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10. A merchant in Boston has effects at Amsterdam, Holland, to the amount of $3530, which he can remit by way of Lisbon at 840 rees per dollar, and thence to Boston at 8s. ld. per milree (=1000 rees); or, by way of Nantz, at 5 livres per dollar, and thence to Boston at 6s. 8d. per crown of 6 livres. Which is the most advantageous way of remittance, and what is the difference between them?
Ans. By Lisbon £1198 8s. 84,d. By Nantz £1059. 11. If 140 braces at Venice be equal to 150 braces at Leghorn, and 7 braces at Leghorn be equal to 4 American yards : how many American yards are equal to 524 Venetian braces ?
Ans. 32 yards. 12. If 40 lb. at Newburyport make 36 at Amsterdam, and 90 lb. at Amsterdam make 116 at Dantzick, how many lb. at Dantzick are equal to 244 at Newburyport?
Ans. 283.1 lbs. 13. A merchant in Mississippi purchases goods to the value of $1500 from a merchant in New York. He sells the goods in his store at Jackson, and receives his pay in cotton, which he sends to his correspondent in New Orleans, who forwards it to Liverpool, where it is sold, and the net proceeds remitted to a banker in London, and placed to the credit of the Mississippi merchant. For how much sterling money must the latter draw a bill of exchange on London in favor of the New York merchant, allowing him 3 per cent. interest for the credit on his goods, exchange on London being 7 per cent. advance ?
Ans. £324 17s. 7d. 14. A banker received 759 ducats at 7s. 6d. per ducat, and 579 dollars, at 4s. 8d. per dollar, which he exchanges for Flemish marks, at 14s. 3d. each. How many ought he to receive?
Ans. 589 MT: 15. A bill of exchange for £100 sterling was accepted at London, for an equal value delivered at Amsterdam, at £1 13s. 6d. Flemish, for £1 sterling. How much money
was delivered at Amsterdam ?
Ans. £670 Flemish. 16. A merchant delivered at London £120 sterling, to receive £147 Flemish in Amsterdam. How much was £1 sterling valued at in Flemish money? Ans. £1 4s. 6d.
17. A factor sold goods at Cadiz for 1468 pieces of eight, valued at 4s. 64d. sterling each. How much sterling money do those pieces of eight amount to? Ans. £333 7s. 2d.
18. A traveller wished to have an equal number of crowns,
at 5s. 6d. per crown, and dollars, at 4s. 5d. each. How many of each sort may he have for £309 8s.? Ans. 624 of each.
19. A man wished to exchange £527 17s. 6d. for dollars at 4s. 6d. each, ducats at 5s. 8d. each, and crowns at 6s, id. each; and wanted 2 dollars for every ducat, and 3 dollars for every 2 crowns. How many of each should he receive ?
Ans. 927 dollars, 4634 ducats, and 618 crowns. 20. A banker is to receive £500. He is offered crowns, at 6s. 11d. per crown, which are worth but 6s., or he may have dollars at 4s. 5d, each, which are worth but 4s. 4d. Which of these should he receive to have the least loss? and how much will he lose in the payment ?
Ans. The smallest loss will amount to £9 8s. 895d.
When questions are of a complicated nature, which frequently happens in mercantile exchange, where the circulating medium of several foreign countries enter into the computation, they may be solved, perhaps, more simply by what is called Conjoined Proportion than by the usual method, as follows:
Exemplification for the Black-board. 1. If the exchange of London on Genoa be at 47 pence sterling per pezza, and that of Amsterdam on Genoa at 86 grotes per pezza ; what is the proportional exchange between London and Amsterdam, through Genoa ; that is, how many shillings and grotes Flemish (that is Amsterdam money, 12 grotes to a shilling) are equal to one pound sterling ?
1£ Sterling. £1=240 pence.
d.47= 1 pezza.
Suggestive Questions. In the above statement of 4 lines we have 3 sums of money on the left, given equal to 3 on the right, and if we knew how many grotes were equal to £1 sterling (the first line) all the 4 would be equal. Now, supposing the deficiency on the left to be supplied, would the products of these equal values be also equal ? But the product may be found complete on the right, while one factor is wanting on the left; how, then, may that factor be found ?
2. If the exchange of London on Madrid be at 42 pence sterling per dollar, or 272 maravedis, and that of Amsterdam on Madrid at 96 grotes Flemish per ducat=375 maravedis, what is the exchange between London and Amsterdam, through Madrid, in shillings and grotes, per pound sterling, allowing 12 grotes for a shilling ?
m. 375= 96 grotes.
