# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/२६२

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66
GAŅITASĀRASAŃGRAHA.

The rule for finding out any unknown fraction in other required places (than the beginning) :-

125. The optionally split up parts of the (given) sum when divided in order by the simplified known quantities (in the intended Bhāgānubandha (fractions), and (then) diminished by one become the unknown (fractional quantities) in the required places of our choice.

Thus ends the Bhāgānubandha class (of fractions).

Bhāgāpavāha Fractions.

Then (comes) the rule for the (simplification of) Bhāgāpavāha (or the dissociated) variety (in fractions):--

126. In the operation concerning (the simplification of) the Bhāgāpavāha class (of fractions), subtract the numerator from the (product of the dissociated) whole number as multiplied by the denominator. (When, however, the dissociated quantity is not integral, but is fractional,) multiply (respectively) the numerator and the denominator of the first (fraction to which the other fraction is negatively attached) by the denominator diminished by the numerator, and by the denominator (itself, of this other fraction).

Examples of Bhāgāpavāha fractions containing dissociated integers.

127. Karșās 3, 8, 4 and 10, diminished by ${\displaystyle {\tfrac {1}{4}},{\tfrac {1}{2}},{\tfrac {1}{12}}}$ and ${\displaystyle {\tfrac {1}{6}}}$ of a karșā, are offered by certain men for the worship of tīrthańkaras. What is (the sum) when they are added ?

125. The method given in this rule is similar to what is explained under stanza No. 122: only the results thus obtained have to be, in this case, each diminished by one.

126. Bhāgāpavāha literally means fractional dissociation. As in Bhāgānubandha, there are two varieties here also. When an integer and a fraction are in Bhāgāpavāha relation, the fraction is simply subtracted from the integer.

Two or more fractions may also be in such relation, as for example, ${\displaystyle {\tfrac {2}{5}}}$ dissociated from ${\displaystyle {\tfrac {1}{6}}}$ of itself or ${\displaystyle {\tfrac {6}{7}}}$ dissociated from ${\displaystyle {\tfrac {1}{6}},{\tfrac {1}{8}}}$ and ${\displaystyle {\tfrac {1}{9}}}$ of itself. It is meant here that ${\displaystyle {\tfrac {1}{6}}{\text{ of }}{\tfrac {2}{5}}}$ is to be subtracted from ${\displaystyle {\tfrac {2}{5}}}$ in the first example; and the second example comes to ${\displaystyle {\tfrac {6}{7}}-{\tfrac {1}{8}}{\text{ of }}{\tfrac {6}{7}}-{\tfrac {1}{8}}{\text{ of }}\left({\tfrac {6}{7}}-{\tfrac {1}{6}}{\text{ of }}{\tfrac {6}{7}}\right)-{\tfrac {1}{9}}}$ of ${\displaystyle {\Big \{}{\tfrac {6}{7}}-{\tfrac {1}{6}}{\text{ of }}{\tfrac {6}{7}}-{\tfrac {1}{8}}{\text{ of }}\left({\tfrac {6}{7}}-{\tfrac {1}{6}}{\text{ of }}{\tfrac {6}{7}}\right){\Big \}}}$.