Mixture and Alligation Explained with Formulas and Examples - Aptitude Questions & Answers

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Mixture and Alligation Explained with Formulas and Examples is one of the most important topics in Quantitative Aptitude. In this lesson, you will learn concepts, formulas, shortcuts, solved examples, and aptitude questions with answers. This topic is useful for exams like SSC, Bank, CAT, TCS, and other competitive exams.

Mixture and Alligation: Complete Guide with Formulas & Examples

What are Mixtures and Alligation?

Mixture and Alligation are mathematical concepts used to solve problems involving the mixing of two or more ingredients at different prices or concentrations. These concepts are particularly useful in determining proportions, costs, and concentrations in various mixtures.

Example: Mixing 10 liters of milk (₹50/liter) with 5 liters of water (₹0/liter) creates a 15-liter mixture whose cost per liter needs to be calculated.

Basic Concepts and Terminology

Mixture

A substance formed by combining two or more different ingredients. The properties of the mixture (like cost, concentration) lie between the properties of its ingredients.

Example: Tea with sugar, Alloys (brass = copper + zinc), Medicine mixtures, etc.

Alligation

A rule that helps find the ratio in which two or more ingredients at given prices must be mixed to produce a mixture of desired price or concentration.

Example: Finding in what ratio ₹10/kg and ₹15/kg rice should be mixed to get rice at ₹12/kg.

Mean Price/Value

The average price or concentration of the final mixture. It always lies between the prices/concentrations of the ingredients being mixed.

Example: When mixing ₹10/kg and ₹20/kg items, mean price will be between ₹10 and ₹20.

Alligation Rule - The Core Formula

For two ingredients with prices/concentrations:
Cheaper Price (C), Dearer Price (D), Mean Price (M)

Quantity of Cheaper : Quantity of Dearer = (D - M) : (M - C)

Visual Representation - Alligation Cross

Alligation Cross Method:
Cheaper (C)      Dearer (D)
      \\       /
       Mean (M)
      /       \\
(D - M)      (M - C)

Ratio = (D - M) : (M - C)

Example 1: In what ratio must rice at ₹10/kg be mixed with rice at ₹12/kg to get mixture at ₹11/kg?

Cheaper (C) = ₹10, Dearer (D) = ₹12, Mean (M) = ₹11
D - M = 12 - 11 = 1
M - C = 11 - 10 = 1
Ratio = 1 : 1

Mix in ratio 1:1

Example 2: Mix pulses at ₹15/kg and ₹20/kg to get mixture at ₹18/kg.

C = ₹15, D = ₹20, M = ₹18
D - M = 20 - 18 = 2
M - C = 18 - 15 = 3
Ratio = 2 : 3

Mix in ratio 2:3

Types of Mixture and Alligation Problems

Type 1: Finding Ratio for Desired Mean Price

Example Problem: In what ratio must a grocer mix two varieties of tea costing ₹300/kg and ₹400/kg so that the mixture costs ₹360/kg?

C = ₹300, D = ₹400, M = ₹360
D - M = 400 - 360 = 40
M - C = 360 - 300 = 60
Ratio = 40 : 60 = 2 : 3

Interpretation: For every 2 kg of ₹300 tea, mix 3 kg of ₹400 tea.

Example 2: How much sugar at ₹15/kg should be mixed with 20 kg of sugar at ₹20/kg to get mixture at ₹18/kg?

C = ₹15, D = ₹20, M = ₹18
D - M = 20 - 18 = 2
M - C = 18 - 15 = 3
Ratio = 2 : 3
Given: Dearer sugar = 20 kg (corresponds to 2 parts)
2 parts = 20 kg → 1 part = 10 kg
Cheaper sugar needed = 3 parts = 30 kg

Need 30 kg of ₹15/kg sugar

Type 2: Finding Missing Price When Ratio is Given

Example Problem: Two varieties of sweets costing ₹400/kg and ₹x/kg are mixed in ratio 7:5 to get mixture at ₹650/kg. Find x.

C = ₹400, D = ₹x, M = ₹650, Ratio = 7:5
Using formula: (D - M) : (M - C) = 7 : 5
(x - 650) : (650 - 400) = 7 : 5
(x - 650) : 250 = 7 : 5
(x - 650)/250 = 7/5
5(x - 650) = 7 × 250
5x - 3250 = 1750
5x = 5000
x = ₹1000

Example 2: A mixture of 30 kg contains rice and wheat in ratio 2:3. If rice costs ₹25/kg and mixture costs ₹30/kg, find wheat price.

