Mixtures & Alligations

Master the concepts of mixtures and allegations with our comprehensive guide. Learn about weighted averages, replacement problems, and complex problem-solving techniques.

Table of Contents

Simplified Quantitative Formulas: Mixtures and Alligations

  • Mixture Problems: Involve combining substances (like mixing water and juice) and finding ratios or concentrations. Simple mixture: direct mix; Compound mixture: mix of mixtures.
  • Alligation Method: Quick way to find the ratio of two ingredients in a mixture when you know their individual values and the target mixture value.
    Formula: (M₂ – M_A) : (M_A – M₁)
  • Weighted Average: The mixture's overall value is the weighted average of parts.
    M_A = (Q₁M₁ + Q₂M₂) / (Q₁ + Q₂)
  • Mixing Two Mixtures: If you mix two existing mixtures with different compositions, the final ratio can be found using an extended formula or by finding the fraction of each original mixture in the new one and then computing new composition.
  • Repeated Replacement: If you repeatedly remove some mixture and replace it with another substance, the original substance's remaining amount follows a pattern.
    Remaining = A × ((A – B)/A)^n
  • Fraction After Replacements: The fraction of the new liquid after n operations: 1 - ((A - B)/A)^n
  • Advanced: Mixing multiple mixtures, handling concentration changes, and profit/loss in mixtures are also tested in CAT.

What do these mean? (Super Simple Explanations & Examples)

  • Mixture Problems: Imagine you have a glass of orange juice and a glass of water. If you pour them together, you get a mixture! If you mix two different juices, that's a simple mixture. If you mix a juice with another mixture (like juice + milk), that's a compound mixture.
    Example: Mix 1 cup of apple juice and 2 cups of water. The mixture has 1 part juice and 2 parts water, so the ratio is 1:2.
  • Alligation Method: This is a shortcut to find out how much of each ingredient you need to get a mixture of a certain value (like price or concentration).
    Example: You want to mix 1% and 5% salt water to get 3%. The ratio is (5-3):(3-1) = 2:2 = 1:1. So, mix equal parts of each!
  • Weighted Average: This tells you the "average" value of the mixture, based on how much of each part you have.
    Example: Mix 2 kg of ₹50/kg tea and 3 kg of ₹70/kg tea. Average price = (2×50 + 3×70) / (2+3) = (100+210)/5 = ₹62/kg.
  • Mixing Two Mixtures: If you mix two mixtures (like salty water and sweet water), you can find out how much of each is in the new mix by adding up the parts and using the weighted average.
    Example: Mix 1 liter of 10% juice with 1 liter of 20% juice. The new mixture is (1×10 + 1×20)/(1+1) = 30/2 = 15% juice.
  • Repeated Replacement: If you keep taking out some mixture and adding something new, the amount of the original stuff goes down in a pattern.
    Example: You have 100 L of water. Remove 10 L and add 10 L of milk, five times. Water left = 100 × (90/100)^5 ≈ 59 L.
  • Fraction After Replacements: This tells you how much of the new thing is in the mixture after several swaps.
    Example: After 5 swaps above, fraction of milk = 1 - (0.9)^5 ≈ 0.41 (or 41% milk).
  • Advanced: If you mix three or more mixtures, use the weighted average for all. For profit/loss, calculate cost price and selling price for the mixture.
    Example: Mix 2 kg of ₹30/kg rice, 3 kg of ₹40/kg rice, and 1 kg of ₹50/kg rice. Average price = (2×30 + 3×40 + 1×50)/(2+3+1) = (60+120+50)/6 = ₹38.33/kg.

1. Basic Concepts

(a) Understanding Mixtures

Understanding the basic concepts of mixtures and their properties.

Key Terms:

  • Mixture: Combination of two or more substances
  • Alligation: Rule to find the ratio of quantities in a mixture
  • Weighted Average: Average considering the weights of components
  • Replacement: Process of removing and adding components

Important Points:

  • Mixture problems involve finding quantities or ratios
  • Alligation rule helps find the ratio of mixing
  • Weighted average is used for unequal quantities
  • Replacement problems involve removing and adding components

Example 1: Basic Mixture

Two types of rice: ₹30/kg and ₹40/kg

Mixed in ratio 2:3

Find average cost per kg

Solution:

Average = (30×2 + 40×3)/(2+3) = ₹36/kg

2. Alligation Rule

(a) Alligation Method

Using the alligation rule to find mixing ratios.

