Elements of Algebra

156 ELEMENTS SECT. III. $= 16$ ducats; and, by means of this comparison, we may change any sum of old louis into ducats, and vice-versa. Thus, suppose it were required to find how many ducats there are in $1000$ old louis, we have this proportion: Lou. Lou. Duc. Duc. As $9 : 1000 :: 16 : 1777\frac{7}{9}$, the number sought. If, on the contrary, it were required to find how many old louis d'or there are in $1000$ ducats, we have the following proportion: Duc. Duc. Lou. As $16 : 1000 :: 9 : 562\frac{1}{2}$ louis. Ans. 479. At Petersburgh the value of the ducat varies, and depends on the course of exchange; which course determines the value of the ruble in stivers, or Dutch halfpence, $105$ of which make a ducat. So that when the exchange is at $45$ stivers per ruble, we have this proportion: As $45 : 105 :: 3 : 7$; and hence this equality, $7$ rubles $= 3$ ducats. Hence again we shall find the value of a ducat in rubles; for Du. Du. Ru. As $3 : 1 :: 7 : 2\frac{1}{3}$ rubles; that is, $1$ ducat is equal to $2\frac{1}{3}$ rubles. But if the exchange were at $50$ stivers, the proportion would be, As $50 : 105 :: 10 : 21$; which would give $21$ rubles $= 10$ ducats; whence $1$ ducat $= 2\frac{1}{10}$ rubles. Lastly, when the exchange is at $44$ stivers, we have As $44 : 105 :: 1 : 2\frac{17}{44}$ rubles; which is equal to $2$ rubles, $38\frac{2}{7}$ copecks. 480. It follows also from this, that we may compare different kinds of money, which we have frequently occasion to do in bills of exchange. Suppose, for example, that a person of Petersburgh has $1000$ rubles to be paid to him at Berlin, and that he wishes to know the value of this sum in ducats at Berlin. The exchange is at $47\frac{1}{2}$; that is to say, one ruble makes $47\frac{1}{2}$ stivers; and in Holland, $20$ stivers make a florin; $2\frac{1}{2}$ Dutch florins make a Dutch dollar: also, the exchange of Holland with Berlin is at $142$; that is to say, for $100$ Dutch dollars, $142$ dollars are paid at Berlin; and lastly, the ducat is worth $3$ dollars at Berlin. 481. To resolve the question proposed, let us proceed $24$ drachms $= 1$ ris dollar $72$ do $= 3$ ris dollars $= 1$ ducat $128$ do $= 5\frac{1}{3}$ do $= 1\frac{7}{9}$ do