Elements of Algebra

158 ELEMENTS SECT. III. arrive at the same results by multiplying at once the sum proposed by the product of all the third terms, and dividing by the product of all the first terms: or, which amounts to the same thing, that we have only to make the following proportion: As the product of all the first terms, is to the given number of rubles, so is the product of all the second terms, to the number of ducats payable at Berlin. 484. This calculation is abridged still more, when amongst the first terms some are found that have common divisors with the second or third terms; for, in this case, we destroy those terms, and substitute the quotient arising from the division by that common divisor. The preceding example will, in this manner, assume the following form. As $(2.20.5.100.3.105) : 1000 :: (95.2.142.100) : \frac{1000.95.2.142.100}{2.20.5.100.3.105}$; and after cancelling the common divisors in the numerator and denominator, this will become $\frac{10.19.142}{3.21} = \frac{26980}{63} = 428\frac{16}{63}$ ducats, as before. 485. The method which must be observed in using the Rule of Reduction is this: we begin with the kind of money in question, and compare it with another which is to begin the next relation, in which we compare this second kind with a third, and so on. Each relation, therefore, begins with the same kind as the preceding relation ended with; and the operation is continued till we arrive at the kind of money which the answer requires; at the end of which we must reckon the fractional remainders. 486. Let us give some other examples, in order to facilitate the practice of this calculation. If ducats gain at Hamburgh $1$ per cent. on two dollars banco; that is to say, if $50$ ducats are worth, not $100$, but $101$ dollars banco; and if the exchange between Hamburgh and Konigsberg is $119$ drachms of Poland; that is, if $1$ dollar banco is equal to $119$ Polish drachms; how many Polish florins are equivalent to $1000$ ducats? It being understood that $30$ Polish drachms make $1$ Polish florin, Here $1 : 1000 :: 2$ dollars banco $100$ -- $101$ dollars banco $1$ -- $119$ Polish drachms $30$ -- $1$ Polish florin; therefore,