पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/२५४

58

GAŅIITASĀRASAŃGRAHA.

Examples an illustration thereof.

88. Say what the denominators are of three (different fractional) quantities each of which has 1 for its numerator, when the sun (of those quantities) is ${\tfrac {23}{12}}$ .

84. Say what the denominators are of three (fractional quantities) which have 3, 7 and 9 (respectively) for their numerators, when the sum (of those quantities) is ${\tfrac {73}{48}}$.

The rule for arriving at the denominators of two (fractional) quantities which have one for each of their numerators, when the sum (of those quantities) has one for its numerator:--

85. The denominator of the (given) sum multiplied by any chosen number is the denominator (of one of the intended fractional quantities); and this (denominator) divided by the (previously) chosen (number) as lessened by one gives rise to the other (required denominator). Or, when in relation to the denominator of the (given) sum (any chosen) divisor (thereof) and the quotient (obtained therewith) are (each) multiplied by their sum, they give rise to the two (required) denominators,

Examples in illustration thereof.

86. Tell me, O you who know the principles of arithmetic, what the denominators of the two (intended fractional) quantities are when their sum is either ${\tfrac {1}{8}}$ or ${\tfrac {1}{10}}$.

The first rule for arriving at the denominators of two (intended fractions) which have either one or (any number) other

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85. Algebraically, when ${\tfrac {1}{n}}$ is the sum of two intended fractions, the fractions according to this rule are - ${\tfrac {1}{pn}}$ and ${\tfrac {1}{\tfrac {pn}{p-1}}}$, where p is any clnosen quantity. It will be seen at once that the sum of these two fractions is ${\tfrac {1}{n}}$.

Or, when the sum is ${\tfrac {1}{ab}}$, the fractions may be taken to be ${\tfrac {1}{a(a+b)}}$ and ${\tfrac {1}{b(a+b)}}$.