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एतत् पृष्ठम् परिष्कृतम् अस्ति

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

53. The first term is , the common ratio is , 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.

Simple fractions: (addition and subtraction).

The rule of operation in connection with simple fractions therein :--

55. If, in the operations relating to sinmple fractions, the numerator and the denominator (of each of two given simple fractions) are multiplied in alternation by the quotients obtained

55. The method of reducing fractions to common denomenators described in this rule applies only to paris of fractions. This rule will be clear from the following worked out example:--

To simplify . 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 .

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 . Now .