पृष्ठम्:Ganita Sara Sangraha - Sanskrit.djvu/२५२

54 GANITASARASANGRABA. which, (while having the given denominators), have one for the numerators and (are then reduced so as to) have a common deno- minator, becomes the (first required) numerator among those which (successively rise in value by one (aud are to be found out) On the remainder (obtained in this division) being divided by the sum of the other numerators (having the commou denominator as above), it, (i e., the resulting quotient), becomes another (viz., the second required) numerator (it added to the first one already obtained) In this manner (the problem has to be worked out) to the end. An e cumple in illustration thereof. 74. The sum of certain numbers which are divided (respect- ively) by 9, 10 and 11 is 877 as divided by 990 Give out what the numerators are (in this operation of adding fractions) The rule for arriving at the required denominatois (is as follows) :- 75. When the sun of the (different fi actional) quantities having one for their numerators is one, the required) denominators are such as, beginning with one are in order multiplied successively) by obtained in this division is then divided by the sum of the romaming provisional numeratos, e, 189, giving the quotient 1, which, combined with the uumerator of the hist fraction, namely 2, becomes the numerator in relation to the second denominator. The remainder in this second division, viz., 90, is divided by the provisional numerator 90 of the last fiaction, and the quotient 1, when corubined with the numerator of the previous fraction, namely 3, gives rise to the numera- tor in relation to the last denominator Hence the factions, of which is the sum, are, to and ii. It is noticcable here that the numielaiols successively found out thus become the required numerators in relation to the given denominatois in the order in which they are given. Algebraically also, given the denominatois a, b & c, in respect of 3 fractions bec+ (+1) ac + ( + 2) ab abc the numerators a, 1 and · whose sum is + 2 are easily found out by the method as given above. will be found that when there are 75. In working out an example according to the method stated herein, it fractions, there anc, after leaving out the first and the last fractions, - 2 teinis in geometrical progression with as the first term and as the common ratio The sum of these - 2 tems is 3 [1-6] 1=1 which when reduced becomes 11 3-2 which is the same