पाटीगणित (श्रीधराचार्य - प्राचीन संस्कृत टीका एवं प्रो. कृपाशंकर शुक्ल सविस्तार व्याख्या)

36 DETERMINATIONS
(ii) ALLIGATION
Rule for finding the varṇa of the alloy obtained by melting
together a number of pieces of gold of given weights and
varṇas :
52(ii). The sum of the products of weight and varṇa
of the several pieces of gold, being divided by the sum of the
weights of the pieces of gold, gives the varṇa (of the alloy).¹
That is, if n pieces of gold of weights w₁, w₂, w₃, ..., wₙ and
varṇas v₁, v₂, v₃, ..., vₙ respectively are melted together, then the varṇa
v of the alloy is given by
v = (w₁ v₁ + w₂ v₂ + ... + wₙ vₙ) / (w₁ + w₂ + ... + wₙ) .
The term varṇa used above is analogous (though not equivalent) to
the modern term ‘carat’. It is used to denote the fineness of gold. Gold
of 1 varṇa contains 1 part of pure gold and 15 parts of impurities (in the
form of baser metals) ; gold of 2 varṇas contains 2 parts of pure gold and
14 parts of impurities ; and so on. Pure gold, called kalyâṇa-suvarṇa, is of
16 varṇas (or 24 carats). Thus the amount of pure gold in a piece of
gold of weight w and varṇa v is equal to wv/16.
The rationale of the above rule is as follows : The amount of pure
gold in the n pieces of gold mentioned above is equal to
(w₁ v₁)/16 + (w₂ v₂)/16 + ... + (wₙ vₙ)/16 , (1)
and the amount of pure gold in the alloy of those pieces of gold is equal to
((w₁ + w₂ + ... + wₙ) v) / 16 (2)
Since (1) and (2) are equal, therefore
((w₁ + w₂ + ... + wₙ) v) / 16 = (w₁ v₁)/16 + (w₂ v₂)/16 + ... + (wₙ vₙ)/16 .
Hence
v = (w₁ v₁ + w₂ v₂ + ... + wₙ vₙ) / (w₁ + w₂ + ... + wₙ) .
Ex. 61. Of what varṇa do 9, 5, and 17 mâṣas of
gold of 12, 10 and 11 varṇas (respectively) become when
melted together ?

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¹ Cf. BM, III, H 1, 16 verso ; GSS, vi. 169 (i), 172½ ; L (ASS),
p. 99, vs. 103 (i); GK, I, p. 76, lines 5-7.