3. A wine merchant mixes 12 gallons .of wine, at 45. rod. per gallon, with 24 gallons, at 55. 6d, and 10 gal. lons, at 6s. 3.d.-What is a gallon of this composition worth? An. 55.7d. 4. A goldsmith melced together 8 oz. of gold of 22 carats fine, il 8 oc. of 21 carats fine, and 10 oz. of 18 carats fine: Pray what is the quality, or fineness, of the compofition? 8X22 +20 X 21 7-10 X 18 8+20+10 =201, carats fire, Anf. 5. A refiner melts 5 15 of gold of 20 carats fine, with 8th of 18 carats fine : How much alloy must he put to it, to make it 22 carats fine ? 22-5 X 20+8X18:5+8=31 Ans. It is not fine enough by 37's carats, so that no alloy muf be added, but more gold. ALLIGATION ALTERNATE* Is the method of finding what quantity of each of the ingredients, whose rates are given, will compose a mixture of a given rate : So that it is the reverse of Alligation Medial, and may be proved by it. CASE By connecting the less rate with the greater, and placing the difference between them and the mean rate alternately, or one after the other in turn, the quantities resulting are such that there is precisely as much gained by one quantity as is lost by the other ; and therefore the gain and loss upon the whole, are equal, and are exıctly the proposed rate. In like manner, let the number of fimples be what it may, and with how many soever each one is linked, since it is always a less with a greztir than the mean price, there will be an equal balance of loss and gain between every two, and consequently an cqual bal. ance on the whole. It is obvious from the rule, that questions of this sort admit of a great varicty of answers ; for having found 1 answer, we may find as many more as we please by only multiplying or dividing each of the quantities found, by 2, 3, 4, &c. If any one of the simples be of little or no value with repeat to che reft, its rate is supposed to be nothing, as water mixed with wins, and alloy with gold and lilyer. EE CA SE I. The whole work of this case consists in linking the extremes truly together, and taking the differences between them and the mean price, which differences are the quanrities sought. Rule 1.--Place the several prices of the simples, being reduced to one denomination, in a column under each other, the least uppermost, and so gradually downward, as they increase ; with a line of connection at the left hand, and the mean price at the left hand of all. 2. Connect, with a continued line, the price of each fimple, or ingredient, which is less than that of the com. pound, with one or any number of thofe, which are greater than the compound, and each greater rate or price with one, or any number, of the less. 3. Place the difference between the mean price (or mixture rate) and that of each of the fimples, opposite to the rates with which they are connected. 4. Then, if only one difference ftand' against any rate, it will be the quantity belonging to that rate ; but if there be feveral, their sum will be the quantity: Ex A M P L E s. 1. A merchant has spices, some at is. 6d per. It, some at 2s. fome at 45. and some at 5s. per ft.-How much of each fort muít he mix, that he may fell the mixture at 35. 4d. per ib: d. d. . s. 18 Mean 24 rite 40d. 148 20 at 1 d. 8 at I 20 - 2 22 - 4 16 per t. 24 48 160 per It 160 4 22 5 IÓ - 5 40d. 24 48 22 s, d. 24 4od. 48 -20 + 8 28 400. 48 d. th: s. do per ti 5 ib. 18 20 20 at 16 8+20 28 2 per tb. 16 16 4 60. 22 +16 38 5 d. 15. si á 18 28 at I 24 8 +20 2 16+22 38 per fb: 4 5 Note. These five answers arise from as many various ways of linking the rates of the ingredients together. 2. * A merchant has Canary wine, at 35. per gallon ; Sherry at 2s. id. and Claret at is. 5d. per gallon : Hov much of each sort. muit he take to sell it at 25. 4i.. per gallon? 36 3+11 14 at 30 Mean rate 28d. 25 8 8 15 3. A goldsmith would mix gold of 19 carats fine, with some of 16, 18, 23 and 24 carats fine, so that the compound may be 21 carats fiue : What quantity of each muit he take? 16 { per gall. 18 * Note. The 2d question admits but one way of linlsing, and so but of one answer; yet all numbers in the same proportion between themselves, as the numbers which compose the answer, will likewise fatisfy the condition of the question. 4. It is required to mix several sorts of wine at 35. 55. and 75. per gallon, with water, that the mixture may be worth 4s. per gallon : How much of each fort must the mixture confist of ? § Anfw. 3 gal. water, I gal. wine, at 35. I 2 ditto, at 55. and 4 ditto, at 75. CAS E II. IVhen the rules of all the ingredients, the quantity of but one of them, and the mean rate of the whole mixtures are give in, ia find the several quantities of the rest, in proportion to the quantity given. Rule.--Take the differences between cach price, and the mean rate, and place them alternately; as in Cafe 1. Then, as the difference tanding againt that fimple, whose quantity is given, is to that quantity ; so is cach of the otlar differences, severally, to the several quantities re. quired. E x A M P LE S. 1. A merchant has 40lb of tea, at 6s. per It, which he will mix with some at 5s. 3d. fome at 5s, 2.l. and some at-45. 6.--How much of each fort must he take, to mix with the 40it, that he may fell the mixture at 55. 5d. per id ? 10 (tity per ib: 14:40 As 14 : 40 :: 55. 2 d. - 55. 8d. 2. How much gold, of 16, 20 and 24 carats fine, and how much alloy, must be mixed with 10 oz. of 18 carats fine, that the composition may be 22 carats fine ? Anf. 10 oz. of 16 carats fine, 10 of 20, 170 of 24, and 10 of alloy. ALTERNATION TOT AL. CA SE III. When the rales of the several ingredients, the quantity to be compounderl, and the mean rate of the whole mixture are given, to find how much of each fort will make up the quantity. RULE.- Place the differences between the mean rate, and the several prices alternately, as in Cafe ift. Then, As the sum of the quantities, or differences thus deterinined, is to the given quantity or whole compofition; fo is the difference of each rate, to the required quantity of each rate. Ex A M P L E s. 1. Suppose I have. 4. forts of currants, at 8d. 12d. 18d. and 22d. per it--the worst would not sell, and the best. were too dear ; 1 therefore concluded to mix 120lbs and. so much of each fort, az to sell them at 16d. per itHow much of each fort must I take ? d. tHi: 6:36 at 16d. : 12d. 1b. ib. 8:48 22d. 8d.) 2 2:12 2. A goldsmith has several forts of gold, viz. of 15; 17, 20, and 22 carats fine, and would melt together, of all these forts, so much as may make a mals E E 2 af |