# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/३०६

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110
GAŅIITASĀRASAŃGRAHA.

Proportionate Division.

Hereafter we shall expound in (this) chapter on mixed problems the working of proportionate division:-

${\displaystyle 79{\tfrac {1}{2}}}$[*]. The operation of proportionate division is that wherein the (given) collective quantity (to be divided) is first divided by the sum of the numerators of the common-denominator-fractions (representing the various proportionate parts), donominators of which fractions are struck off out of consideration; and (then it) has to be multiplied (respectively in each case) by (these) proportional numerators. This is called kuṭṭīkāra by the learned.

Examples in illustration thereof.

${\displaystyle 80{\tfrac {1}{2}}}$. Here, (in this problem,) 120 gold pieces are divided among 4 servants in the (respective) proportional parts of ${\displaystyle {\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {1}{4}}}$ and ${\displaystyle {\tfrac {1}{6}}}$. O arithmetician, tell me quickly what they obtained.

${\displaystyle 81{\tfrac {1}{2}}}$. (The sum of) 363 dīmāras was divided among five, the first one(among them) getting 3 parts, and 3 being the common ratio successively (in relation to the shares of the others). What was the state of each ?

${\displaystyle 82{\tfrac {1}{2}}}$ to ${\displaystyle 85{\tfrac {1}{2}}}$. A certain faithful śrāvaka took a number of lotus flowers, and going into the Jina temple conducted (therein) with devotion the worship of the chief Jinas that were worthy of worship. He offered ${\displaystyle {\tfrac {1}{4}}}$ part to Vṛṣabha, ${\displaystyle {\tfrac {1}{6}}}$ to worthy Pārśva, and ${\displaystyle {\tfrac {1}{12}}}$ to Jinapati, and ${\displaystyle {\tfrac {1}{3}}}$ to sage Suvrata.; he devotedly gave ${\displaystyle {\tfrac {1}{8}}}$ to Aristanémi who dostroyed all the eight kinds of karmas' and ${\displaystyle {\tfrac {1}{6}}{\text{ of }}{\tfrac {1}{4}}}$ to Jinaśānti. 480 lotuos were brought (for this purpose. ) By adopting the operation known

${\displaystyle 79{\tfrac {1}{2}}}$.^  In working the example in stanza ${\displaystyle 80{\tfrac {1}{2}}}$ according to this rule we get ${\displaystyle {\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {1}{4}},{\tfrac {1}{6}}={\tfrac {6}{12}},{\tfrac {4}{12}},{\tfrac {3}{12}},{\tfrac {2}{12}}}$. After removing the denominators here, we have 6, 4, 3 and 2. These are also called prakṣēpas or proportional numerators. The sum of these is 15, by which the amount to be distributed, viz., 120 is divided; and the resulting quotient 8 is separately multiplied by the proportional numerators,6,4,3 and 2. Then the amounts thus obtained are 6 x 8 or 48, 4 x 8 or 32, 3 x 8 or 24, 2 x 8 or 16. It is worthy of note that prakṣēpas means both the operation of proportionate division and a proportional numerator .