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.