ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

A PROBLEM ON INTEREST 185
Brahmagupta (628 A.D.) gives a more general rule : He
enunciates his problem thus :
The principal (p) is lent out for t₁ months and the
unknown interest on this (=x) is lent out for t₂ months
at the same rate and becomes A. To find x.
This evidently gives the quadratic :
x² + (pt₁ / t₂) x - (Apt₁ / t₂) = 0
whose solution is
x = ± √[ Apt₁/t₂ + (pt₁ / 2t₂)² ] - pt₁ / 2t₂
The negative value of the radical does not give a solution
of the problem, so it is discarded.
Brahmagupta states the formula thus :
Multiply the principal (p) by its time (t₁) and divide
by the other time (t₂) (placing the result) at two places.
Multiply the first of these by the mixture (A). Add to
this the square of half the other. Take the square-root of
this (sum). From the result subtract half the other.
This will be the interest (x) on the principal.¹
A Problem on Interest
Brahmagupta gives a solution of a problem on interest :
In what time will a given sum s, the interest on which
for t months is r, become k times itself ?
The rule for the solution of this problem as given by Brah-
magupta is :
The given sum multiplied by its time and divided by
the interest (phala), being multiplied by the factor
(guṇa) less one, is the time (required).²
Miscellaneous Problems
Brahmagupta in his Gaṇitādhyāya of the Brāhmasphuṭa-
siddhānta gives numerous solutions in relation to miscellaneous
problems. Here I shall be quoting a few of the problems which

1. कालप्रमाणघातः परकालहृतो द्विधाऽऽद्यमिश्रवधात् ।
   अन्यार्धकृतियुतात् पदमन्यार्धोनं प्रमाणफलम् ॥ —BrSpSi. XII. 15.
2. कालगुणितं प्रमाणं फलभक्तं व्येकगुणहतं कालः ।
   स्वफलयुतरूपभक्तं मूलफलैक्यं भवति मूलम् ॥ —BrSpSi. XII. 14.