पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/२६१

एतत् पृष्ठम् परिष्कृतम् अस्ति
65
CHAPTER III-FRACTIONS.

120. O friend, subtract (the following) from 3: ${\displaystyle {\tfrac {1}{4}}}$ associated with ${\displaystyle {\tfrac {1}{2}}}$ of itself and with ${\displaystyle {\tfrac {1}{3}}}$ of this (assicuated quantity), ${\displaystyle {\tfrac {2}{9}}}$ associated with ${\displaystyle {\tfrac {1}{8}},{\tfrac {1}{3}}}$ and ${\displaystyle {\tfrac {1}{2}}}$ of itself (in additive consecution), ${\displaystyle {\tfrac {1}{9}}}$ (similarly) associated with (fractions thereof) commencing with ${\displaystyle {\tfrac {1}{8}}}$ and ending with ${\displaystyle {\tfrac {1}{2}}}$, and ${\displaystyle {\tfrac {1}{3}}}$ associated with ${\displaystyle {\tfrac {1}{2}}}$ of itself.

121. O friend, you, who have a thorough knowledge of Bhāgānubandha, give out (the result) after adding ${\displaystyle {\tfrac {1}{4}}}$ associated with ${\displaystyle {\tfrac {1}{2}}}$ of itself, ${\displaystyle {\tfrac {1}{10}}}$ associated with ${\displaystyle {\tfrac {1}{4}}}$ of itself, ${\displaystyle {\tfrac {1}{6}}}$ associated with ${\displaystyle {\tfrac {1}{2}}}$ of itself, ${\displaystyle {\tfrac {1}{2}}}$ associated with ${\displaystyle {\tfrac {1}{3}}}$ of itself, and ${\displaystyle {\tfrac {1}{3}}}$ associated with ${\displaystyle {\tfrac {1}{3}}}$ of itself.

Now the rule for finding out the one unknown (element) at the beginning (in each of a number of associated fractions, their sum being, given):--

122.[१] The optionally split up parts of the (given) sum, which are equal (in number) to the (intended) component elements (thereof), when divided in order by the resulting quantities arrived at by taking one to be the associated quantity (in relation to these component elements), give rise to the value of the (required) unknown (quantities in association).

Examples in illustration thereof.

123. A certain fraction is associated with ${\displaystyle {\tfrac {1}{2}},{\tfrac {1}{3}}}$ and ${\displaystyle {\tfrac {1}{4}}}$ of itself (in additive consecution); another (is similarly associated with ${\displaystyle {\tfrac {1}{5}},{\tfrac {1}{4}}}$ and ${\displaystyle {\tfrac {1}{9}}}$ of itself; and another (again is similarly associated) with ${\displaystyle {\tfrac {3}{5}},{\tfrac {1}{9}}}$ and ${\displaystyle {\tfrac {1}{2}}}$ of itself; the sum of these (three fractions so associated) is 1: what are these fractions ?

124. A certain fraction, when associated (as above) with ${\displaystyle {\tfrac {1}{2}},{\tfrac {1}{5}},{\tfrac {3}{4}}}$ and ${\displaystyle {\tfrac {1}{9}}}$ of itself, becomes ${\displaystyle {\tfrac {1}{2}}}$. Tell me, friend, quickly the measure of this unknown (fraction),

1. 122. This rule will be clear from the working of example No. 123 as explained below:--
There are three sets of fractions given and splitting up the sum, 1, into three fractions according to rule No. 75, we get ${\displaystyle {\tfrac {1}{2}},{\tfrac {1}{3}}}$ and ${\displaystyle {\tfrac {1}{6}}}$. By dividing these fractions by the quantities obtained by simplifying the three given sets of fractions wherein 1 is assumed as the unknown quantity, we obtain and ${\displaystyle {\tfrac {1}{5}},{\tfrac {1}{4}}}$ and ${\displaystyle {\tfrac {1}{15}}}$ which are the required quantities.