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100
GAŅIITASĀRASAŃGRAHA.

(connected with the loan) are not known. Their sum however is 82. What is the capital, what is the time, and what the interest ?

The rule for arriving separately at the various amounts of interest accruing on various capitals for various periods of time from the mixed sum of (those) amounts of interest:-

37. Let each capital amount, multiplied by the (corresponding) time and multiplied (also) by the (given) total (of the various amounts) of interest, be separately divided by the sum of the products obtained by multiplying each of the capital amounts by its corresponding time, and let the interest (of the capital so dealt with) be (thus) declared.

An example in illustration thereof.

88. In this (problem), the (given) capitals are 40, 30, 20 and 50; and the months are 5, 4, 8 and 6 (respectively). The sum of the amounts of interest is 34. (Find out each of these amounts.)

The rule for separating the various capital amounts from their mixed sum:-

30. Let the quantity representing the mixed sum of the various capitals lent out be divided by the sum of those (quotients) which are obtained by dividing the various amounts of interest by their corresponding periods of time and let the (resulting) quotient be multiplied (respectively) by (the various) quotients obtainod by

37. Symbolically, ${\displaystyle {\tfrac {c_{1}t_{1}m}{c_{1}t_{1}+c_{2}t_{2}+c_{3}t_{3}\dots }}=i_{1}}$;

and ${\displaystyle {\tfrac {c_{2}t_{2}m}{c_{1}t_{1}+c_{2}t_{2}+c_{3}t_{3}\dots }}=i_{2}}$; where ${\displaystyle m=i_{1}+i_{2}+i_{3}+...}$ and ${\displaystyle c_{1},c_{2},c_{3}}$ etc., are the various capitals, and ${\displaystyle t_{1},t_{2},t_{3}}$ etc., are the various periods of time.

39. Symbolically, ${\displaystyle {\tfrac {m}{{\tfrac {i_{1}}{t_{1}}}+{\tfrac {i_{2}}{t_{2}}}+{\tfrac {i_{3}}{t_{3}}}+...}}\times {\tfrac {i_{1}}{i_{2}}}=c_{1}}$;

and ${\displaystyle {\tfrac {m}{{\tfrac {i_{1}}{t_{1}}}+{\tfrac {i_{2}}{t_{2}}}+{\tfrac {i_{3}}{t_{3}}}+...}}\times {\tfrac {i_{1}}{i_{2}}}=c_{2}}$