Averages

🎯 The Magic of Averages: Finding the “Middle Ground”

The Pizza Party Story 🍕

Imagine you and your friends ordered 3 pizzas for a party. One pizza has 8 slices, another has 6 slices, and the last one has 4 slices. That’s 18 slices total!

If you wanted to share them equally among the 3 pizzas, each pizza would have… 6 slices!

That’s what average is — finding the fair share or the typical value in a group of numbers.

🧮 Simple Average (Arithmetic Mean)

What Is It?

Simple Average = Add all the numbers together, then divide by how many numbers you have.

Think of it like this: You’re redistributing all the values equally.

The Formula

Average = Sum of all values ÷ Number of values

🍎 Real Example

Problem: Riya scored 80, 70, 90, and 60 in her 4 tests. What’s her average score?

Solution:

1. Add all scores: 80 + 70 + 90 + 60 = 300
2. Count the tests: 4 tests
3. Divide: 300 ÷ 4 = 75

✅ Riya’s average score is 75!

💡 Quick Tip

If all numbers are the same, the average equals that number!

• Average of 5, 5, 5 = 5 ✨

⚖️ Weighted Average

When Simple Average Isn’t Enough

What if some things matter more than others?

Imagine your final grade depends on:

• Homework (20% of grade)
• Midterm (30% of grade)
• Final Exam (50% of grade)

A simple average won’t work here because the final exam counts more than homework!

The Formula

Weighted Average = (Value₁ × Weight₁) + (Value₂ × Weight₂) + ...
                   ÷ (Total of all Weights)

📚 Real Example

Problem: Sam scored:

• Homework: 90 (weight = 2)
• Project: 80 (weight = 3)
• Exam: 70 (weight = 5)

What’s Sam’s weighted average?

Solution:

graph TD
    A["Step 1: Multiply each score by its weight"] --> B["90 × 2 = 180"]
    A --> C["80 × 3 = 240"]
    A --> D["70 × 5 = 350"]
    B --> E["Step 2: Add them up"]
    C --> E
    D --> E
    E --> F["180 + 240 + 350 = 770"]
    F --> G["Step 3: Add all weights"]
    G --> H["2 + 3 + 5 = 10"]
    H --> I["Step 4: Divide"]
    I --> J["770 ÷ 10 = 77"]

✅ Sam’s weighted average is 77!

🎯 Notice This!

Sam’s exam score (70) pulled the average DOWN because it had the highest weight (5).

🔢 Average of Consecutive Numbers

What Are Consecutive Numbers?

Numbers that follow each other in order, with the same gap between them.

Examples:

• 1, 2, 3, 4, 5 (gap = 1)
• 2, 4, 6, 8, 10 (gap = 2, these are consecutive even numbers)
• 5, 10, 15, 20 (gap = 5)

The Magic Shortcut ✨

For consecutive numbers, the average is simply the MIDDLE number!

If there are two middle numbers (even count), average them!

🎪 Example 1: Odd Count

Problem: Find the average of 3, 4, 5, 6, 7

Solution:

• There are 5 numbers
• The middle (3rd) number is 5
• ✅ Average = 5

Verify: (3+4+5+6+7) ÷ 5 = 25 ÷ 5 = 5 ✓

🎪 Example 2: Even Count

Problem: Find the average of 10, 20, 30, 40

Solution:

• There are 4 numbers
• Two middle numbers: 20 and 30
• Average of middle: (20 + 30) ÷ 2 = 25
• ✅ Average = 25

🌟 Another Shortcut!

Average = (First Number + Last Number) ÷ 2

Example: Average of 3, 4, 5, 6, 7 = (3 + 7) ÷ 2 = 10 ÷ 2 = 5 ✓

📊 Change in Average Problems

The Detective Work 🔍

These problems ask: “What happens when something changes?”

• A new person joins a group
• Someone leaves a group
• One value is replaced by another

Type 1: New Member Joins

Problem: The average age of 4 friends is 20 years. A new friend joins, and the average becomes 22. What is the age of the new friend?

Think About It:

• Old total age = 4 × 20 = 80 years
• New total age = 5 × 22 = 110 years
• New friend’s age = 110 - 80 = 30 years ✅

graph TD
    A["4 friends, avg = 20"] --> B["Total age = 80"]
    C["5 friends, avg = 22"] --> D["Total age = 110"]
    B --> E["New friend = 110 - 80"]
    D --> E
    E --> F["New friend is 30 years old!"]

Type 2: Someone Leaves

Problem: The average weight of 5 students is 50 kg. If one student weighing 60 kg leaves, what’s the new average?

Solution:

• Old total = 5 × 50 = 250 kg
• After leaving = 250 - 60 = 190 kg
• New average = 190 ÷ 4 = 47.5 kg ✅

Type 3: Replacement

Problem: The average of 6 numbers is 30. If one number (18) is replaced by 42, what’s the new average?

Solution:

• Change in value = 42 - 18 = +24
• Change spread across 6 numbers = 24 ÷ 6 = +4
• New average = 30 + 4 = 34 ✅

🧠 Golden Formula for Replacement

New Average = Old Average + (New Value - Old Value) ÷ Count

🎯 Quick Summary

🌈 Why Averages Matter

Averages help us make sense of messy data:

• Your average screen time tells you daily habits
• Average temperature helps plan your outfit
• Average speed tells you how fast your trip was

You now have the superpower to find patterns in numbers! 🦸

🎮 Try This Yourself!

Mini Challenge: Your 5 friends have ₹10, ₹20, ₹30, ₹40, and ₹50.

1. What’s the simple average?
2. If they pool their money equally, how much does each person get?

Hint: Both answers are the same! 😉

Remember: Average is just finding the “fair share” — the number that represents the whole group! 🌟