# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/३५४

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158
GAŅITASĀRASAŃGRAHA.

thrice, five times and four times (that money which the others together had on hand. What is the money in the purse, and what the money on hand with each of them ?)

249-250${\displaystyle {\tfrac {1}{2}}}$. Three merchants saw on the way a purse containing money. The first among them said, "If I get ${\displaystyle {\tfrac {1}{4}}}$ of this money in the purse, I shall (with what I have on hand) become (possessed of) twice the (money on hand with) both of you." Another said that, if he secured ${\displaystyle {\tfrac {1}{3}}}$ part of the money in the purse, ho would with the money on hand with him (become possessed of) thrice (tho money on hand with the others). The third man said, " If I obtain ${\displaystyle {\tfrac {1}{2}}}$ of this (money in the purse), I shall become possessed of four times the money (on hand with both of you)." Tell me quickly, O mathematician, what the money on hand, with each of them was, and what was the money in the purse.

The rule for arriving at the money on hand, which, with the moneys begged (of others), becomes a specified multiple (of the money on hand with the others)

251${\displaystyle {\tfrac {1}{2}}}$-252${\displaystyle {\tfrac {1}{2}}}$.[*] The sums of the moneys begged are multiplied each by its own corresponding multiple quantity as increased by one. With the aid of these (products) the moneys on hand are arrived at according to the rule given in stanza 241. These quantities ( so obtained) are reduced so as to have a common denominator. Then they are (severally) divided by the sum as diminished by unity of the specified multiple quantities (respectively) divided by (those same) multiple quantities as increased by one. (The resulting quotients) themselves should be understood to be the moneys on hand , with the various persons) .

251${\displaystyle {\tfrac {1}{2}}}$-252${\displaystyle {\tfrac {1}{2}}}$. ^  Algebraically,

${\displaystyle x-{\Bigg \{}{\tfrac {(a+b)(m+1)+m(c+d)(n+1)}{n+1}}+{\tfrac {(a+b)(m+1)+m(f+g)(p+1)}{p+1}}+}$

${\displaystyle {\text{ etc. }}-(s-2)(a+b)(m+1){\Bigg \}}\div (m+1)]+}$
${\displaystyle \left({\tfrac {m}{m+1}}+{\tfrac {n}{n+1}}+{\tfrac {p}{p+1}}-1\right)}$

Similarly for y,s, etc. Here a,b,c,d,f,g, are sums of money begged of each other.