402. To find the value in Federal money of Bills of Exchange on England, at a given per cent. premium.

I. Reduce the shillings, pence, and farthings to the decimal of a pound, and annex it to the given pounds. (Art. 200.)

II. Multiply this sum by the nominal par value of £1, increased by the given per cent. exchange, and the product will be the value of the bill in Federal money.

Or, multiply the pounds and decimal of a pound by 40; then find the per cent. exchange and add it to this product; the result will be the value of the bill in Federal money.

OBS. In negotiating foreign bills, it is customary to draw three of the same date and amount, which are called the First, Second and Third of Exchange; and collectively, a Set of Exchange. These are sent by different ships or convey. ances, and when the first is accepted or paid, the others are void.

2. What is the value of a bill of exchange on London for £1568, 5s., at 9 per cent. premium ?

3. When exchange is 9 per cent. premium, what is the value of a bill for £2573, 15s. 8d.?

4. What is the Federal value of £3181, 7s. 6d., at 10 per cent. premium?

per cent., it is plain £1 $4.80)$2730.00 00

=

5. A traveler paid $2730 for a bill of exchange on London, at 8 per cent. premium: what was the amount of the bill? Suggestion.-Since the rate of exchange was 1.08 x $40, or $4.80. Now if $4.80 will buy £1, $2730 will buy as many pounds as $4.80 are contained times in $2730, which is £568.75. The decimal .75 is equal to 15s. Hence,

Operation.

£568.75

20

s.15.00

Ans. £568, 15s.

403. To find the value in Sterling money of any sum of Federal money, at a given per cent. exchange.

Divide the given sum by the nominal par value of £1 in Federal money increased by the per cent. exchange, and the quotient will be the value in pounds and the decimal of a pound. Reduce the decimal to shillings, pence, and farthings, annex them to the pounds, and the result will be the answer. (Art. 210.)

QUEST.-402. How do you find the value in Federal money of bills of exchange on England at a given per cent. premium?

6. What is the Sterling value of $1560.75, at 9 per cent. exchange?

7. What is the Ster. value of $4368.50, at 10 per cent. Ex.? 8. What is the Ster. value of $5280.60, at 81 per cent. Ex.? 9. What is the Ster. value of $56000, at 91 per ct. exchange? 10. A merchant consigned 2560 bbls. of flour to his agent in Liverpool, who sold it at £1, 8s. 6d. per barrel, and charged 2 per cent. commission: what is the net amount of the flour in Federal money, allowing 8 per cent exchange?

12. A merchant in N. Y. shipped 1000 bales of cotton weighing 360 lbs. per bale, to his agent in London, who sold it at 81d. per pound, paid d. a pound for freight, and charged 21 per cent. commission. What was the net proceeds in Federal money, allowing 8 per cent. exchange?

404. In France, accounts are kept in francs and centimes. The franc is the unit money. It is subdivided into decimes or tenths, and centimes or hundredths, according to the decimal notation, in the same manner as Federal money.

10 centimes make 1 decime; 10 decimes make 1 franc.

OBS. In business matters, decimes are expressed in centimes. Thus, in quoting exchanges, instead of saying 5 francs, 2 decimes, and 3 centimes are worth a dollar, it is customary to say, exchange is 5 francs and 23 centimes per dollar; in circulars and prices current, it is written, Francs, 5.23 to 5.18, &c.

405. In negotiating Bills of Exchange on France, it is customary to make the dollar the fixed sum or basis of calculation, while the French currency is the variable sum.

406. To find the value in Federal money of Bills of Exchange on France.

Divide the given sum by the exchange value of $1 expressed in francs and centimes, and the quotient will be the value of the bill in Federal money.

OBS. Formerly exchange on France was calculated at a certain per cent. on a nominal par of exchange, which assumed a dollar to be equal to 5 francs. This method, though still adhered to in books, is discarded by business men.

13. What is the value of a bill of exchange on Paris for 4515 francs, exchange at 5.25 francs per dollar? Ans. $860.

14. What is the value of a bill of exchange on Havre for 10640 francs, exchange at 5.16 francs per dollar.

15. What is the value of 25265 f. exchange 5.20 f. per $1? 16. What is the value of 68432 f. exchange 5.16 f. per $1?

ARITHMETICAL PROGRESSION.

408. PROGRESSION is continued proportion. It is of two kinds, arithmetical and geometrical.

409. Arithmetical progression is a series of numbers, which increase or decrease by a common difference; as 3, 5, 7, 9, 11, 13, &c.; or 13, 11, 9, 7, 5, &c.

OBS. If the series increases, it is called ascending; if it decreases, descending. Arithmetical progression is sometimes called equidifferent series.

410. When four numbers are in arithmetical progression, the sum of the extremes is equal to the sum of the means. Thus, in the series 3, 5, 7, 9, the sum 3+9=5+7.

Again, if three numbers are in arithmetical progression, the sum of the extremes is double the mean.

