase, the population of a town became 7500. What was the original population?

*Solution*:

• Let original population = x
• After 25% increase: x + 25% of x = 7500
• x × (1 + 25/100) = 7500
• x × 1.25 = 7500
• x = 7500 / 1.25
• x = 6000

**Answer: Original population was 6000**

Common Mistakes

• **Wrong base in percentage change**: Students calculate percentage increase/decrease on the new value instead of the original value. *Fix*: Always divide by the original (starting) value.

• **Confusing "percentage" with "percentage points"**: An increase from 20% to 25% is a 5 percentage point increase, but a 25% increase in the rate itself [(25−20)/20 × 100]. *Fix*: Read the question carefully to see what is being compared.

• **Adding successive percentages directly**: Students think 20% increase followed by 10% decrease equals 10% increase. *Fix*: Use the successive change formula: a + b + (ab/100), which gives 8%, not 10%.

• **Forgetting to convert percentage to decimal/fraction before multiplying**: Writing 25% × 200 = 5000 instead of (25/100) × 200 = 50. *Fix*: Always convert % to decimal or fraction first.

• **Reversing the fraction**: When asked "what percentage is 15 of 60", students write 60/15 instead of 15/60. *Fix*: Remember — the number after "of" goes in the denominator.

Quick Reference

• **Percentage = (Part ÷ Whole) × 100** — the master formula.

• **X% of Y = (X × Y) ÷ 100** — for finding a part.

• **To increase by P%, multiply by (100 + P)/100**.

• **To decrease by P%, multiply by (100 − P)/100**.

• **Successive changes: a + b + (ab/100)** — saves time on two-step problems.

• **Memorise: 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%** — speeds up mental calculation.