33s. 114% gr. Ans.
Exercises for the Slate or Black-board. 1. If the exchange from Philadelphia to London was 4 per cent. above par (104 per 100) and from London to Paris 23 liv. 8 sous per pound sterling, what would be the proportional exchange from Philadelphia to Paris, through the medium of London ? and how many dollars would purchase a bill on Paris for 1100 livres 15 sous, allowing 20 sous to be equal to 1 livre, and £1 sterling to be equal to $4,0?
Ans. 5 liv. 2 sous per dollar. $217-43. 2. If, at New York, bills on London are at 5 per cent. above
par; the exchange of London on Amsterdam 34s. 4gr. per pound sterling; and Amsterdam on Paris 54gr. for 3 livres; what is the proportional exchange between New York and Paris in francs per dollar, 80 francs being equal to 81 livres ?
Ans. 4-882 fr.
dollar. 3. If the rate of exchange were, Boston on Paris 5:30 francs per dollar, Paris on Lisbon 464 rees per ecu of 3 livres,
what would be the proportional exchange between Boston and Lisbon, viz., how many rees per dollar ? Ans. 830 rees.
4. If the exchange of London on Lisbon be at 68 pence sterling per milree (=1000 rees), and that of Genoa on Lisbon‘at 718 rees per pezza ; what is the proportional exchange between London and Genoa, through Lisbon, in pence sterling per pezza ?
CONTRACTED MULTIPLICATION AND DIVISION OF DECIMAL
1. CONTRACTED MULTIPLICATION.
IT frequently happens, when one decimal fraction is multiplied by another, that the fractional part of the product extends to numbers altogether insignificant. Thus, if it were required to multiply 4*233 by 6*287, the product would extend to six decimal places, the last figure to the right being one-millionth part of 1, a number devoid of worth, even if it related to gold. To save the tedious labor of producing such worthless numbers, then, is frequently a matter of some consequence, especially where the computations are numerous, as in some of the articles in this Appendix. This may easily be effected by proceeding as follows:
1. Place the multiplier under the multiplicand in an inverted order, putting the unit's place of the multiplier under that decimal place in the multiplicand, which is the lowest meant to be retained in the product.
2. În multiplying, begin each line of partial products with that figure in the multiplicand which stands directly over the multiplying figure, increasing it by the tens that would have been produced (if any) by multiplying another figure to the right ; and also increasing it by one, if the right hand figure would have been 5 or upwards ; and let the first figures on the right of all the partial products stand directly under each other.
3. When it is desirable to be absolutely certain that the last figure retained is that nearest to the truth, the work should be extended to one place more than is wished to be retained.
4. The local value of the total product should be ascertained by an inspection of the two factors.
In general, when a decimal fraction is abbreviated by striking off, or omitting, some of the places on the right hand, in order that the last figure retained may be the nearest to the truth, whether too great or too little, it should be increased by one when the right hand figure is 5 or upwards. Thus, ‘1246, abridged to three decimal places, would be ‘125, while ‘1244 would only be ‘124.
Exemplifications for the Black-board. 1. Multiply 4127643 by 6*25135, retaining only four decimal places in the product.
Suggestive Questions on the Contractea Method.-In the first partial product, how many tens are carried to the first figure on the right? Would, or would not, the adjoining omitted figure have been 5 or upwards ? By how much, then, has the first figure standing on the right been increased ? By how much has the second partial product been increased ? Why? By how much has the third? Why? [The answer to these two “ whys” is different.] By how much has the fifth been increased ? Why? By how much the sixth? Why?
2. Multiply 36425 by 724325, retaining 5 decimal places in the product, so that 4 places may be absolutely certain of being nearest to correctness.
21730 14 48650
4346 289 7300
626384, or “2638 nearest 4 places.
Suggestive Questions on the Contracted Method.-Why is the first partial product increased by one ? Why the second by two? Why the third by two? Why the fifth by one?
To those who may not perceive why the figures of the multiplicand are placed in an inverted order, and in a rather unusual place, it may be remarked, that both form a mere mechanical contrivance to save time and labor, by enabling the student instantly to decide where the multiplication by each several figure of the multiplier is to begin. The order in which those figures are taken is of no moment, as has been shown, p. 257.
Exercise for the Slate or Black-board. 1. Multiply 34'265 by 4'396, true to three decimal places, and prove by multiplication in the usual manner.