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Mixtures and Alligations

03 min read

Mixtures
Mixtures typically contain two kinds of substances that do not react with each other. Mixtures can be solid in solid, solid in liquid (solution) or liquid in liquid. In a mixture of A and B present in the ratio 3 : 2, we can say that the concentration of A is 3/5 and that of B is 2/5. For solutions, the concentration may be expressed in terms of weight per volume (gm/ml).

Mixing two mixtures
Let p1, p2 and p be the concentrations of the first, second and final mixtures respectively. Let q1, q2 and q be their quantities respectively.
Then the final concentration can be determined using the rule for weighted averages which gives:
p = (p1q1 + p2q2) / (q1 + q2)
This formula can be extended even if more than 2 mixtures are present.

Alligation
The equation above can be rewritten as q1/q2 = (p2 - p) / (p - p1)
A simpler way to understand this would be to consider the mixing of a cheaper and dearer (costlier) item in order to obtain an item of intermediate price. In that case, the equation of alligation can be descriptively written as:

Quantity of cheaper / quantity of dearer = (rate of dearer - average rate) / (average rate - rate of cheaper)

This can be represented pictorially as follows:


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