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

MIXTURES. PURCHASE AND SALE 49
or RC₁x - P₁D² ( R/D y - 1 ) = RC₂x - P₂D² ( R/D y - 1 )
= ... = RCₙx - PₙD² ( R/D y - 1 ) = N, say. (1)
Let x = N = D²( R/D y - 1 ), then
RCᵣ - Pᵣ = 1, (2)
for r = 1, 2, ..., n.
But ((Cᵣ/D)x) / y = Pᵣ + Aᵣ/y,
where Aᵣ is the remainder when Cᵣx/D is divided by y, so that, when
x = D²( R/D y - 1 ),
Pᵣ = Cᵣx / Dy - Aᵣ / y
= [ Cᵣ D² ( R/D y - 1 ) ] / Dy - Aᵣ / y
= RCᵣ - (Cᵣ D + Aᵣ) / y. (3)
Therefore we must have
y > CᵣD, for all r,
or y > C_λD,
where C_λ is the greatest of all C's.
Hence a solution of (1) is
x = D²( R/D y - 1 ),
y = D(C_λ + k),
k being an arbitrary integer.
Ex. 77. The capitals of four men are 1½, 2, 3, and 5
rûpas. By purchasing and selling (certain articles) at the same
rates, and by selling the remnant articles at the rate of 1 for ½
(of a rûpa), they become possessed of equal riches. (Find the
rates of purchase and sale).¹

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¹ Cf. GSS, vi. 108½, 110½; GK, I, p. 99, lines 2-5 and 15-18.