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ALLIGATION ALTERNATIVE Is when the price of several things are given, to find such quantities of them to make a mixture, that may bear a price. propounded.
In ordering the Rates and the given Price, observe, 1. Place them one under the other 18 and the propunded price «f mean
22 rate at the left hand of them, thus
28 2. Link the several rates together, by 2, and 2, always observing to join a greater and a less than the mean..
3. Against each extreme place the difference of the mean and its yoke-fellow.
When the prices of the several simples and the mean rate are griera without any quantity, to find how much of each simple is required to compose the mixture.
RULE. Take the difference between each price and the mean rate, and set them alternately, they will be the answer required.
Proof. By alligation medial.
1. A vintner would mix four sorts of wine together, of 18d. 200. 24d. and 28d. per quart, what quantity of cach must he have to sell the mixture at 22d. per quart ? Answer. Proof
or thus, Proof. 18. 2 of 18:1. = 36d. 18- 6 of 18d. = 1080. 20- 6 of 20d.
20- 12 of 2ud. 40 22 2414 of 24.11. = 96
24 2 of 24d. = 48 282 of 28d. = 56
28. - 4 of 28d. = 112
22d. NOTE. Questions in this Rulé admit of a great Variety of Anncers, according to the manner of linking them.
2. A grocer would mix sugar at, 41. 6d. and 10d. per lb. so as to sell the compound for 8d. per ib. What quantity of each must he take?
Ans. 216. at 4d. 210. at 6d. and 6lb. at 10d. 3. I desire to know how much tea, at 169. 148. 9s. and 8s. per lb. will compose a mixture worth 10s. per Ib ?
Ans. lib, at 16s. 216, at 14s. 616, at Is. and '4lb. at 8s. 4. A farmer would mix as much barley at 3s. 6d. per bushel, rye at 4s. per bushel, and oats at 2.1. per bushel, as to make a mixture worth 28, 6d. per bushel. How much is that of each sort?
Ans. 6 of barly, 6 of rye, and 30 of oats. 5. A grocer would mix raisins of the sun at 7d. per lö. with
Malagas at 6d. and Smyrnas at 4d. per ty. I d’sire to kino what quantiły of each sort he must take to sell them at 5d. per D?
Ans. llb. of raisins of the gin, ilb. of Malayai,
and 316. of Smyrilns. 6. A tobacconist would mix tobacco at 2x. To. Od. and 1s. 3d per lb. ou as the compound may bear a price of 1s. 8d. per 16. What.quantity of each sort must he take?
Ans. 71b, at 2s. 41b at 1s. 6d, and 416. at Is. 3d
ALTERNATION PARTIAL 15
of thein, and the mean rate, are given, to find the several quantities of the rest in proportion to that given.
Rule. Take the difference between each price, and the mean rate, as before. Then,
As the difference of that simple, whose quantity is given, is to the rest of the differences severally: so is the quantity given to the several quantities required. :
1. A tobacconist being determined to mix 2016. of tobacco at 151. per ib. with others at 16d. per 16. 184. per lb. and 22d. per 1b. how many pounds of each sort must he take to make one pouud of tkat mixture worth 17d. Answer.
As 5:2 :: 20:8 22
8lb. at 22d.—176d.
As 5 : 1
As 3616. : 612d. :: llb. 17d. 2. A farmer would mix 20 bushels of wheat at 60d. per bushel, with rye at 30d. barley at 24d. and oats at 18d. per bushel. How much must he take of each sort, to make the composition worth 32d. per bushel ?
Ans, 20 bushels of wheat, 35 bushels of rye, 70 bushels
of barley, and 10 bushels of oats. 3. A distiller would mix 40 gallons of French brandy, at 19s. per gallon, with English at 78. and spirits at 48. per gallon. Whüt quantity of each sort must he take to afford it for 88. per gallon ?
Ans. 10 gallons French, 32 English, and 32 Spirits. 4. A grocer would mix teas of 12.. 10s. and Cis. with 2017, at 4.5. per lb. How much of each sort must he take to make the composition worth es. per i?
Ans. 2016, at 45. 10lb. at 63, 1016. at T0s. 2017, at 121.
$. A wine merchant is désirous of mixing 18 gallons of Ca. pary at 68. 9d. per gallon, with Malaga at 78. 6d. per gallon ; Sherry at 5s. per gallon; and white wine at 45. 3d. per gailon. How much of each sort must he take that the mixture may be sold for 6s. per gallon ?
Ans. 18 gallons of Canary, 314 of Malaga, 13 of Sherry, and 37 of white wine.
ALTERNATION TOTAL Is when the price of each simple, the quantities to be compounded, and the mean rate art given, to find how much of each sort will make that quantity.
Rule. Take the difference between each price, and the mean rate as before. Then,
As the sum of the differences : is to each particular difference :: so is the quantity given : to the quantity required.
