Mixtures and Alligation

🧪 Mixtures and Alligation: The Art of Blending

Imagine you’re a master chef. You have sweet honey and plain water. How do you mix them to get the perfect sweetness? That’s exactly what mixtures and alligation teach us!

🎯 What Are We Learning?

Think of this like making the perfect lemonade. You have:

• Sweet syrup (expensive/stronger)
• Plain water (cheaper/weaker)

How much of each do you need to get just the right taste at the right price?

Three superpowers you’ll gain:

1. 🧩 Mixture Problems – Combining things to get what you want
2. ⚖️ Alligation Rule – A magic shortcut to find the perfect ratio
3. 🔄 Replacement and Dilution – What happens when you swap parts out

🧩 Part 1: Mixture Problems

The Big Idea

A mixture is when you combine two or more things together.

Real-life examples:

• Mixing milk with water
• Blending two types of tea
• Combining gold of different purities

The Simple Formula

When you mix two things:

Total Value = (Amount₁ × Rate₁) + (Amount₂ × Rate₂)

🌟 Example: The Milk Shop

A shopkeeper has:

• Milk A: 10 liters at ₹50/liter
• Milk B: 15 liters at ₹40/liter

What’s the price of the mixture?

Step 1: Find total cost

• Milk A: 10 × 50 = ₹500
• Milk B: 15 × 40 = ₹600
• Total: ₹1100

Step 2: Find total quantity

• 10 + 15 = 25 liters

Step 3: Find mixture price

• 1100 ÷ 25 = ₹44 per liter

🎨 Visual Flow

graph TD
    A["Milk A: 10L × ₹50"] --> C["Total Cost: ₹1100"]
    B["Milk B: 15L × ₹40"] --> C
    C --> D["Total: 25 Liters"]
    D --> E["Price = ₹1100 ÷ 25"]
    E --> F["₹44 per liter"]

💡 Key Insight

The mixture’s price always falls between the two original prices. It’s like mixing hot and cold water – you get something in the middle!

⚖️ Part 2: The Alligation Rule

The Magic Shortcut

Alligation is like a seesaw. The heavier side goes down!

The Rule: To find the mixing ratio, use this simple cross pattern:

        Cheaper          Dearer
           ↘              ↙
            Mean Price
           ↙              ↘
    (Dearer - Mean)  :  (Mean - Cheaper)

🌟 Example: Tea Blending

A shopkeeper wants to mix:

• Tea A: ₹60/kg (cheaper)
• Tea B: ₹90/kg (dearer)

He wants the mixture at ₹72/kg

Using Alligation:

     ₹60              ₹90
        ↘            ↙
          ₹72
        ↙            ↘
   90-72=18     72-60=12

Ratio = 18 : 12 = 3 : 2

So mix 3 parts of Tea A with 2 parts of Tea B!

🎨 Visual Flow

graph TD
    A["Cheaper: ₹60"] --> M["Mean: ₹72"]
    B["Dearer: ₹90"] --> M
    M --> R1["90-72 = 18"]
    M --> R2["72-60 = 12"]
    R1 --> F["Ratio 18:12 = 3:2"]
    R2 --> F

✅ Quick Check

If mixing at ₹75/kg instead:

• Difference from ₹60 = 15
• Difference from ₹90 = 15
• Ratio = 1:1 (equal amounts!)

💡 Why Does This Work?

Imagine a seesaw:

• The mean price is the balance point
• Items farther from the mean need less quantity
• Items closer to the mean need more quantity

The math captures this balance perfectly!

🔄 Part 3: Replacement and Dilution

The Big Idea

What happens when you remove some mixture and add something new?

This is like:

• Removing some tea from a pot and adding water
• Taking out some paint and mixing fresh color

The Magic Formula

When you replace x liters from an n-liter container k times:

Final Concentration = Initial × (1 - x/n)^k

🌟 Example: The Wine Barrel

A barrel has 20 liters of pure wine.

Every time, we remove 4 liters and add 4 liters of water.

After 1 replacement:

Wine left = 20 × (1 - 4/20)
         = 20 × (4/5)
         = 16 liters

After 2 replacements:

Wine left = 20 × (4/5)²
         = 20 × 16/25
         = 12.8 liters

After 3 replacements:

Wine left = 20 × (4/5)³
         = 20 × 64/125
         = 10.24 liters

🎨 Visual Flow

graph TD
    A["Start: 20L Wine"] --> B["Remove 4L, Add Water"]
    B --> C["16L Wine + 4L Water"]
    C --> D["Remove 4L, Add Water"]
    D --> E["12.8L Wine + 7.2L Water"]
    E --> F["Remove 4L, Add Water"]
    F --> G["10.24L Wine + 9.76L Water"]

🌟 Example: Finding Replacements Needed

A container has 64 liters of milk. Each time, 8 liters is replaced with water.

After how many operations will milk be 27 liters?

Using the formula:

27 = 64 × (1 - 8/64)^k
27 = 64 × (7/8)^k
27/64 = (7/8)^k

Since (7/8)³ = 343/512 ≈ 0.67 and 27/64 ≈ 0.42…

Let’s check: (7/8)³ = 0.669… (too high)

Actually: 27/64 = (3/4)³

So if replacement fraction = 1/4 (which is 8/32, not 8/64)…

Better approach: 27/64 = (3/4)³ tells us k = 3 times

(When 1 - x/n = 3/4, meaning x/n = 1/4)

💡 Key Insights

1. Each replacement dilutes – You can never fully remove the original!
2. Exponential decay – The original substance decreases rapidly
3. Same formula for all – Works for wine, milk, paint, anything!

🎯 Quick Reference

🏆 Success Tips

1. Draw the alligation cross – It makes ratios super easy!
2. Check your answer – Mixture value is always BETWEEN the originals
3. For replacement – Remember it’s exponential, not linear
4. Units matter – Keep everything in same units (liters, kg, etc.)

🎪 Fun Fact

Ancient traders used alligation to:

• Mix gold of different purities
• Blend spices for the perfect flavor
• Create the right wine strength

They didn’t have calculators – just this clever cross method!

Now you know the secrets of mixing! Whether it’s tea, milk, or gold – you can find the perfect blend. 🌟