Ratio = 2:3 → Rice = 12 kg, Wheat = 18 kg
Let wheat price = ₹x/kg
Total cost = (12×25) + (18×x) = 300 + 18x
Mixture cost = 30×30 = ₹900
300 + 18x = 900
18x = 600
x = ₹33.33/kg

Wheat costs ₹33.33/kg

Type 3: Mixture Replacement Problems

When part of mixture is removed and replaced with another ingredient.

Formula for Replacement:

If a container contains \'x\' units of liquid from which \'y\' units are replaced:
Quantity of original liquid left = x(1 - y/x)^n
Where n = number of replacements

Example Problem: A 40-liter mixture contains milk and water in ratio 3:1. How much water must be added to make ratio 1:1?

Current mixture: Milk = 30L, Water = 10L (from 3:1 ratio in 40L)
Let water added = x liters
New water = 10 + x liters
New ratio = Milk : Water = 30 : (10 + x) = 1 : 1
30/(10 + x) = 1/1
10 + x = 30
x = 20 liters

Example 2: A vessel contains 60 liters of milk. 12 liters drawn and replaced with water. This is repeated once more. Find final milk concentration.

Initial milk = 60L
After first replacement:

Milk left = 60 × (1 - 12/60) = 60 × (4/5) = 48L
Water = 12L

After second replacement:

Milk left = 48 × (1 - 12/60) = 48 × (4/5) = 38.4L
Total mixture = 60L
Milk percentage = (38.4/60)×100 = 64%

Quick Formula: Milk left = Initial × (1 - fraction replaced)^n = 60×(1-12/60)² = 60×(4/5)² = 60×0.64 = 38.4L

Type 4: Alligation with More Than Two Ingredients

Method: Apply Alligation Pairwise

When mixing more than two ingredients, use alligation for pairs or use weighted average method.

Example Problem: Three varieties of rice costing ₹20, ₹25, and ₹30/kg are mixed. The mixture costs ₹26/kg and contains equal quantities of two varieties. Which varieties were mixed equally and in what ratio with the third?

Since two varieties mixed equally, let those be A and B in 1:1 ratio
Average of A and B = (A + B)/2
Check possibilities:

If A=20, B=25 → Average = 22.5
Mix with C=30 to get 26 → Ratio = (30-26):(26-22.5) = 4:3.5 = 8:7

Other combinations can be checked similarly

Example 2: Milk with 20% water mixed with milk having 30% water to get milk with 25% water. Find ratio.

This is concentration problem: Cheaper = 20% water, Dearer = 30% water, Mean = 25%
Using alligation: (30-25):(25-20) = 5:5 = 1:1
Interpretation: Mix in 1:1 ratio

Mix in ratio 1:1

Type 5: Profit/Loss on Mixture

Example Problem: A grocer mixes 26 kg of rice at ₹20/kg with 30 kg at ₹36/kg and sells at ₹30/kg. Find profit percentage.

Total cost = (26×20) + (30×36) = 520 + 1080 = ₹1600
Total quantity = 26 + 30 = 56 kg
Selling price = 56 × 30 = ₹1680
Profit = 1680 - 1600 = ₹80
Profit % = (80/1600)×100 = 5%

Example 2: A dishonest dealer mixes 20% water in milk and sells at cost price. Find gain percentage.

Assume milk cost = ₹100/liter (for easy calculation)
Dealer mixes 1 liter milk (₹100) with 0.2 liter water (₹0)
Total mixture = 1.2 liters
Cost price = ₹100 for 1.2 liters
Selling at milk price: Sells 1.2 liters as milk = 1.2 × 100 = ₹120
Profit = 120 - 100 = ₹20
Profit % = (20/100)×100 = 20%

Quick Formula: Gain % = (water added/pure milk)×100 = (0.2/1)×100 = 20%

Special Cases and Shortcuts

1. Mixture of Two Mixtures

Example: Mix 40% alcohol solution with 60% alcohol solution to get 50% solution.

Ratio = (60-50):(50-40) = 10:10 = 1:1

2. Removal and Replacement

After n replacements of x liters from V liters:
Pure liquid left = V[(V-x)/V]^n

Example: 10L milk, 2L removed and water added, repeated 3 times.

