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6. A grocer hath several sorts of sugar, viz. one sort at 8 dols. per cwt. another sort at 9 dols. per cwt. a third sort at 10 dols. per cwt. and a fourth sort at 12 dols. per cwt..and he would mix an equal quantity of each togeth. er; I demand the price of 3 cwt. of this wixture?

ns. $34 12cts. 5m. 7. A Goldsmith melted together 5 lb. of silver bullion, of 8 oz. fine, 10 lb. of 7 02.. tine, and 15 lb. of 6 0% fine; pray what is the quality, or fiueness of this composition

Ans. 6uz. 15pwt. 8gr. fine. 8. Suppose 5 lb. of gold of 22 carats tine, 2 db. of 21 carats tine, and 1 lb. of alloy be melted together; what is the quality, or fineness of this mass ?

Ans. 19 carats fine.

ALLIGATJON ALTERNATE, IS the method of finding what quantity of each of the ingredients, whose rates are given, will compose a mix. ture of a given rate; so that it is the reverse of alligation medial, and may be proved by it.

CASE. I. When the mean rate of the whole mixture, and the rates of all the ingredients are given without any limited quantity.

RULE. 1. Place the several rates, or prices of the simples, be. ing reiluced to one denomination, in a column under each other, and the mean price in the like name, at the left hand.

2. Connect, or link, the price of each simple or ingre. dient, which is less than that of the mean rate, with one or any number of those', which are greater than the mean rate, and each greater rate, or price with oue, or any nuinber of the less.

3. Place the difference, between the mean price (or mixture rate) and that of each of the simples, opposite to the rates with which they are connected.

4. Then, if only one difference stands against any rate, it will be the quantity belonging to that rate, but if there be several, their sum will be the quantity.

EXAMPLES. 1. A merchant has spices, some at 9d. per Ib. some at 18. sume at 2s. and some at 25. 6d. per lb. how much of each sort inust he mix, that he may sell the mixture at Is. 8d. per pound ? 1. d. d. 26

do lb. r 10 at 97

97 47 | 12 14 12 Gives the .d. 12to 24 18 24 Answer. 01 20 24 | 11 (30iL 30

(502 85 %. Agrocer would mix the following quantities of sucar; vil.. at 10 cents, 13 cents, and 16 cts. per lb. ; what quantity of each sort must be taken to make a mixture worth 12 cents per pound : Ans. 5lb. at lucts. 2lb. at 13cts, and 2lb. at 16 cts. per lb.

3. A grocer has two sorts of tea, viz. at 9s. and at 158. per lb. how must lie naix them so as to afford tiie cumposition for 12g. per Ib. ? . Ans. He must mix an equal quantity of each sort.

4. A goldsinith would mix gold of 17 carats fine, with some of 19, 21, and 24 carats fine, so that the compourid Hay be 22 carats fine; wliat quantity of each must he take.

Ans. 2 of each of the first three sorts, and 9 of the last.

5. It is required to mix several sorts of rum, viz. at Is. 78. and 9s. per gallon, with water at. O per gallon together, so that the mixture may be worth 6s. per gallon ; how much of each sort must the mixture consist of ? Ans. I gal.of Rum at 5s. I do. at 7s. 6 du at 9s. and 3 . gals. water. Or, 3 gals. rum at 5s. 6 do. at 7s. 1

du, at 9s. and I gal. water. ' 6. A grucer hath several sorts of sugar, viz. one sort at 12 ctx. per Ib. another at 11 cts. a third at 9 cts. and a fwrth at 8 cts. per Ib. ; I demnand how much of each sort must he mix together, that the whole quantity may be alorded at 10 conts per pound ?

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1b. cts.

lb. cts. . lb. cts. 2 at 12 r1 at 12

rs at 12 J1 at 11 1st. Ans.

2 at 9 (2 at 8 (1 at 8

3 at 8 4th Ans. Slb. of each sort.*

CASE II. ALTERNATION PARTIAL. Or, when one of the ingredients is limited to a certain quantity, thence to find the several quantities of the rest, in proportion to the quantity given.

RULE. Take the difference between each price, and the mean rate, and place them alternately as in Case I. Then, as the difference standing against that simple whose quantity is given, is to that quantity : so is each of the other differences, severally, to the several quantities required.

