पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/२५९

CHAPTER III--FRACTIONS.

quantities, as, (being equal in number to the given compound fractions), have their sum equal to the given sum (of the partially given compound fractions) : then, divide these (optionally chosen) values of the unknown (compound-fractional) quantities by (their) known (elements) respectively

An example in illustration thereof.

110. (The following partially known compound fractions, viz., ) ${\tfrac {1}{2}}$ of a certain quantity, ${\tfrac {3}{5}}{\text{ of }}{\tfrac {1}{3}}$ of another (quantity), and ${\tfrac {1}{2}}{\text{ of }}{\tfrac {2}{3}}$ of (yet) another (quantity give rise to) ${\tfrac {1}{2}}$ as (their) sum. What are the unknown (elements here in respect of these compound fractions)?

Examples in complex fractions

111. (Given) ${\tfrac {1}{\tfrac {6}{7}}},{\tfrac {1}{\tfrac {3}{8}}}$ and ${\tfrac {1}{\tfrac {2}{4}}}$; say what the sum is when these are added.

112. After subtracting ${\tfrac {1}{\tfrac {2}{3}}},{\tfrac {2}{\tfrac {3}{4}}}$, and also ${\tfrac {2}{\tfrac {3}{4}}}$ and ${\tfrac {1}{\tfrac {2}{3}}}$, from 9, give out the remainder.

Thus end Compound and Complex Fractions.

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Bhāgānubandha Fractions.

The rule in respect of the (simplification of Bhāgānubandha or associated fractions :--

113. In the operation concerning (the simplification of) the Bhāgānubandha class (of fractions), add the numerator to the

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partially known compound fractions, we divide them in order by ${\tfrac {1}{2}},{\tfrac {3}{5}}{\text{ of }}{\tfrac {1}{8}}$, and ${\tfrac {1}{2}}{\text{ of }}{\tfrac {2}{3}}$ respectively. The fractions thus obtained, viz. ${\tfrac {1}{3}},{\tfrac {10}{9}}$ and ${\tfrac {3}{4}}$, are the quantities to be found out.

118. Bhāgānubandha literally means an associated fraction. This rule contemplates two kinds of associated fractions. The first is what is known as a mixed number, i.e., a fraction associated with an integer. The second kind consists of fractions associated with fractions, e.g.,${\tfrac {1}{2}}$ associated with ${\tfrac {1}{3}}$, ${\tfrac {1}{2}}$ associated with its own ${\tfrac {1}{3}}$, and with ${\tfrac {1}{4}}$ of this associated quantity. The expression "${\tfrac {1}{2}}$ associated with ${\tfrac {1}{3}}$" means ${\tfrac {1}{?}}+{\tfrac {1}{3}}{\text{ of }}{\tfrac {1}{2}}$. The meaning of the other example here ${\tfrac {1}{2}}+{\tfrac {1}{2}}$ of ${\tfrac {1}{3}}+{\tfrac {1}{4}}{\text{ of }}{\tfrac {1}{2}}+{\tfrac {1}{2}}{\text{ of }}{\tfrac {1}{3}}$. This kind of relationship is what is denoted by association in additive consecntion.