Mixture & Alligation Guide & Practice

1. Fundamentals & Definitions

• Mixture: A combination of two or more substances (which could be solids, liquids, or gases) mixed together in any proportion such that the components do not chemically react and retain their individual properties.
• Alligation: A mathematical rule or method that enables the calculation of the ratio in which two or more ingredients at given prices must be mixed to produce a mixture of a desired price.
• Mean Price ($m$): The cost price per unit quantity of the final mixture. It is the weighted average price of the mixture.
• Cheaper Ingredient ($c$): The ingredient with the lower cost price per unit.
• Dearer Ingredient ($d$): The ingredient with the higher cost price per unit.
• Quantity of Cheaper ($q_c$): The amount of the cheaper ingredient in the mixture.
• Quantity of Dearer ($q_d$): The amount of the dearer ingredient in the mixture.
• Antecedent and Consequent: In a ratio $x:y$, the first term ($x$) is called the antecedent, and the second term ($y$) is called the consequent.
• Proportion: An equation stating that two ratios are equal (expressed as $a:b :: c:d$).
  • Mean Proportional: If $a:x :: x:b$, then $x = \sqrt{ab}$ is the mean proportional.
  • Third Proportional: If $a:b :: b:x$, then $x = \frac{b^2}{a}$ is the third proportional.

2. Core Concepts & Formulas

The Rule of Alligation (Two-Component Mixture)

When two ingredients of different unit costs are mixed, the ratio of their quantities is inversely proportional to the differences between their individual costs and the mean price.

$$\frac{\text{Quantity of Cheaper } (q_c)}{\text{Quantity of Dearer } (q_d)} = \frac{\text{Cost Price of Dearer } (d) - \text{Mean Price } (m)}{\text{Mean Price } (m) - \text{Cost Price of Cheaper } (c)}$$

This is represented graphically using the Cross Method:

Cheaper Cost (c)                   Dearer Cost (d)
                \                 /
                 Mean Price (m)
                /                 \
        (d - m)         :         (m - c)

$$\text{Ratio } (q_c : q_d) = (d - m) : (m - c)$$

Note: The value of $m$ must always lie between $c$ and $d$ ($c \le m \le d$).

Weighted Average Formula

If $k$ ingredients having quantities $n_1, n_2, \dots, n_k$ and unit costs/values $A_1, A_2, \dots, A_k$ are mixed together, the weighted mean price $A_w$ (or $m$) is:

$$A_w = \frac{n_1 A_1 + n_2 A_2 + \dots + n_k A_k}{n_1 + n_2 + \dots + n_k}$$

Profit and Loss on Selling a Mixture

If a mixture is sold at a given Selling Price ($SP$) with a gain or loss percentage, the Mean Cost Price ($m$) must be calculated first before applying the Alligation Rule. Alligation must be applied to Cost Prices.

Successive Replacement and Dilution

If a container initially contains $a$ units of a pure liquid, and $b$ units are drawn out and replaced by water (or another solvent), and this operation is repeated $n$ times:

$$\text{Quantity of pure liquid remaining} = a \left(1 - \frac{b}{a}\right)^n$$

$$\frac{\text{Quantity of pure liquid remaining}}{\text{Total Capacity}} = \left(1 - \frac{b}{a}\right)^n$$

$$\text{Ratio of Liquid to Solvent after } n \text{ operations} = \frac{\text{Liquid Left}}{\text{Total Capacity} - \text{Liquid Left}}$$

Compound Alligation (Three or More Components)

To mix three or more ingredients with unit costs $x, y, z$ (where $x < y < z$) to obtain a mean price $M$:

1. Group the ingredients into pairs such that one ingredient in the pair is cheaper than the mean price ($< M$) and the other is dearer ($> M$).
2. Apply the rule of alligation to each pair individually.
3. Sum the ratios for any common ingredients across pairs.

Typical Exam Weightage

The alligation cross-method turns most of these into 10-second questions once practiced — high value for low prep time.

Figures are typical ranges based on recent-year patterns, not a guarantee for any specific upcoming paper — always cross-check against the latest official syllabus and previous-year papers for Mixture & Alligation.