📊 Percentage and Average Calculations

📌 Part 1: Percentage Calculations

What Is a Percentage?

A percentage is a way of expressing a number as a fraction of 100. The word "percent" means "per hundred" and is denoted by the symbol %.

Formula:
Percentage = (Part / Whole) × 100

Understanding the Terms

• - Part: The portion or amount you want to find the percentage of.
• - Whole: The total or complete quantity.

🔢 How to Calculate Percentage

Example 1: If 20 students out of 50 pass an exam:

Percentage = (20 / 50) × 100 = 40%

Example 2: What is 25% of 200?

= (25 / 100) × 200 = 50

📚 Types of Percentage Problems

• 1. Finding percentage of a number:
  What is 30% of 150? → (30/100) × 150 = 45
• 2. Finding the whole from part and percentage:
  40 is 20% of what? → Whole = (40 × 100) / 20 = 200
• 3. Finding the part:
  What is 15% of 80? → (15/100) × 80 = 12
• 4. Percentage Increase:
  % Increase = (Increase / Original) × 100
• 5. Percentage Decrease:
  % Decrease = (Decrease / Original) × 100

💹 Percentage Change Examples

• Increase Example: From $200 to $250 → Increase = 50
% Increase = (50 / 200) × 100 = 25%

• Decrease Example: From $150 to $120 → Decrease = 30
% Decrease = (30 / 150) × 100 = 20%

📈 Compound Percentage

When percentages are applied successively:

• Formula:
Total Change = p + q + (p × q) / 100

Example: Price increases by 10%, then 20%:
= 10 + 20 + (10×20)/100 = 32%

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📌 Part 2: Average Calculations

What Is an Average?

An average (arithmetic mean) is the central value of a set of numbers, calculated as the sum divided by the count.

📊 Types of Averages

• • Arithmetic Mean: Sum / Count
• • Median: Middle value in ordered data
• • Mode: Most frequent value

🔢 Arithmetic Mean Formula

Average = (Sum of all observations) / (Number of observations)

Example 1: Average of 5, 10, 15, 20, 25
Sum = 75, Count = 5 → Average = 75 / 5 = 15

Example 2: If average marks of 8 students is 75:
Total = 8 × 75 = 600

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⚖️ Weighted Average

Formula:
Weighted Average = (∑ wi × xi) / ∑ wi

Example: Marks = 80 in three subjects with weights 2, 3, and 1:
= (80×2 + 80×3 + 80×1) / (2+3+1) = 480 / 6 = 80

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👥 Combining Averages

Formula:
Combined Average = (Avg₁ × n₁ + Avg₂ × n₂) / (n₁ + n₂)

Example: Class A: 30 students, Avg = 70
Class B: 25 students, Avg = 80
= (70×30 + 80×25) / (30 + 25) = 4100 / 55 = ~74.55

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📌 Practical Uses of Percentage and Average

• 📈 Finance: Profit/loss, interest rates
• 🏢 Business: Sales growth, KPIs
• 📚 Education: Exam scores, attendance
• 🏥 Health: Body fat %, average calories
• 🛒 Everyday life: Discounts, fuel efficiency

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❗ Common Mistakes to Avoid

• • Confusing percentage with amount change
• • Not converting % to decimals before calculating
• • Mixing up part and whole
• • Using simple average when weighted is needed
• • Incorrectly merging group averages

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🧩 Practice Questions

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🏁 Conclusion

Mastering percentage and average calculations empowers you to analyze data effectively, make informed decisions, and solve practical problems efficiently. With practice and clarity in concepts, you’ll gain speed, accuracy, and confidence in these essential mathematical tools.