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

48 DETERMINATIONS
Ex. 76. The capitals of (three) men are 1, 3, and 5
(rûpas) or ⅓, ¼ and ½ (rûpas) (respectively). By purchasing and
selling (certain articles) at the same rates and by selling the
remnant articles at the rate of 1 for 3 (rûpas), they become
possessed of equal riches. (Find the rates of purchase and
sale).¹
Sub-rule for finding the rates of purchase and sale when each
of the remnant articles is sold for a fractional price :
62. When the remnant-price (i.e., the price at which
each remnant article is sold) is fractional (lit. has a denomi-
nator), reduce it along with the capitals to a common
denominator (and remove the denominators as before), and
multiply the rates of purchase and sale resulting from them
(by applying Rule 60) by the common denominator.²
That is, if the capitals of the n persons and the remnant-price
reduced to a common denominator, be as follows
Capitals : C₁/D , C₂/D , C₃/D , ... , Cₙ/D rûpas
Remnant-price : (R/D) rûpas,
then x = D² ( (R/D)y - 1 ) ,
and y = D (C_λ + k) ,
C_λ being the greatest of all C's and k an arbitrary integer.
The rationale of this rule is as follows :
As before, equating the wealths of the n persons after all the articles
have been sold away, we get [see equations (1) of Rule 60 (notes)]
(R/D) · C₁x - P₁D ( (R/D)y - 1 ) = (R/D) · C₂x - P₂D ( (R/D)y - 1 )
= ... = (R/D) · Cₙx - PₙD ( (R/D)y - 1 ) ,
¹ Cf. GSS, vi. 103½, 104½; GK, I, p. 96, lines 2-6, and p. 97, lines 7-9.
² Cf. GSS, vi. 107½. For another rule, see GSS, vi. 109½; GK, I, p. 98,
lines 2-7.