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cent. that

How many francs must the banker receive, supposing

£20 125 rubles,

=

60 rubles 144 marks of Hamburgh,

=

95 marks 46 dollars of plate (Spanish),
11 dollars=41 francs?

Solution by fractions: . £20=125 rubles, it follows that £1 = 125 rubles; similarly, . 624 rubles=144 marks, it follows that

1 ruble: =

marks, and.. £1=1,25 of marks; but . 95

20

144 621 marks 46 dollars, it follows that 1 mark=

=

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144

624

dollar, and .. £1

=

francs; and therefore that £1=

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And commission at 1 per cent. 312 francs.

.. the banker receives 31186.68+312=31498.68 francs, By the chain-rule method, we have:

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x=

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123 × 144 × 46 × 41 × 1200____ 6518016 francs

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=31186.68 frs.

312 com.

31498.68 frs.

The principle of arbitration of exchange may be extended to a variety of questions, which an example will illustrate.

Ex. 4. 14 Welshmen perform a piece of work in 8 days.

6 Irishmen in 2 days do as much as 5 Englishmen in 3 days.

3 Englishmen in 4 days

4 Scotchmen in 5 days

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2 Scotchmen in 6 days. 7 Welshmen in 21 days.

How long will 10 Irishmen, 12 Englishmen, and 16 Scotchmen be doing the same work respectively?

...6 I. in 2 d., or 12 I. in 1 d.=5 E. in 3 d., or 15 E. in 1 d.

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or 10, d.=14 W. in 8 days. 1st answer.

W. in 1 d., or W. in 1 day;

... 10 I. in

.. 10 I. in

25
32 × 8
25

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W. 8 days;

or 10 d. 14 W. in 8 days.

2 W. in 8 days;

W. in 1 day;

2nd answer.

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..16 S. in 8 d. 14 W. in 8 days. 3rd answer.

=

315. EXERCISES.

1. If London exchanges with Russia, at £24 for 150 rubles; between Russia and Hamburgh, at 275 rubles for 600 marks; between Hamburgh and Venice, at 112 marks for 42 piastres; and between Venice and Paris, at 12 piastres for 65 francs. Find how many francs there are in £600.

2. Of four different kinds of cloth, 4 yards of the 1st kind are worth 7 yards of the 2nd kind, 10 yards of the 2nd are worth 8 yards of the 3rd, and 3 yards of the 3rd are worth 5 yards of the 4th. Required, how many yards of the 4th kind are worth 51 yards of the 1st.

3. If the course of exchange be £8 for 1124 florins Amsterdam, 56 florins for 64 marks of Hamburgh, and 19 schillings of Hamburgh for 2 rubles of Russia. Determine how many pounds sterling are equivalent to 4000 rubles, and also how many rubles to £4000.

4. The rates of exchange are at £1 for every 25.15 francs, at 30 francs for 16 marks of Hamburgh; also at £ for 1 thaler of Berlin, and at 51 thalers for 100 marks of Hamburgh; also at £7 for 100 marks. Which is the most advantageous way to exchange 1500 marks in pounds sterling, through Paris, Berlin, or direct from Hamburgh to London?

5. If 12 lbs. at London = 10 lbs. at Amsterdam.
100 lbs. at Amsterdam = 120 lbs. at Toulouse.

How

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49 lbs. at Antwerp-58 lbs. at Dantzick.

many lbs. at Dantzick=540 lbs. at London?

6. Admit that 48 Russians perform a piece of work in 17 days, and 27 Poles in 14 days=38 Russians in 13 days,

11 Prussians in 8 days=16 Austrians in 6 days,
9 Austrians in 15 days 10 Italians in 7 days,
18 Italians in 4 days= 17 Dutchmen in 5 days,
13 Dutchmen in 19 days 25 Russians in 9 days.

How long will 17 Poles, 19 Prussians, 7 Austrians, 26 Italians, and 11 Dutchmen be in doing the same work respectively? 7. The exchange between Amsterdam and Cadiz is at 90d. per 1 ducat. 1 florin is worth 40 pence. 1 ducat is worth 375 maravedis. 34 maravedis are worth 1 real. 8 reals are worth 1 piastre.

4 piastres are worth 1 pistole.

The exchange between England and Spain is 12 shillings for 1 pistole. It is required to find how much 1088 florins of Holland are worth in pounds sterling.

8. Convert 4.3 inches decimals into duodecimals, granting that 100 inches decimal=144 inches duodecimal.

9. How much will 26 square feet 2 square inches 8 square lines in decimals amount to in duodecimals, when 10000 square lines decimal=20736 square lines duodecimal ?

10. Reduce 6 cubic feet 520 cubic inches 50 cubic lines decimals into duodecimals, when 100 x 100 x 100 cubic lines decimal= 144 x 144 x 144 cubic lines duodecimal.

11. A merchant in England has to receive 1240 piastres from Venice, for which he can obtain directly 50d. per piastre; by the circular way, he remits first to Leghorn, at 48 piastres for 51 ducats; thence to Madrid, at 325 maravedis per ducat; thence to Oporto, at 626 rees per piastre of 272 maravedis; thence to Amsterdam, at 51 pence per crusado of 400 rees; thence to Paris, at 55 pence per 3 francs; and thence to London, at 30 pence per 3 francs. How much more profitable is the circular way than the direct, allowing commission at per cent.?

ALLIGATION.

316. It often happens that in business, goods of the same kind are mixed together, either to improve an inferior kind by mixing it with a superior one, or in order to sell goods of a superior quality, the price of which alone is too high for sale.

When several ingredients of different values are mixed together, a mixture of a certain rate is obtained. Thus, if 10 lbs.

of sugar at 4d. were mixed with 6 lbs. at 6d., and 8 lbs. at 6ąd., what is the price of the mixture per lb. ?

Here 10 lbs. at 4d. per lb. = 40d.

6 lbs. at 6d. per lb. =36d.

8 lbs. at 63d. per lb. = 54d. ..24 lbs. are worth 130d.

.. 1 lb. of mixture 30d., or 5d. nearly.

317. We may also have to determine the quantity of each of the ingredients which are to be mixed together.

Ex. How much wine, at 4s., 5s., and 7s. must be mixed together, so that the mixture may be worth 6s. ?

Since the 4s. wine is sold at 6s., the gain is 2s; the 5s. wine being sold at 6s., there is another gain of 1s.; but the 7s. being sold at 6s., the loss is 1s. Thus, by mixing 1 measure of each sort, the gain is 3s., and the loss ls.; therefore, the 7s. wine must be increased, so that the loss equal the gain; then taking 3 measures of the 7s. wine to 1 measure of each of the others, we have a composition in which there is neither gain nor loss. The proof is evident.

318. The inference from these considerations is, that when several ingredients are mixed together, we may have to find the rate of the mixture when the price and the quantity of each is given; and we may also have to find how much of each kind must be mixed, when the rates of each ingredient, and of the mixture are known. Questions treating of this subject belong to a section of arithmetic called Alligation.

319. Treated algebraically, alligation offers no difficulty whatever; but its principles are not so easily investigated by arithmetic at least, most arithmeticians have failed in their way of handling it, We shall present the solutions of various cases, hoping to render the subject intelligible to the learner.

Ex. 1. A mixture being made of 8 lbs. of tea, at 6s. 6d. per lb.; 10 lbs., at 4s. 6d. per lb.; and 12 lbs., at 5s. 8d. per lb. What is 1 lb. of it worth?

8 lbs. of tea, at 6s. 6d. per lb.=52s.

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