# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/२४६

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50
GANŅITASĀRASAŃGRAHA.

Example in illustration of vyutkalita in relation to a series in geometrical progression.

53. The first term is ${\displaystyle 7{\tfrac {1}{3}}}$, the common ratio is ${\displaystyle {\tfrac {2}{3}}}$, and the number of torms is 8; and the chosen-off number of torms is 3, 4 or 5. What are the first term, the sum and the number of terms in relation to the (rospective) remainder-series ?

Thus ends the vyutkalita of fractions.

The six varieties of fractions.

Hereafter we shall expound the six varieties of fractions.

54. Bhāga (or simple fractions), Prabhāga (or factions of fractions), then Bhāgābhāga (or complex fractions), then Bhāgānubandha or (or fractions in association), Bhāgāpavāha(or fractions in dissociation), together with Bhāgamātŗ (or fractions consisting of two or more of the above-mentiomed fractions)--these are here said to be the six varieties of fractions.

To simplify ${\displaystyle {\tfrac {a}{xy}}+{\tfrac {b}{yz}}}$. Here, a and xy are to be multiplid by z which is the quotient obtained by dividing yz, the donominator of the other fraction, by y which is the common factor of the denomenators. Thus we get ${\displaystyle {\tfrac {az}{xyz}}}$.
Similarly in the second fraction, by multiplying b and yz by x which is the quotient obtained by dividing the first denominator xy by y the common factor. we get ${\displaystyle {\tfrac {bx}{xyz}}}$. Now ${\displaystyle {\tfrac {az}{xyz}}+{\tfrac {bx}{xyz}}=={\tfrac {az+bx}{xyz}}}$.