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Thus, if bills on Paris were at a premium of } per cent. in St. Petersburg, and bills on London at a premium of i per cent. in Paris, whilst bills on London were at a premium of only 5 per cent. in St. Petersburg, a London merchant who wanted to send a bill on London to St. Petersburg would find it more advantageous to transmit the bill through Paris than to make a direct transmission. For the sake of simplicity, let us suppose that the amount of the bill about to be transmitted is £100. If sent directly, this bill would realize only £1005, or £ 100 125. 6d., in St. Petersburg; whereas, if sent through a correspondent in Paris,--who would sell or exchange it for £ion of Paris “paper,” and then send this paper to St. Petersburg,—the bill would realize £101 58.

211. When a foreign bill is transmitted circuitously-instead of directly-from one country to another, the price ultimately realized by the bill is termed an “arbitratedrate of Exchange between the two countries; and the calculation of such rates, from the necessary data, belongs to what is called ARBITRATION of Exchange."

In the case just supposed, £ 100 of British money—although, according to the "course" of Exchange, equivalent to only £100 123. 6d. of Russian money-can, by an “arbitrated” Exchange, be converted into £10i 5s. Russian currency.

212. Arbitration of Exchange is said to be SIMPLE or COMPOUND—according as the number of places to be taken into account is three or more than three; in other words—according as the number of “intermediate” places is one or more than one.

EXAMPLE I. (Par of Exchange.)-Find the par of Exchange between London and Paris : the data being—that the franc (the French standard of value) contains, out of 10 parts, 9 of pure silver and i of alloy ; that 200 francs weigh a kilogramme (15,434 grains) ; that the napoleon, which contains, out of 10 parts, 9 of pure gold and i of alloy, is equivalent to 20 francs; that 155 napoleons (which are equivalent to 3,100 francs) weigh a kilogramme; that the mint price of the sovereign (the British standard of value), which contains, out of 12 parts, II of pure gold and i of alloy, is £3 178. 10}d., or 934-5 pence, an ounce Troy; and that the market value of English standard silver, which contains, out of 40 parts, 37 of pure silver and 3 of alloy, is (about) 5 shillings an ounce Troy.

Here we have£1

240d. 934:5d.

i oz. English standard gold 12 oz. English standard gold)

II oz. pure gold 9 oz. pure gold 10 oz. French standard gold I oz. French }

standard gold) 15,434 grs.

i kilogramme i kilog. French

standard gold) =3,100 francs ? francs

£I

480 grs.

Hence, by the Chain Rule (p. 179), £1=(240 XII X 10 X 480 X 3,100=934'5 X 12X9 X 15,434=)25-22 francs, or 25 francs 22 centimes, and this is the par of Exchange-the gold par, its it is sometimes termed between London and Paris.

Note. - From a comparison of the standard silver of Eng. land with that of France, another par, called the silver par, can be obtained as follows: £1

=24011. bod.

i oz. English standard silver 40 oz. English standard silver

37 oz. pure

silver 9 oz. pure

silver Io oz. French standard silver I oz. French standard silvers

=480

grs. 15,434 grs.

1 kilogramme i kilog. French

==200 francs ? francs £I=(240 X 37 X 10 X 480 X 200=60 X 40 X 9 X 15,434=) 25.57 francs, or 25 francs 57 centimes; and this is the silver par between London and Paris.

EXAMPLE II. (Course of Exchange.)—If the course of Exchange between England and France were 25 francs 46 centimes for £1, what amount of British money would be equivalent to 567 francs 38 centimes ? The answer is evidently the fourth term of the proportion

francs.
25:46 : 567.38 : : £I
=(567.38:25:46=)£22.285, or £22 55. 8]d.

francs.

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EXAMPLE III. (Course of Exchange.)—If the course of Exchange between England and Russia were 38. 3}d. for i rouble, what amount of Russian money would be equivalent to £162 11s. British ? The proportion is 38. 31d. : £162 IIS.

I rouble 39:5d. : 39,012d. :: I rouble a=(39,012-39.5=)987.65 roubles (nearly), or 987 roubles 65 copecks [i rouble=100 copecks].

NOTE.—In the courses of Exchange between England and other countries, a variable amount of foreign money is, in some cases, allowed for a fixed amount of British ; whilst, in other cases, a variable amount of British is allowed for a fixed amount of foreign money. Thus, in the Exchange between England and France the "fixed” rate (£1) is given by England, and the “ variable” rate by France; but in the Exchange between England and Russia the fixed rate (i rouble) is given by Russia, and the variable rate---sometimes 38. 2d., sometimes 35. 4d., &c.—by England.