Thus, in the series 9, 6, 3, the sum 9+3=6+6.

411. In an ascending series, each succeeding term is found by adding the common difference to the preceding term. Thus, if the first terin is 3, and the common difference 2, the series is 3, 5, 7, 9, 11, 13, &c.

In a descending series, each succeeding term is found by subtracting the common difference from the preceding term. Thus, if the first term is 15, and the common difference 2, the series is 15, 13, 11, 9, 7, &c.

412. In arithmetical progression, there are five parts to be considered, viz: the first term, the last term, the number of terms, the common difference, and the sum of all the terms. These parts have such a relation to each other, that if any three of them are given, the other two may be easily found.

413. To find the sum of all the terms, when the extremes and the number of terins are given.

Multiply half the sum of the extremes by the number of terms, and the product will be the sum of the given series.

QUEST.-408. What is progression? Of how many kinds is it? 409. What is arithmetical progression? Obs. If the series increases, what is it called? If it decreases, what? 410. When four numbers are in arithmetical progression, to what is the sum of the extremes equal? 411. In an ascending series, how is each successive term found? 413. When the extremes and number of terms are given, how find the sum of all the terms?

1. If the extremes of a series are 2 and 14, and the number of terms is 7, what is the sum of all the terms? Ans. 56.

2. If the extremes of a series are 36 and 4, and the number of terms 9, what is the sum of all the terms?

3. How many strokes would a clock which goes to 24 o'clock, strike in a day?

415. To find the common difference, when the extremes and the number of terms are given.

Divide the difference of the extremes by the number of terms less 1, and the quotient will be the common difference required. 4. If the extremes are 2 and 38, and the number of terms 13, what is the common difference? Ans. 3.

5. The extremes are 3 and 19, the number of terms 9: what is the com. dif. and the sum of the series?

416. To find the number of terms, when the extremes and common difference are given.

Divide the difference of the extremes by the common difference, and the quotient increased by 1 will be the number of terms.

6. If the extremes are 2 and 53, and the common difference 3, what is the number of terms? Ans. 18.

7. A man spent $3 the first holiday, $45 the last, and each day $3 more than on the preceding: how many holidays did he have, and how much did he spend?

417. When the sum of the series, the number of terms, and one of the extremes are given, to find the other extreme.

Divide twice the sum of the series by the number of terms, and from the quotient take the given extreme.

8. If one extreme is 10, the number of terms 6, and the sum of the series 42, what is the other extreme? Ans. 4.

9. If the sum of the series is 1924, one extreme 27, and the number of terms 26, what is the other extreme?

QUEST.-415. When the extremes and number of terms are given, how find the common difference? 416. When the extremes and common difference are given, how find the number of terms? 417. When the sum of the series, the number of terms, and one of the extremes are given, how find the other extreme?

GEOMETRICAL PROGRESSION.

418. Geometrical progression is a series of numbers which increase by a common multiplier, or decrease by a common divisor; as 2, 4, 8, 16, 32, &c.; or 32, 16, 8, 4, 2.

OBS. 1. If the series increases, it is called ascending; if it decreases, descending. The numbers which form the series, are called the terms of the progression. The common multiplier, or divisor, is called the ratio.

2. In an ascending series, each succeeding term is found by multiplying the preceding by the ratio. Thus, if the first term is 2, and the ratio 3, the series is 2, 6, 18, 54, &c.

In a descending series, each succeeding term is found by dividing the preceding by the ratio. If the first term is 54 and the ratio 3, the series is, 54, 18, 6, 2.

3. If the first term and ratio are the same, the progression is simply a series of powers; as 2; 2×2; 2×2×2; 2×2×2×2, &c.

4. In Geometrical as well as in Arithmetical progression, there are five parts to be considered, viz: the first term, the last term, the number of terms, the ratio, and the sum of all the terms. These parts have such a relation to each other, that if any three of them are given, the other two may be easily found.

419. To find the last term, when the first term, the ratio, and the number of terms are given.

Multiply the first term into that power of the ratio whose index is 1 less than the number of terms, and the product will be the last term required.

OBS. The several amounts in compound interest, form a geometrical series of which the principal is the 1st term; the amount of $1 for 1 year the ratio and the number of years +1 the number of terms. Hence the amount of any sum at compound interest, may be found in the same way as the last term of a geometrical series.

1. If the first term of a geometrical progression is 4, and the ratio 3, what is the 5th term? Ans. 324.

2. If the first term is 48, and the ratio, what is the 5th term.

3. The first term of a series is 3, the ratio 4: what is the 7th term?

QUEST.-418. What is geometrical progression? Obs. When the series increases, what is it called? When it decreases, what? What are the terms of the progression? In an ascending series, how is each succeeding term found? How in a descending series? If the first term and the ratio are the same, what is the series? 419. When the first term, the ratio, and the number of terms are given, how do you find the last term? Obs. How find the amount of any sum at compound interest by geometrical progression ?