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. 1d. 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. 13d. 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. 8d.

CONJOINED PROPORTION.

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?

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?

1£ Sterling.

£1-240 pence.

d. 42=272 maravedis.
m. 375= 96 grotes.

Left hand prod. 1575 0)626688|0(39711+3=143
75

15418

33s. 1157 gr. Ans.

12438

1413

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 $40 ?

9 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. per 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? Ans. 4819%.

SUPPLEMENT.

CONTRACTED MULTIPLICATION AND DIVISION OF DECIMAL

FRACTIONS.

1. CONTRACTED MULTIPLICATION.

Ir 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. In 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 4'127643 by 6'25135, retaining only four decimal places

in the product.

In full.

4'127643

6'25135 2,0638215 12 382929 41/27643

2063 8215 8255 286 247658 58

25 803314106805

Contracted.

4'127643

531526

247659

8255

2064

41

12

2

25 8033

Suggestive Questions on the Contracted 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 cor

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.

2. Multiply '008 by 3'796, true to three decimal places, and prove as above.

3. Multiply '5264 by '0428, true to three decimal places, and prove. 4. Multiply 1'729 by 7'218, true to four decimal places, and prove. 5. Multiply 26'45 by 39'46, true to two decimal places, and prove.

2. CONTRACTED DIVISION.

When it is desirable in division to limit the number of decimal places in the quotient, it may be done as follows:

1. Take as many figures only, on the left hand side of the divisor, as the whole number of figures required to be in the quotient, and cut off the rest.

2. Let each remainder successively be a new dividend, without bringing down any figure from the original dividend, but, instead thereof, let another figure be continually cut off from the divisor for each quotient figure, till the whole is exhausted, observing, however, as in contracted multiplication, to increase each particular product by the nearest number of tens in the product of the quotient figure, into the figure last cut off in the divisor.

3. When the whole divisor does not contain as many figures as are required to be in the quotient, no figure should be cut off till the figures in the divisor shall be equal to the remaining figures required to be in the quotient, when the cutting off should commence as above directed.

Exemplifications for the Black-board.

1. Divide 74'33373 by 1'346787, retaining three decimal places only, or five places in the quotient altogether.

Suggestive Questions.-How many tens are carried into the first partial product? How many into the second? the third? the fourth? the fifth ?

The pupils should not write the partial product, but make the subtraction, as usual, mentally.

2. Divide ‘07567 by 2-32467, true to four decimal places; or three significant figures, the first being a cipher.