Alligation Rule:

  • Cheaper Quantity : Dearer Quantity = (d - m) : (m - c)
  • Where:
  • c = Cost of cheaper item
  • d = Cost of dearer item
  • m = Mean price

Example 2: Alligation

Cheaper rice = ₹30/kg

Dearer rice = ₹40/kg

Mean price = ₹36/kg

Find mixing ratio

Solution:

Ratio = (40-36):(36-30) = 4:6 = 2:3

3. Weighted Averages

(a) Weighted Average Calculations

Calculating weighted averages for mixtures.

Formulas:

  • Weighted Average = Σ(Value × Weight)/Σ(Weight)
  • For two components:
  • Average = (a₁w₁ + a₂w₂)/(w₁ + w₂)
  • Where:
  • a₁, a₂ = Values
  • w₁, w₂ = Weights

Example 3: Weighted Average

Component A: 20% (3 kg)

Component B: 30% (2 kg)

Find weighted average

Solution:

Average = (20×3 + 30×2)/(3+2) = 24%

4. Replacement Problems

(a) Replacement Calculations

Solving problems involving replacement of components.

Formulas:

  • Final Quantity = Initial Quantity × (1 - r)^n
  • Where:
  • r = Replacement ratio
  • n = Number of replacements

Example 4: Replacement

Initial mixture: 40L (20% alcohol)

Replace 25% with water

Find final concentration

Solution:

Final alcohol = 40 × 0.2 × 0.75 = 6L

Final concentration = (6/40) × 100 = 15%

5. Multiple Mixtures

(a) Multiple Mixture Problems

Solving problems involving multiple mixtures.

Formulas:

  • For n components:
  • Average = Σ(aᵢwᵢ)/Σ(wᵢ)
  • Where:
  • aᵢ = Value of component i
  • wᵢ = Weight of component i

Example 5: Multiple Mixtures

Mixture A: 20% (2 kg)

Mixture B: 30% (3 kg)

Mixture C: 40% (1 kg)

Find final concentration

Solution:

Average = (20×2 + 30×3 + 40×1)/(2+3+1) = 28.33%

6. Profit & Loss

(a) Profit & Loss in Mixtures

Solving profit and loss problems involving mixtures.

Formulas:

  • Profit = Selling Price - Cost Price
  • Profit % = (Profit/Cost Price) × 100
  • Cost Price = Σ(Cost × Quantity)

Example 6: Profit & Loss

Component A: ₹30/kg (2 kg)

Component B: ₹40/kg (3 kg)

Selling price = ₹38/kg

Find profit percentage

Solution:

Cost price = (30×2 + 40×3)/5 = ₹36/kg

Profit = 38 - 36 = ₹2/kg

Profit % = (2/36) × 100 = 5.56%

7. Advanced Concepts

(a) Important Theorems

Key Theorems:

  1. If a mixture of n components has values a₁, a₂, ..., aₙ and weights w₁, w₂, ..., wₙ:
    Average = Σ(aᵢwᵢ)/Σ(wᵢ)
  2. For replacement problems:
    Final Quantity = Initial Quantity × (1 - r)^n
  3. For alligation:
    Cheaper : Dearer = (d - m) : (m - c)
  4. For profit and loss:
    Profit % = ((SP - CP)/CP) × 100

Example 7: Complex Problem

Initial mixture: 60L (30% alcohol)

Replace 20% with water

Add 10L of 40% alcohol

Find final concentration

Solution:

After replacement: 60 × 0.3 × 0.8 = 14.4L alcohol

After adding: 14.4 + (10 × 0.4) = 18.4L alcohol

Final volume = 60 × 0.8 + 10 = 58L

Final concentration = (18.4/58) × 100 = 31.72%

(b) Special Cases

Special Scenarios:

  • When all components have same weight:
    Average = (a₁ + a₂ + ... + aₙ)/n
  • When replacement ratio is constant:
    Final = Initial × (1 - r)^n
  • When mixing equal quantities:
    Average = (a₁ + a₂)/2

Example 8: Special Case

Equal quantities of 20%, 30%, and 40% solutions

Find final concentration

Solution:

Average = (20 + 30 + 40)/3 = 30%

Practice Questions

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