SATHEE: Alligation & Mixture - All Formulas & Shortcuts

Alligation & Mixture - All Formulas & Shortcuts

Quick reference guide with the powerful Alligation Rule and mixture replacement techniques

📘 Basic Definitions

1. Key Concepts

Alligation:

Method to find ratio of quantities when two or more ingredients at different prices are mixed to get a mixture at a given price (mean price)

Mixture:

Combination of two or more components/elements

Mean Price:

Average price/value of the mixture

Types:

Simple Mixture: Two ingredients mixed
Compound Mixture: Multiple ingredients OR repeated operations
Replacement: Removing part and adding another substance

⚡ The Alligation Rule (MOST IMPORTANT!)

2. Alligation Formula

When two ingredients at prices P₁ and P₂ are mixed to get mean price M:

Rule:

    P₁ (cheaper)
          \
           \    (M - P₁)
            M
           /    (P₂ - M)
          /
    P₂ (costlier)

Ratio = (P₂ - M) : (M - P₁)

The ratio of quantities is:

Cheaper : Costlier = (P₂ - M) : (M - P₁)

Quantity₁ : Quantity₂ = (P₂ - M) : (M - P₁)

Key Point: Cross differences give the ratio!

3. Alligation Rule - Worked Example

Q: Mix milk at ₹60/L with milk at ₹40/L to get mixture at ₹50/L. Find ratio.

Solution:

 60 (costlier)
      \
       \    (50 - 40) = 10
        50 (mean)
       /    (60 - 50) = 10
      /
 40 (cheaper)

Ratio = 10:10 = 1:1

Answer: Equal quantities of both

4. Finding Mean Price

If quantities are in ratio a:b and prices are P₁ and P₂:

Mean Price:

M = (a × P₁ + b × P₂)/(a + b)

This is weighted average!

Example: Mix 2L at ₹60 with 3L at ₹45

Mean = (2×60 + 3×45)/(2+3) = (120 + 135)/5 = 255/5 = ₹51/L

🔥 Mixture & Replacement

5. Basic Mixture Formula

When mixing two quantities with different concentrations:

Final concentration:

C = (Q₁ × C₁ + Q₂ × C₂)/(Q₁ + Q₂)

Where:

Q = Quantity
C = Concentration/Price/Value

6. Replacement Rule (CRITICAL!)

From a vessel of V liters, x liters is removed and replaced with another liquid:

After 1 replacement:

Remaining original = V × (1 - x/V) = V - x New concentration = Original × (V - x)/V

After n replacements:

Remaining original = V × [(V - x)/V]ⁿ

Remaining = V × (1 - x/V)ⁿ

7. Replacement - Worked Example

Q: 80L vessel contains milk. Replace 8L with water 3 times. Find final milk quantity.

Solution:

After 1 replacement: 80 × (80-8)/80 = 80 × 72/80 = 72L After 2 replacements: 80 × (72/80)² = 80 × 0.81 = 64.8L After 3 replacements: 80 × (72/80)³ = 80 × 0.729 = 58.32L

OR using formula: = 80 × [(80-8)/80]³ = 80 × (72/80)³ = 80 × (9/10)³ = 80 × 729/1000 = 58.32L milk

💡 Advanced Formulas

8. Mixing Three or More Ingredients

For n ingredients with prices P₁, P₂, …, Pₙ in quantities Q₁, Q₂, …, Qₙ:

Mean Price:

M = (Q₁P₁ + Q₂P₂ + … + QₙPₙ)/(Q₁ + Q₂ + … + Qₙ)

Example: Mix 10L at ₹50, 20L at ₹60, 30L at ₹70

Mean = (10×50 + 20×60 + 30×70)/(10+20+30) = (500 + 1200 + 2100)/60 = 3800/60 = ₹63.33/L

9. Milk & Water Mixture Formulas

Initial mixture has milk:water = a:b Add x liters of milk:

New ratio:

Milk = a + x Water = b (unchanged) New ratio = (a+x):b

Add x liters of water:

Milk = a (unchanged) Water = b + x New ratio = a:(b+x)

10. Finding Original Quantity

After mixing x liters, ratio becomes a:b:

Original milk (if x liters water added):

Original Milk = (Current Total × a)/(a+b)

Example: After adding 5L water, ratio becomes 3:2 (milk:water). Final volume = 20L

Current Milk = 20 × 3/5 = 12L Current Water = 20 × 2/5 = 8L Original Water = 8 - 5 = 3L Original Milk = 12L Original Ratio = 12:3 = 4:1

🎯 Special Cases

11. Ratio to Percentage Conversion

If mixture has components in ratio a:b:

Percentage of first:

% = [a/(a+b)] × 100

Percentage of second:

% = [b/(a+b)] × 100

Example: Milk:Water = 3:2

Milk% = 3/5 × 100 = 60% Water% = 2/5 × 100 = 40%

12. Two Vessels Mixed

Vessel A: V₁ liters with ratio a₁:b₁ Vessel B: V₂ liters with ratio a₂:b₂

When mixed:

Total Milk = V₁ × [a₁/(a₁+b₁)] + V₂ × [a₂/(a₂+b₂)] Total Water = V₁ × [b₁/(a₁+b₁)] + V₂ × [b₂/(a₂+b₂)]

