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 63d., 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 6 d. per lb. =54d.

..24 lbs. are worth

130d.

.. 1 lb. of mixture= d., 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.

Ex. 2. How many carats fine in a mixture of 3 lbs. of gold bullion, 18 carats fine; and 5 lbs., 22 carats fine?

[Note. Pure gold contains 24 carats, but if a composition is said to be 20 carats fine, there is in it 4 parts of alloy and 20 parts of pure gold.]

In 3 lbs. there are 3 x 18, or 54 carats fine.

.. in 1 lb of mixture there is 164, or 20.5 carats fine.

Ex. 3. A merchant has wines at 12s., 15s., 18s., and 20s. per gallon, which he mixes to make a composition worth 16s. per gallon. How much of each sort must be taken?

The 12s. wine being sold at 16s., the gain is 4s. per gallon.

..by mixing one gallon of each sort there is 5s. gain, and 6s. loss but the gain must equal the loss, then we perceive that by mixing 2 gallons of the 15s. wine, we increase the gain by 1s. which was wanted. Therefore we obtain a proper composition by putting 1 gallon of the first, third, and fourth kind, and 2 gallons of the second.

Which is easily proved to be correct, for

1 gallon at 12s. = 12s.

2 gallons at 15s. =30s.

1 gallon at 18s.=18s.

1 gallon at 20s.=20s.

..5 gallons of the mixture=80s.

and..1 gallon of the mixture=0 or 16s.

The process is as follows:

16

(12+4x1 gallon, or 4 gain.

15+1×2 gallons, or 2 gain.

18-2×1 gallon, or 2 loss.
20-4 x 1 gallon, or 4 loss.

It need scarcely be mentioned that in questions of this kind the number of answers is unlimited.

Ex. 4, How much alloy must be added to gold, 10 oz. fine, to bring it to 7 oz. fine?

There are 10 oz. fine in 1lb. or 12 oz.

..there is 1 oz. fine in 10 x 12 oz,, or 120 oz.

and ..1633-12-423 oz. alloy to be added.

Ex. 5. How much gold of 21 and 23 carats fine, must be mixed with 30 oz. of 20 carats fine, to bring it to 22 carats fine? (21+1× 1oz.= 1 gain.

2223-1 x 61 oz.=61 loss.

(20+2×30 oz.=60 gain.

We observe that by mixing 1 oz. of each, the gain in fineness is 3, and the loss 1, but as 30 oz. of 20 carats fine are to be mixed, the gain is 61, therefore taking 61 oz. of the 23 carats fine, we get a composition where there is neither gain nor loss.

It will be noticed that the number of answers is unlimited.

Ex. 6. My labourers consist of men at 1s. 6d., and women at 1s. per day, and the amount of the whole wages is the same as if each of them received 1s. 4d.; the number of women is 20, find the number of men.

The loss of each man is 2d., and the gain of 20 women 80d., it is evident the number of men must be 40, because 40 × 2d. = 80d.; the men's loss the women's gain.

Ex. 7. What quantity of tea, worth 8s., 7s. 6d., and 6s. 6d, per lb., must be mixed together to form a parcel containing 60lbs. worth 7s. 4d. per lb.

8s. Od.- 8x1 = 8 loss.

7s. 4d. 7s. 6d.— 2 x1 = 2 loss.

6s. 6d. +10x1=10 gain.

When 1lb. of each kind is taken, we perceive that the gain= the loss, and as a parcel of 60lbs. is to be mixed, 0 or 20lbs. Express how many lbs. of each sort are required.

Ex. 8. A dealer in spirits has 200 gallons, worth 12s. 8d. per gallon, which he mixes with three other kinds, worth 12s. 4d., 15s. 6d., and 16s. 8d. per gallon, in order to sell the whole at 16s. 2d. How much of each must be taken?

The gain on the 200 gallons, is 8400d., and since 6d. is lost on 1 gallon of the last kind, 8400d. will be lost on 400 or 1400 gallons, also since 48d. are gained on 1 gallon of the second, and 4d. lost on 1 gallon of the third, therefore 48 or 12 gallons, will give a loss of 48d. Here then the loss = the gain as required. Questions of this kind admit of many answers.

320. EXERCISES.

1. A labourer performs 34.2 yards on Monday, 37.8 yards on Tuesday, 36.9 yards on Wednesday, 35.7 yards on Thursday, 36.6 yards on Friday, and 34.8 yards on Saturday. What is the average daily work?

2. Brass is made by casting 3 parts of zinc to 7 parts of copper. if 1 lb. of zinc cost 4d., and 1 lb. of copper cost ls. 3d. What is the price of 1 lb. of brass?

3. Bronze for cannons is obtained by melting 11 parts of tin, to 100 parts of copper, the value of tin is 11d. per lb., and of copper, 1s. 4d. per lb. The price of 1 lb. of bronze is required? 4. A bell is cast by melting together 220 lbs. of tin, 780 lbs. of copper, 10 lbs. of zinc, and 8 lbs. of lead. Tin is 1s. per lb.; copper, 1s. 2d. per lb.; zinc, 5d. per lb.; and lead, 2d. per lb. Find the value of the bell, and also of 1 lb. of this bronze.

5. Printing types are composed of 20 parts of antimony, 80 parts of lead, and 5 parts of copper. What is the value of

1 lb. of this composition, if antimony is 4s. 6d. per lb., lead 2d. per lb., and copper 1s. 3d. per lb.?

6. What quantity of wines at 24s., 21s., 18s. and 15s. per gallon, must be mixed so as to make a composition of 1000 gallons at 20s. per gallon?

7. How many gallons of water must be added to 20 gallons of wine at 18s. per gallon, so that the mixture be worth 15s. per gallon?

8. Some sea water contains 1 lb. of salt in 32 lbs of water. How much spring water must be added to it so that in 32 lbs. there may be only 2oz. of salt?

9. A dealer has spirits at 32 degrees which he desires to be reduced to 21 degrees, by mixing water with it. How must he do that?