Mixture and Alligation

1. Basic Concepts

Formula: Weighted Average = (Quantity₁ × Value₁ + Quantity₂ × Value₂) / (Quantity₁ + Quantity₂)

• Example 1: Mix 3L of ₹10/litre milk with 2L of ₹20/litre. Find average cost.
  Solution: (3×10 + 2×20)/(3+2) = (30+40)/5 = ₹14/litre
• Example 2 (SSC CGL 2020): A mixture of 60L has milk and water in 2:1. How much water?
  Solution: 2+1 = 3 parts ⇒ Water = (1/3) × 60 = 20L

2. Alligation Rule

Formula:
Cheaper Quantity / Dearer Quantity = (Mean Price - Dearer Price) / (Cheaper Price - Mean Price)

• Example 1: Mix ₹30/kg rice with ₹50/kg rice to get ₹42/kg. Find ratio.
  Solution: (50 - 42):(42 - 30) = 8:12 = 2:3
• Example 2 (RRB NTPC 2019): Mix milk at ₹6/litre with water (₹0) to get ₹4/litre. Find ratio.
  Solution: (6 - 4):(4 - 0) = 2:4 = 1:2 ⇒ Milk:Water = 1:2

3. Replacement Problems

Involves removing and replacing part of the solution repeatedly.

• Example 1 (SSC CHSL 2018): A 40L solution has 25% alcohol. 10L is removed and replaced with water. Final % alcohol?
  Solution: Remaining alcohol = 30L × 25% = 7.5L ⇒ % = (7.5/40)×100 = 18.75%
• Example 2: A container has 20L milk. 4L removed and replaced with water. Repeated twice. Final milk = ?
  Solution: Final milk = 20 × (1 - 4/20)² = 20 × (16/25) = 12.8L

4. Profit-Based Mixing

Used when items are mixed for profit (common in shopkeeper problems).

• Example 1 (SSC CGL 2019): A seller mixes ₹60/kg sugar with ₹30/kg sugar and sells at ₹54/kg. What is the profit %?
  Solution: Let ratio = 1:1 ⇒ Avg CP = ₹(60+30)/2 = ₹45 ⇒ Profit = 9 ⇒ (9/45)×100 = 20%
• Example 2: Mix ₹8/litre milk with water (₹0) and sell at ₹10/litre. If selling price is average, find profit %.
  Solution: Avg CP < SP ⇒ Use alligation or assume ratios to find % profit.

5. Previous Year Exam Questions

• Example 1 (SSC GD 2019): A 70L mixture contains milk and water in 3:4. How much water is in it?
  Solution: 3+4 = 7 parts ⇒ Water = 4/7 × 70 = 40L
• Example 2 (RRB ALP 2018): In what ratio should water be mixed with milk to gain 25% by selling at cost price?
  Solution: Profit = 25% ⇒ Ratio = 1:4 (Water:Milk)
• Example 3 (SSC MTS 2021): A container has 80L of milk. 20L is replaced with water. What is % of milk?
  Solution: Remaining milk = 60L ⇒ % = (60/80) × 100 = 75%
• Example 4 (RRB JE 2019): Mix ₹15/kg sugar with ₹25/kg to get ₹20/kg. What is ratio?
  Solution: (25-20):(20-15) = 5:5 = 1:1
• Example 5 (SSC CGL 2022): How much water must be added to 50L of milk to dilute it to 80%?
  Solution: Let x be added ⇒ 50 = 0.8(50 + x) ⇒ x = 12.5L
• Example 6 (RRB NTPC 2020): 10L of mixture contains milk and water in 7:3. Add 5L water. New ratio?
  Solution: Milk = 7L, Water = 3+5 = 8L ⇒ Ratio = 7:8
• Example 7 (HSSC 2019): Mix two types of oil ₹60/litre and ₹90/litre in 2:1. Find price/litre.
  Solution: Avg = (2×60 + 1×90)/(2+1) = ₹70/litre
• Example 8 (SSC CHSL 2022): Mix 4L of 10% acid with 6L of 40% acid. Find final %.
  Solution: (4×10 + 6×40)/10 = 280/10 = 28%
• Example 9 (RRB Group D 2022): A mix is 40% alcohol. Add 5L water to 20L mix. New alcohol %?
  Solution: Alcohol = 8L, Total = 25L ⇒ % = (8/25) × 100 = 32%
• Example 10 (SSC CPO 2020): 25L of milk replaced with water twice from 100L milk. Final milk left?
  Solution: Final = 100 × (1 - 25/100)² = 100 × (3/4)² = 56.25L

Tips to Remember

• Use the alligation method for quick ratio mixing without lengthy calculations.
• Always label "cheaper" and "dearer" correctly to apply alligation formula.
• Replacement formula: Final Quantity = Initial × (1 - removed/total)^n