Final Ratio = Total Milk : Total Water

13. Wine & Water Problems

Standard pattern: Replace x liters n times

Formula:

Remaining wine = Initial × (1 - x/V)ⁿ

Quick values to remember:

Replace 1/2 once: 1/2 remains
Replace 1/2 twice: 1/4 remains
Replace 1/3 once: 2/3 remains
Replace 1/4 once: 3/4 remains

📊 Quick Shortcuts

Shortcut 1: Equal Price Difference

If P₁, M, P₂ are in AP (equal differences):

Ratio = 1:1

Example: ₹40, ₹50, ₹60

Difference both sides = 10 Therefore ratio = 1:1

Shortcut 2: Finding Cheaper/Costlier Quantity

When total quantity is T and ratio is a:b:

Cheaper quantity:

= T × a/(a+b)

Costlier quantity:

= T × b/(a+b)

Shortcut 3: Percentage to Ratio

45% milk, 55% water:

Ratio = 45:55 = 9:11

67% solution, 33% water:

Ratio = 67:33 ≈ 2:1

Shortcut 4: Replacement Percentage

If x% is replaced n times:

Remaining:

= Original × (1 - x/100)ⁿ

Example: Replace 20% twice

Remaining = (0.8)² = 0.64 = 64%

🔥 Exam Patterns

Pattern 1: Classic Alligation

Q: In what ratio must tea at ₹60/kg be mixed with tea at ₹45/kg to get mixture at ₹50/kg?

Solution:

 60
   \
    \   (50-45) = 5
     50
    /   (60-50) = 10
   /
 45

Ratio = 10:5 = 2:1 (For every 2 parts of cheaper, 1 part costlier)

Pattern 2: Multiple Replacements

Q: A vessel contains 100L milk. Replace 10L with water. Repeat this process 3 times. Find final milk.

Solution:

Remaining = 100 × [(100-10)/100]³ = 100 × (0.9)³ = 100 × 0.729 = 72.9L

Pattern 3: Finding Quantity Added

Q: 40L mixture has milk:water = 3:1. How much water to add to make ratio 2:1?

Solution:

Initial: Milk = 30L, Water = 10L

Final ratio 2:1 means: Milk : Water = 2:1 30 : (10+x) = 2:1

Cross multiply: 30 = 2(10+x) 30 = 20 + 2x 2x = 10 x = 5L water to add

Pattern 4: Concentration Problems

Q: Mix 20L of 30% alcohol with xL of 50% alcohol to get 40% alcohol. Find x.

Solution:

Using alligation: 50
\ (40-30) = 10 40 / (50-40) = 10 / 30

Ratio = 10:10 = 1:1 Therefore x = 20L

💎 Mental Math Tricks

Trick 1: Quick Mean Price

For ratio 1:1:

Mean = (P₁ + P₂)/2

For ratio 2:1:

Mean = (2P₁ + P₂)/3

For ratio 1:2:

Mean = (P₁ + 2P₂)/3

Trick 2: Replacement Once

Replace 1/n of vessel:

Remaining = (n-1)/n

Examples:

Replace 1/4: Remaining = 3/4
Replace 1/5: Remaining = 4/5
Replace 1/10: Remaining = 9/10

Trick 3: Water Added to Pure Milk

Add x liters water to y liters pure milk:

Water% = x/(x+y) × 100 Milk% = y/(x+y) × 100

Example: 20L milk + 5L water

Water% = 5/25 × 100 = 20% Milk% = 20/25 × 100 = 80% Ratio = 4:1

🎓 Golden Rules

Alligation cross-differences give ratio (never same-side differences!)
In replacement, use (1 - fraction)ⁿ for n replacements
Mean price always between the two prices (P₁ < M < P₂)
Ratio can be reversed: a:b same as quantity relation
For equal differences from mean, ratio is 1:1

🔍 Common Mistakes

❌ Using same-side differences in alligation (should be cross) ❌ Not using power n for n replacements ❌ Adding pure ingredient without adjusting ratio ❌ Forgetting mean must be between the two values ✅ Always draw alligation diagram ✅ Remember: (V-x)/V for one replacement, raise to power for multiple ✅ Check if mean price is logical (between extremes) ✅ Convert ratio to same units before calculation

📝 Summary Table

Scenario Formula

Alligation Ratio (P₂ - M) : (M - P₁)

Mean Price (Q₁P₁ + Q₂P₂)/(Q₁ + Q₂)

One Replacement V × (V-x)/V

n Replacements V × [(V-x)/V]ⁿ

Mixture Concentration (Q₁C₁ + Q₂C₂)/(Q₁ + Q₂)

Ratio to % [a/(a+b)] × 100

Scenario	Formula
Alligation Ratio	(P₂ - M) : (M - P₁)
Mean Price	(Q₁P₁ + Q₂P₂)/(Q₁ + Q₂)
One Replacement	V × (V-x)/V
n Replacements	V × [(V-x)/V]ⁿ
Mixture Concentration	(Q₁C₁ + Q₂C₂)/(Q₁ + Q₂)
Ratio to %	[a/(a+b)] × 100