Percentage

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Percentage — Key Facts for NABE (Pakistan)

• Percentage: Per cent means “per hundred” — express fractions as parts of 100
• Formula: X% = X/100; To find %: (Value/Total) × 100
• Increase/Decrease: New % = Original ± (Change%/100) × Original
• Successive % Change: Multiply factors (1 ± r/100) for each change
• ⚡ Exam tip: Percentage questions often combine with profit/loss, CI, and population problems

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Percentage — NABE (Pakistan) Study Guide

Basic Concept

Percentage is a way of expressing a number as a fraction of 100. The symbol ”%” means “per hundred.”

Conversions:

• Fraction to %: Multiply by 100
  • 3/4 = (3/4) × 100 = 75%
• Decimal to %: Multiply by 100
  • 0.45 = 45%
• % to Decimal: Divide by 100
  • 32% = 32/100 = 0.32

Common Percentages and Their Fraction Forms

Percentage	Fraction	Decimal
50%	1/2	0.5
25%	1/4	0.25
75%	3/4	0.75
10%	1/10	0.1
20%	1/5	0.2
5%	1/20	0.05

Percentage Change

Percentage Increase:

% Increase = (Increase Amount / Original Value) × 100

Percentage Decrease:

% Decrease = (Decrease Amount / Original Value) × 100

Example: Price increases from Rs. 80 to Rs. 100

• Increase = 100 - 80 = Rs. 20
• % Increase = (20/80) × 100 = 25%

Percentage of a Number

Formula: X% of Y = (X/100) × Y

Examples:

• 15% of 400 = (15/100) × 400 = 60
• 8% of 250 = (8/100) × 250 = 20

Finding the Original Value

When you know the increased/decreased value and the percentage:

Example: After a 20% increase, a price becomes Rs. 360. Find original price.

• Let original = 100%
• New value = 120% of original
• 120% of original = 360
• Original = 360 × (100/120) = Rs. 300

NABE Exam Pattern

Common question types:

1. Basic percentage calculations
2. Percentage increase/decrease
3. Finding original value after change
4. Successive percentage changes
5. Comparison of quantities using percentages

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Percentage — Comprehensive NABE (Pakistan) Notes

Detailed Theory

1. Fundamental Concepts

Origin of Percentage: The concept of percentage evolved from the Babylonian system ofsexagesimal fractions. The Latin term “per centum” (by the hundred) gave us the modern symbol ”%.”

Base, Portion, and Rate Model:

• Base: The original or total amount (100%)
• Portion: The part of the base being considered
• Rate: The percentage (%)
• Formula: Portion = Base × Rate

2. Percentage Point vs. Percent Change

These are commonly confused:

Percentage Point: The arithmetic difference between two percentages

• Example: Interest rate changes from 8% to 10%
• Change = 10 - 8 = 2 percentage points

Percent Change: The relative change as a percentage

• Change = (2/8) × 100 = 25% increase

Why It Matters: A change from 50% to 60% is 10 percentage points but a 20% relative increase.

3. Successive Percentage Change

When percentage changes occur one after another:

Formula for multiple changes:

• Final = Initial × (1 + r₁/100) × (1 + r₂/100) × (1 + r₃/100)…

Example: A quantity increases by 20%, then decreases by 10%

• Start: 100
• After 20% increase: 100 × 1.20 = 120
• After 10% decrease: 120 × 0.90 = 108
• Net change = +8%

Short Formula: Multiply the factors (1 ± r/100)

Common Trap: Two successive 20% increases do NOT equal 40% increase

• 100 × 1.2 × 1.2 = 144 (not 140!)

4. Percentage in Business Applications

Profit and Loss:

• Profit % = (Profit/Cost Price) × 100
• Loss % = (Loss/Cost Price) × 100
• Selling Price = Cost Price × (1 ± Profit or Loss%/100)

Example: A shopkeeper buys at Rs. 800 and sells at 15% profit

• Profit = 800 × 15/100 = Rs. 120
• Selling Price = 800 + 120 = Rs. 920

Discount:

• Discount % = (Discount/Marked Price) × 100
• Selling Price = Marked Price - Discount

Example: Article marked at Rs. 500, discount of 12%

• Discount = 500 × 12/100 = Rs. 60
• Selling Price = 500 - 60 = Rs. 440

5. Population and Depreciation Problems

Population Growth Formula:

• Population after n years = Initial × (1 + r/100)^n

Depreciation Formula:

• Value after n years = Initial × (1 - r/100)^n

Example: Population of a city is 50,000, growing at 4% per year. Find population after 3 years.

• P = 50,000 × (1 + 4/100)³
• P = 50,000 × (1.04)³
• P = 50,000 × 1.1249 ≈ 56,245

6. Alligation Method with Percentages

When mixing two solutions of different concentrations:

Example: Mix 20% and 60% alcohol solutions to get 40% solution

• Alligation:
  • 60% solution — 20 parts
  • 20% solution — 20 parts
  • (since 40-20 = 20 and 60-40 = 20)
• Parts of 60% : Parts of 20% = 20 : 20 = 1 : 1

7. Error Analysis — Common Mistakes

1. Assuming Simple Addition: Successive increases don’t simply add
2. Wrong Base: Always identify whether you’re calculating from original or current value
3. Percentage of vs. Percentage More Than: “50% more” = 150% of original
4. Neglecting Negative Changes: A decrease followed by equal increase doesn’t restore original value

8. Quick Calculation Tricks

Finding 10%: Move decimal one place left

• 10% of 450 = 45

Finding 5%: Find 10% and halve

• 5% of 450 = 22.5

Finding 1%: Move decimal two places left

• 1% of 450 = 4.5

Finding 25%: Quarter the number

• 25% of 400 = 100

Finding 33.33%: Divide by 3

• 33.33% of 90 = 30

Practice Questions for NABE

1. If 35% of a number is 245, find the number.
2. A town’s population was 25,000 in 2020 and increased to 30,000 in 2023. Find the percentage increase.
3. The price of an article is reduced by 20%. By what percentage must the new price be increased to return to the original price?
4. In an examination, 65% passed in Math, 55% passed in English, and 40% passed in both. What percentage passed in at least one subject?
5. A car purchased for Rs. 500,000 depreciates at 10% per year. What will be its value after 3 years?

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