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54
GANITABĀRASANGRAUA.
which, (while having the given denominators), have one for the numorators and are then reducel so as to have a common denominator, becomes the first required) numerator among thoso which (successively) rise in value hy one (and are to be found out). On the remainder (obtained in this division) being divided by the aim of the other numerators (having the common denominator as above), it, (i.e., the resulting quotient), becomes another (viz., tho second required) numerator (if added to the first one already obtained). In this manner the problem has to be worked out) to the end.
Un example in illustration thereof. 74. The sum of certain numbers which are divided (respectively) hy 9, 10 and 11 is 871 as divided hy 990. Give out what the numerators are in this operation of adding fractions).
The rule for arriving at the required denominators (is as follows) :
75. When the sum of the different fractional) quantities having one for their numerators is one, the required) denominators are such AH, beginning with one, are in order multiplied (nuccessively) by
obtained in this division in the divided by the son of the remaining provisional numerators, i... 189, giving the ypotient 1, which, combined with the numerator of the tirat fraction, namely %, becomca the numerator in relation to the scoond denominator. The remainder in this second division), viz., 90, is divided by the provisional numerator 90 of the Inst fraction, and the quotient I, when combined with the numerator of the previous fruction, namely 3, gives rise to the numerator in relation to the last denominator. Hence the fractions, of which is the sum, are to and fr.
It is notiovable here that thu honeruto's successively found out thus become the required numeratore in relation to the given denominators in the order in which they are given. Algebruically #lo, given the denominators a, c, in respect of 3 fructions
lica + ( + 1) ac + (x + 2) ab whose sumn is
the numerators + 1 and
abc # + ? are easily found out by the method as given abovy.
75. In working on an example according to the method stated berein, it will be found that when there are fractions, thero nre, after leaving out the first and the last fructions - 2 terni in geometrical progressicy with an the first term and to the conimon l'utio. The bune of these # - 2 terms is
.
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, which when reduced becomies ! -
Ry" which is the same
which is the
