पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/३२६

130

GAŅIITASĀRASAŃGRAHA.

The rule for separating the prices of (an interchangeable) larger and (a similar) smaller number of two different things from the given mixed sums of the prices of these things:-

139${\tfrac {1}{2}}$. From the higher price-sum, as multiplied by the corresponding larger number of one of the two kinds of things, subtract the lower price-number as multiplied by the smaller number relating to the other of the two kinds of thing. Then divide the result by the difference between the squares of the numbers relating to these things. This gives rise to the period of the thing which is larger in number. The other, that is, the price of the thing which is smaller in number, is obtained by interchanging the multipliers.

An example in illustration thereof.

140${\tfrac {1}{2}}$ to 142${\tfrac {1}{2}}$. The mixed price of 9 citrous and 7 fragrant wood-apples is 107; again the mixed price of 7 citrons and 9 fragrant wood-apples is 101. O you arithmetician, tell me quickly the price of a citron and of a wood-apple here, having distinctly separated those prices well.

The rule for separating the prices and the numbers of different mixed quantities of different kinds of things from their given mixed price and given mixed values:-

143${\tfrac {1}{2}}$. The (different) given (mixed) quantities (of the different things) are to be multiplied by an optionally chosen number; the given (mixed) price (of these mixed quantities) is to be diminished (by the value of these products separately). The resulting quantities

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139${\tfrac {1}{2}}$. Algebraically, if

$ax+by=m$

and     $ax+ay=n$,

then    $a^{2}x+aby=am$

and     $b^{2}x+aby=bn$

$\therefore \quad x(a^{2}-b^{2})=am-bn$.

$\therefore \quad x={\tfrac {am-bn}{a^{2}-b^{2}}}$.

143${\tfrac {1}{2}}$. The rule will become clear by the following working of the problem in stanzas 144${\tfrac {1}{2}}$ and 145${\tfrac {1}{2}}$:-

The total number of fruits in the first         heap is 21.
          Do.               do.               second       do.    22.
          Do.               do.               third            do.   23.