1. A grocer has four sorts of sugar, viz. 121. 10d. 6d. and sd. per lb. and would make a composition of 144 lb. worth 8d. per ld. I desire to know what quantity of each he must take ? Answer.
4 :: 48 at 12d. 576 As 12 :: 144 : 48 10
2 :: 21 at 10d. 240 As 12; 2: 144 : 95
2:: 24 at 6d. 147 4
:: 48 at 4d. 192
12 144 911529d. 2. A grocer having four sorts of tea, of 55. 6s. 8s. and 9s. per The would have a composition of 87 lb. worth 78. per lb. what quantity must there be of each ?
Ans. 14 lb. of 5s. 29 lb. of 6s. 29 16. of 8s. and 1 lb, 9.. 3. A vintner had 4 sorts of wine, viz. white wine at 4.. per gallon ; Flemish at 68. per gallon; Malaga at 8s. per gallon; and Canary at 168. per gallon, would make a mixture of 60 gallons, to be worth 5s. per gallor.. What quantity of each must he take ?
Ans. 45 gallons of white wine, 5 gallons of Flemish,
5 gallons of Malaga, and 5 gallons of Canary. 4. A silversmith hath four sorts of gold, viz. of 24 carets fine, of 22, 20, and 15 carets fine, would mix as much of each sort together, so as to have 42 oz. of 17 carets fine. How much must he take of each ?
Ans. 4 of 24, 4 of 22, 4 of 20, and 30 of 15 carets fine. 5. A druggist having some drugs of 85. 58. and 46. per 16. made them into 2 parcels; one of 98 lb. at 6s, per lb. the other of 42 lb. at 70. per 16. How much of each sort did he take for each parcel?
Ano, 12lb. of 8s.
8lb. of 5s,
Ans. 30 of 8s.
6 of 5s.
28 lb. at 6s. per lb.
42 lb. at 7s. per lb.
POSITION, OR THE RULE OF FALSE, IS
a rule that by false or supposed numbers, taken at pleasure, discovers the true one required. It is divided into two parts; SINGLE and DOUBLE.
SINGLE POSITION Is, by using one supposed number, and working with it as the true one, you find the real number required, by the following
Rule. As the total of the errors : is to the true total : : 80 is the supposed number : to the true one required.
PROOF. Add the several parts of the sum together, and if it agrees with the sum, it is right.
EXAMPLES. 1. A schoolmaster being asked how many scholars he had, said, if I had as many, half as inany, and one quarter as many more,
I should have 88. How many had he? Suppose he had 40
As 110 : 88: : 40 32 40
22 2. A person having about him a certain number of Portugal pieces, said, if the third, fourth, and sixth of them were added together they would make 54. I desire to know how many he had ?
Ans. 72. 3. A gentleman bought a chaise, horse, and harness for £60. the horse came to twice the price of the harness, and the chaise to twice the price of the horse and hartiess. What did he give for each ? Ans. Horse £13.. 6...8. Harness £6...13...4. Chaise £40.
4. A, B, and C, being determined to buy a quantity of yoods, which would cost them £120. agreed among themselves that B should have a third part more than A, and C a fourth part more than B. I desire to know what each man must pay ?
Ans. A £20. B £40, and C £50.
5. A person delivered to another a sum of money unknown, to receive interest for the same, at 6, per cent. per annum, sim ple interest, and at the end of ten years received for principal and interest £300. What was the sum lent?
Ans. £187...10...0. DOUBLE POSITION Is, by making use of two supposed numbers, and if both prove
false (as it generally happens) they are, with their errore, to be thus ordered:
Rule 1. Place each error against its respective position 2. Multiply them cross ways.
3. If the errors are alike, i. e. both greater or both less than the given number, take their difference for a divisor, and the difference of their product for a dividend. But if unlike, take their sum for a divisor, and the sum of their product for a divja dend, the quotient will be the answer.
1.: A, B, and C, would divide £200 between them, so that B may have £6 more than A, and C £8 more than B, how much must each have ? Suppose A had 40 Then suppose A had 50
then B had 46 then B must have 56
170 too little by 30h
140 too little by 60
00 Au for A. 2. A man had 2 silver cups of unequal weight, having one cover to both, of 50%. now it the cover is put on the lesser cup, it will double the weight of the greater cup; and set on the greater cup, it will be thrice as heavy as the lesser cup. What is the weight of each cup ? Ans. 3 ounces lesser, 4 greater.
3. A gentleman bought a house with a garden, and a horse in the stable, for £500. now he paid 4 times the price of the horne for the garden, and 5 times the price of the garden for the Jouse. What was the value of the house, garden, and horse, separately?
Ans. Horsé £20. Garden £80. House £400. 4. Ihree persons discoursing concerning their ages ; eays H,