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

MIXTURES, PURCHASE AND SALE 45
shed by 1 and then multiplied by the common denominator
(of the capitals), gives the rate of purchase (i.e., the rate at
which the articles are purchased for one rūpa ; kraya).¹
That is, if n persons, whose capitals reduced to a common deno-
minator are
C₁/D, C₂/D, C₃/D, ..., Cₙ/D rūpas respectively,
invest their capitals in purchasing certain articles at the rate of x articles
per rūpa, and if by selling those articles in multiples of y at the rate of y
articles per rūpa, and the remnant articles, at the rate of R rūpas per
article, they become equally rich, then
x = D(Ry – 1),
and y = C_λ + k,
where C_λ is the greatest of all C's, and k an arbitrary integer.
The rationale of this rule is as follows :
Suppose that by selling the articles in multiples of y, the n persons
earn P₁, P₂, P₃ ..., Pₙ rūpas respectively. Then the total amounts
earned by the n persons by selling all the articles (including the remnant
articles) are respectively
P₁ + R((C₁/D)x – P₁y), P₂ + R((C₂/D)x – P₂y), ... ,
and Pₙ + R((Cₙ/D)x – Pₙy) rūpas,
or (RC₁x/D) – P₁(Ry – 1), (RC₂x/D) – P₂(Ry – 1), ...,
and (RCₙx/D) – Pₙ(Ry – 1) rūpas.
Since these amounts are equal to one another, therefore we have
(RC₁x/D) – P₁(Ry – 1) = (RC₂x/D) – P₂(Ry – 1) = ...
= (RCₙx/D) – Pₙ(Ry – 1).
or RC₁x – P₁D(Ry – 1) = RC₂x – P₂D(Ry – 1) = ...
= RCₙx – PₙD(Ry – 1) = M say. (1)
To solve (1), let x = M = D(Ry – 1);
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¹ Also see GSS, vi. 102½ ; GK, I, pp. 94-97, vv. 36 (ii) – 38(i).