EXAMPLE IV. (Simple ARBITRATION of Exchange.)—The course of Exchange between London and Paris being 24 francs 85 centimes for Z1, and between Paris and St. Petersburg 7 francs for 2 roubles, what is the arbitrated rate between London and St. Petersburg ? Here we have

=24:85 francs 7 francs 2 roubles

2 roubles={1 £1=(24:85X2+7=)7'1 roubles, or 7 roubles 10 copecks.

EXAMPLE V. (Compound ARBITRATION of Exchange.)- The course of Exchange between London and Frankfort being u forins 30 kreutzers [1 florin 60 kreutzers] for £1, between Frankfort and Paris 4 florins 18 kreutzers for 9 francs, and between Paris and Lisbon 13 francs 20 centimes for 3 milrees, what is the arbitrated rate between London and Lisbon ? Here we have

£1

=!1'5 florins 4'3 florins

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9 francs

3 milrees

7 milrees =I £=(1145 X9X3--43X132=)547 milrees, or 5 milrees 470 rees [1 milree = 1,000 rees].

13'2 francs

ALLIGATION. 213. Under this head come questions which relate to the mixing of different kinds of the same commodity—tea, sugar, wine, &c.; the object of such mixing being (a) to improve the quality of an inferior article, or (b) to make a superior article cheaper and more saleable.

The method of working exercises in ALLIGATION—exercises which, it may be observed, seldom occur in practice—will be understood from the following examples :

ExamPLE I.-If 20 lbs. of tea worth 2s. 6d. a pound, 16 lbs. worth 38. a pound, and 12 lbs. worth 35. 43. à pound were mixed together, how much a pound would the mixture be worth, on an average ?

lbs.
Price of 20 @ 2

6 per po'ınd 50
16
3

48
3 4

40

S. d.

S.

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12

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48

I

= 138 Average price of 1 lb. of the mixture 138s. • 48 = 25. 10 d.

EXAMPLE II.-In what proportions does a grocer mix sugar worth 6d. a pound and sugar worth 4d. a pound, when he wants the mixture to be worth, on an average, 51d. a pound ?

On every pound of 6-penny sugar in the mixture, the grocer loses (6d. 51d.=) 3 farthings; whilst on every pound of 4-penny sugar he gains (5 d. -4d.=) 5 farthings. Consequently, as the total loss and the total gain must exactly counterbalance each other, the grocer, instead of mixing the two sugars in equal proportions, makes the 6-penny sugar as much larger in quantity than the 4-penny, as 5 farthings are larger in amount than 3 farthings. So that 5 lbs. of the dearer, and 3 lbs. of the cheaper sugar would form the required mixture; 5 times 3 farthings being the loss on the former, and 3 times 5 farthings the gain on the latter : lbs. d.

d. 5 @ 6 per pound

= 30 3 4

= 12

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8,51

= 42 In practice, the work would assume the following form the numbers indicating the gain and the loss per pound being taken cross-wise for the required proportions :

d.

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d.

[ocr errors]

51d.

Ans.

= 5

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farthings. J6; loss per pound = 3

:

5 @ 6 per pound. 4; guin

3 4 NOTE.-If a particular quantity of the mixture were required, the number denoting the quantity should be divided into parts proportional to 5 and 3. Thus, in 4 lbs. of the mixture there would be 2} lbs. of the 6-penny and 11 lbs. of the 4-penny sugar; in 24 lbs. of the mixture, 15 lbs. of the 6-pienny and 9 lbs. of the 4-penny sugar; &c. :lbs. d. d.

lbs. d. d.
2} @6
= 15

15 @ 6 =

90 i}

9 4 36

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= 21

S.

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4 5

24 , 51 = 126 EXAMPLE III.--In what proportions should a vintner mix four different kinds of sherry--worth, respectively, 158., 208., 30s., and 325. a gallon—in order to form a mixture worth, on an average, 24s. a gallon ? On every gallon of the

S. 15

gain of 9
20

gain 4
sherry, there would be a
30

loss 6
32

loss 8 It is evident, therefore, after what has already been explained, that the gains would counterbalance the losses if (a) the mixture were made to contain 6 gallons of the first sherry for every 9 of the third, and 8 gallons of the second for every 4 of the fourth ; or if (6) 8 gallons of the first were taken for every 9 of the fourth, and 6 gallons of the second for every 4 of the third.

According to the first arrangement, the gains and losses would stand thus :

galls.
On 6 of the 15 sherry, a gain of 6 x 91
9

loss
8

gain 4

loss According to the second arrangement, the gains and losses would be

galls
On 8 of the 15 sherry, a gain of 8 x 91
9

loss
6

gain

6 x 41 4 , 30

s.

3.

30 ܕܕ

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20

8 x 41

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