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

MIXTURES, PURCHASE AND SALE 47
then another rate of purchase is x' articles per rûpa, where
x' = P₁/d + R( C₁x - P₁y )/d
= P₂/d + R( C₂x - P₂y )/d
...
= Pₙ/d + R( Cₙx - Pₙy )/d ,
where x = Ry - 1, y = C_λ + k, C_λ being the greatest of all C's and
k an arbitrary integer; the rate of sale being the same as before
( i.e., C_λ+k articles per rûpa ).
The rationale of this rule is as follows :
When D = 1 , the equations (1) of the previous rule may be
written as
RC₁x - P₁(Ry - 1) = RC₂x - P₂(Ry - 1) = ...
= RCₙx - Pₙ(Ry - 1) = M. (1)
Dividing throughout by d , and writing x' for x/d , P_r' for P_r/d,
and M' for M/d, we have
RC₁x' - P₁'(Ry - 1) = RC₂x' - P₂'(Ry - 1) = ....
= RCₙx' - Pₙ'(Ry - 1) = M'. (2)
The values of x , y satisfying (1) give one set of rates of purchase
and sale, and the values of x' , y satisfying (2) give another set of rates
of purchase and sale.
When M = Ry - 1 , a solution of (1) is, as before,
x = Ry - 1 ⎫
⎬ (3)
y = C_λ+k ⎭
and the corresponding solution of (2) is
x' = (Ry - 1)/d , i.e., M' ⎫
⎬ (4)
y = C_λ+k . ⎭
The rates of purchase and sale given by (3) are those already given
in Rule 60 ; those given by (4) are those mentioned in the present rule.
Since, from (2),
M' = P₁/d + R(C₁x - P₁y)/d = P₂/d + R(C₂x - P₂y)/d
= ... = Pₙ/d + R(Cₙx - Pₙy) ,
it is clear that the value of x' given by (4) is the same as that stated
in the above rule. It can be easily seen that M' is an integer.
Note. When D ≠ 1, then another rate of purchase will be D(Ry - 1)/d
articles per rûpa, where y = C_λ + k.