Important Formulas & Concepts

Alligation or Mixture

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Alligation or Mixture

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Important Formulas & Concepts

Study Material

Alligation or Mixture

Alligation or Mixture is one of the most important topics in Quantitative Aptitude and is frequently asked in SSC, Banking, Railway, Insurance, Defence, CAT, CDS, NDA, and various competitive examinations.

This chapter mainly deals with:

  • Mixing two or more ingredients
  • Finding ratio of mixtures
  • Mean price calculations
  • Replacement of liquids
  • Concentration and dilution
  • Profit and cost-based mixture problems

Understanding percentages, ratios, averages, and replacement concepts helps candidates solve aptitude questions quickly and accurately.

What is a Mixture?

A new product obtained by mixing two or more ingredients in a certain ratio is called a Mixture.

OR

The combination of two or more quantities is known as a mixture.

Example:

Mixing milk and water together forms a mixture.

Important Terms

1. Alligation

Alligation is the rule used to find the ratio in which two or more ingredients at given prices must be mixed to produce a mixture of desired price.

2. Mean Price

The cost price of one unit quantity of the mixture is called the Mean Price.

Important Property:

  • Mean price is always greater than the cheaper price.
  • Mean price is always smaller than the dearer price.

3. Cheaper Quantity

The ingredient having lower cost price is called the Cheaper Quantity.

4. Dearer Quantity

The ingredient having higher cost price is called the Dearer Quantity.

Rule of Alligation

If two ingredients having prices:

  • Cheaper Price = c
  • Dearer Price = d
  • Mean Price = m

Then:

Cheaper Quantity : Dearer Quantity

= (d − m) : (m − c)

Alligation Diagram

      Dearer Price (d)
             \
              \
               \ (d − m)
                \
                 Mean Price (m)
                /
               / (m − c)
              /
             /
     Cheaper Price (c)

Important Formulae

1. Mean Price Formula

Mean Price = Total Cost / Total Quantity

2. Ratio Formula Using Alligation

Cheaper : Dearer

= (Dearer − Mean) : (Mean − Cheaper)

3. Replacement Formula

Suppose a container contains x litres of liquid.

If y litres are removed and replaced by water repeatedly for n operations:

Quantity Left

= Original Quantity × [(x − y)/x]n

Concept of Replacement

When some quantity is removed from a mixture:

  • The removed portion contains all ingredients proportionally.
  • Replacement changes concentration.
  • The total volume remains unchanged.

Important Concentration Formula

If:

  • Total Volume = V
  • Substance Percentage = x%

Then:

Quantity of Substance

= (x/100) × V

Concept of Dilution

Dilution means reducing concentration by adding water or another cheaper ingredient.

Important Effects:

  • Volume increases
  • Concentration decreases
  • Total quantity of original substance remains constant

Concept of Pure Substance

A pure substance contains:

100% Concentration

Example:

Pure milk, pure alcohol, pure wine.

Applications of Alligation

  • Milk and water problems
  • Alcohol concentration problems
  • Profit and loss mixture problems
  • Chemical mixing
  • Commodity blending
  • Percentage and average applications

Important Shortcut Concepts

1. Mean Lies Between Extremes

The mean price always lies between:

  • Cheaper price
  • Dearer price

2. Quantity of Substance Remains Constant

While adding water:

  • Total volume changes
  • Original substance quantity remains unchanged

3. Repeated Replacement

Repeated replacement decreases the original substance exponentially.

Quick Revision Formula Table

Concept Formula
Mean Price Total Cost / Total Quantity
Alligation Ratio (d − m) : (m − c)
Substance Quantity (x/100) × Volume
Replacement Formula [(x − y)/x]n
Pure Substance 100% Concentration

Common Mistakes to Avoid

  • Using incorrect alligation ratios.
  • Confusing cheaper and dearer quantities.
  • Ignoring concentration concepts.
  • Calculation mistakes in percentages.
  • Using wrong replacement formula.

Important Exam Tips

  • Memorize alligation formula thoroughly.
  • Practice percentage calculations daily.
  • Use cross-difference method carefully.
  • Focus on concentration concepts.
  • Practice replacement problems regularly.
  • Improve ratio simplification speed.
  • Practice previous year aptitude questions.

Alligation or Mixture is an important aptitude topic based on ratios, percentages, averages, and concentration concepts. Strong understanding of alligation rules and replacement formulas helps candidates solve competitive examination questions quickly and accurately.

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