# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/३५२

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एतत् पृष्ठम् परिष्कृतम् अस्ति
156
GAŅITASĀRASAŃGRAHA.

by one, are (severally) subtracted. The resulting remainders constitute the several values of the moneys on hand. The value of the money in the purse is obtained by carrying out operations as before and then by dividing by any particular specified fractional part (mentioned in the problem).

An example in illustration thereof.

239–240. Five merchants saw a purse of money. They said one after another that by obtaining ${\displaystyle {\tfrac {1}{6}},{\tfrac {1}{7}},{\tfrac {1}{8}},{\tfrac {1}{9}}{\text{ and }}{\tfrac {1}{10}}}$(respectively) of the contents of the purse, they would each become with what he had on hand three times as wealthy as all the remaining others with what they had on hand together. O arithmetician, (you tell) me quickly what moneys these had on hand (respectively), and what the value of the money in the purse was.

The rule for arriving at the measure of the money contents of a purse, when specified fractional parts (thereof added to what may be on hand with one among a number of persons) makes him a specified number of times (as rich as all the others with what they together have on hand)

241. The specified fractional parts relating to all others (than the person in view) are (reduced to a common denominator, which is ignored for practical purposes. These are severally) multiplied by the specified multiple number (relating to the person in view). To these products, the fractional part (relating to the person) in view (and treated like other fractional parts) is added. The resulting sums are (severally) divided each by its (corresponding specified) multiple quantity as increased by one. Then these quotients are also added. The several sums (so obtained in relation

241. The formula given in the rule is:-

{\displaystyle {\begin{aligned}&x={\Bigg \{}{\tfrac {a+mb}{n+1}}+{\tfrac {a+mc}{q+1}}+{\tfrac {a+md}{r+1}}+\cdots \quad \cdots \quad \cdots -(s-2)a{\Bigg \}}\div (m+1)\\&y={\Bigg \{}{\tfrac {b+na}{m+1}}+{\tfrac {b+nc}{q+1}}+{\tfrac {b+nd}{r+1}}+\cdots \quad \cdots \quad \cdots -(s-2)a{\Bigg \}}\div (n+1)\end{aligned}}}

and so on; where x,y, . . . . are moneys on hand; a,b,c,d, . . . . fractional parts; m,n,q, r,. . . . . various multiple numbers; and s the number of persons concerned in the transaction.