पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/३०४

108

GAŅIITASĀRASAŃGRAHA.

optionally chosen (maximum available amount of an installment) by (whatever happens to be) the outstanding (fractional part of the number of terms in the scrics), to the amount of the (first installment as multiplied by the sum of that series in arithmetical progression, which has (one for the first term, one for the common difference, and has one for the number of terms the integral value of) the quotient obtained by dividing (the above optionally chosen maximum) amount of debt (discharged at an installment) by the (above amount of the first) installment. The interest thoreon is that which accrues for the period of an installment. The time (of an installment) divided by the amount of the (first) installment and multiplied by the (optionally chosen maximum) amount of debt (discharged at an installment) gives rise to the time (which is the time of the discharge of the whole debt).

Examples in illustration thereof.

72 and $73{\tfrac {1}{2}}$. A certain man utilised, (for the discharge of a debt) bearing interest at 5 per cent (per month), 60 (as the available maximum amount) with 7 as the first installment amount, increasing it by 7 in successive installments due every ${\tfrac {3}{5}}$ of a month. He thus gave in discharge of the debt tho sum of a series in arithmetical progression consisting of ${\tfrac {60}{7}}$ terms, and gave also the interest accruing on those multiples of 7. What is the debt amount corresponding to the sum of the series, what is that interest (which he paid), and (what is) the time of discharge of that debt ?

$73{\tfrac {1}{2}}$ to 76. A certain man utilised for the discharge of a debt, bearing interest at 5 per cent (per mensem), 80(as the available maximum amount) with 8 as tho first installment amount, increasing it by 8 in successive installments due every ${\tfrac {1}{2}}$ of a month. He thus

 

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8 represents the number of terms of the series in arithmetical progression, which has 1 for the first term and 1 for the common difference; and ${\tfrac {4}{7}}$ is the agra or the outstanding fractional part. The sum of the above-mentioned series, viz, 36, multiplied by 7, the amount of the first installment, is added to the product of ${\tfrac {4}{7}}$, and 60, which latter is the maximum available amount of an installment. Thus, we get $36\times 7+{\tfrac {4}{7}}\times 60={\tfrac {2004}{7}}$, which is the required capital amount in the due debt. The interest on ${\tfrac {2004}{7}}$ for ${\tfrac {3}{5}}$ of a month at the rate of 5 per cent per mensem will be the interest paid on the whole. The time of discharge will be $\left({\tfrac {3}{5}}\div 7\right)\times 20={\tfrac {36}{7}}$ months .