EXAMPLES. 1. A farmer would mix 10 bushels of wheat, at 70 cts. per bushel, with rye at 48 cts. corn at 36 cts. and barley at 30 cts. per bushel, so that a bushel of the composition may be sold for 38 cents; what quantity of each must be taken.

70% 8 stands against the given quan. Mean rate, 38 36

[tity. 10 (30 32

2 : 2bushels of rye. As 8:10 : : { 10 : 124 bushels of corn.

( 32 : 40 bushels of barley. * These four answers arise from as many various ways of linking the rates of the ingredients together.

Questions in this rule admit of an infinite variety of answers : for after the quantities are found from different methods of linking; any other numbers in the same proportion between themselves, as the numbers which compose the answer, will likewise satisfy the conditions of the question

2. How much water must be mixed with 100 gallons of rum, worth 78. 6d. per gallon, to reduce it te 6s. 3d. per gallon ?

Ans. 20 gallons. 3. A farmer would mix 20 bushels of rye, at 65 cents per bushel, with barley at 51 cts. and oats at 30 cts. per bushel ; how much barley and vats must be mixed with the 20 bushels of rye, that the provender may be worth 41 cents per bushel ?

Ans. 20 bushels of barley, and 61, bushels of oats.

4. With 95 gallons of rum at 8s. per gallon, I mixed other run at 68. 8d. per gallon, and some water; then I found it stood me in 6s. 4d. per galion; I demand how much rum and how much water I took ?

Ans. 95 gals. rum at 6s. 8d. and so gals. water.


When the whole composition is limited to a given quantity.


Place the difference between the mean rate, and the several prices alternately, as in Case I.; then, As the sum of the quantities, or difference thus determined, is to the given quantity, or whole composition : so is the difference of each rate, to the required quantity of each rate.

EXAMPLES. 1. A grocer had four sorts of tea, at 19. 3s. 6s. and 10s. per lb. the worst would not sell, and the best were too dear; he therefore mixed 120 lb, and so much of each sort, as to sell it at 4s. per lb. ; how much of each sort did he take ? s. 16.

- 1b. 1- 6

r6: 60 at 12 2 lb. lb. J2 : 20 1 As 12 : 120 :: 11 : 10 - 6 >per te.

15 : 50 - 10

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2. How much water at 0 per gallon, must be mixed with wine at 90 cents per gallon, so as to fill a vessel of 100 gallons, which may be afforded at 60 cents per gallon ?

Ans. 35} gals. water, and 664 gals. wine. 3. A grocer having sugars at 8 cts. 16 cts. and 24 cts. per pound, would make a composition of 240 lb. worth. 20 cts. per lb. without gain or less ; what quantity of each must be taken ?

Ans. 40 lb. at 8 cts. 40 at 16 cts. and 160 at 24 cts. 4. A goldsmith had two sorts of silver bullion, one of 10 oz. and the other of 5 07.. fine, and has a mind to mix a pound of it so that it shall be 8 oz fine ; how much of each sort must he take ?

Ans. 4 of 5 oz. fine, and 7 of 10 oz. fine. 5. Brandy at 35. 6d. and 5s. 9d. per gallon, is to be mixed, so that a hhd. of 63 gallons may be sold for 121. 12s.; how many gallons must be taken of each ?

Ans. 14 gals. at 5s. 9d. and 19 gals. at 3s. 6d.

: ARITHMETICAL PROGRESSION. ANY rank of numbers more than two, increasing by common excess, or decreasing by common difference, is said to be in Arithmetical Progression. S S 2, 4, 6, 8, &c. is an ascending arithmetical series • WW28, 6, 4, 2, &c. is a descending arithmetical series :

The numbers which form the series, are called the terms of the progression; the first and last terms of which are caller the extremes.*

· PROBLEM I. The first term, the last term, and the number of terms being given, to find the sum of all the terins.

*A series in progression includes five parts, viz. the first torm, last term, number of terms, common difference, and sum of the series.

By having any three of these parts givwn, the other two may be found, which admits of a variety of Problems ; but most of them are best understood by an algebraic process and are here omitted.

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