Milk left = 10×(8/10)³ = 10×0.512 = 5.12L

3. Alligation in Time and Distance

Example: Travel at 40 km/h for half distance, 60 km/h for other half. Find average speed.

Ratio of time = (60-avg):(avg-40) but distances equal → Use harmonic mean: 2/(1/40+1/60)=48 km/h

Mixture and Alligation: Frequently Asked Questions

What is the basic principle behind alligation rule?

The alligation rule is based on the principle of weighted averages. When two ingredients are mixed, the ratio in which they are mixed is inversely proportional to the difference between their prices and the mean price.

How to handle mixture problems with more than two ingredients?

For more than two ingredients, use the alligation rule pairwise. First mix two ingredients to get an intermediate mixture, then mix this intermediate mixture with the third ingredient, and so on.

What\'s the difference between alligation and allegation?

Alligation is a mathematical rule for mixtures. Allegation means a claim or assertion. They are completely different words with different meanings.

How to solve replacement problems quickly?

Use the formula: Final quantity = Initial × (1 - fraction replaced)^n. For concentration: Final concentration = Initial × (1 - fraction replaced)^n.

Can alligation be used for problems other than price mixtures?

Yes, alligation can be used for concentration problems (like alcohol percentage), average speed, average marks, and any situation involving weighted averages of two groups.

Practice Problems

Problem 1

In what ratio must water be mixed with milk costing ₹15/liter to get mixture costing ₹10/liter?

Solution: Water cost = ₹0, Milk = ₹15, Mean = ₹10. Ratio = (15-10):(10-0) = 5:10 = 1:2. Mix 1 liter water with 2 liters milk.

Problem 2

A mixture contains alcohol and water in ratio 4:3. If 5 liters of water is added, ratio becomes 4:5. Find original quantity.

Solution: Let alcohol = 4x, water = 3x. New water = 3x+5. Ratio: 4x/(3x+5)=4/5 → 20x=12x+20 → 8x=20 → x=2.5. Original = 7x = 17.5 liters.

Problem 3

How many kg of rice at ₹8/kg must be mixed with 36 kg at ₹12/kg to get mixture at ₹9/kg?

Solution: C=8, D=12, M=9. Ratio = (12-9):(9-8)=3:1. Dearer=36kg (1 part=36kg). Cheaper needed=3 parts=108kg.

Problem 4

A container has 60L milk. 12L drawn and replaced with water. Then 15L mixture drawn and replaced with water. Find final milk quantity.

Solution: First: Milk left=60×(48/60)=48L. Second: Milk left=48×(45/60)=36L. Or use formula: 60×(48/60)×(45/60)=60×0.8×0.75=36L.

Problem 5

Two alloys A and B have copper and zinc in ratios 2:3 and 3:4. In what ratio should they be mixed to get copper:zinc = 5:8?

Solution: Copper in A=2/5=0.4, in B=3/7≈0.4286. Desired=5/13≈0.3846. Using alligation: (0.4286-0.3846):(0.3846-0.4)=0.044:-0.0154=44:15.4≈20:7.

Important Formulas and Rules

Basic Alligation Rule:
Quantity of Cheaper : Quantity of Dearer = (D - M) : (M - C)

Mean Price Formula:
Mean Price = (Quantity₁×Price₁ + Quantity₂×Price₂) / (Quantity₁ + Quantity₂)

Replacement Formula:
After n replacements: Pure liquid = Initial × [1 - (replaced/total)]^n

Alligation for Three Ingredients:
Use alligation for pairs or set up equations:
(Q₁×P₁ + Q₂×P₂ + Q₃×P₃) / (Q₁+Q₂+Q₃) = Mean Price

Frequently Asked Questions

What is Mixture and Alligation Explained with Formulas and Examples?

Mixture and Alligation Explained with Formulas and Examples is an important aptitude topic used in competitive exams that tests your logical reasoning and problem-solving abilities.

Is Mixture and Alligation Explained with Formulas and Examples important for competitive exams?

Yes, Mixture and Alligation Explained with Formulas and Examples is frequently asked in SSC, Bank, CAT, TCS, and other placement exams. It's essential to master this topic for better scores.

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Key formulas vary by topic, but generally include basic concepts, shortcuts, and standard problem-solving approaches specific to Mixture and Alligation Explained with Formulas and Examples.

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