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 together; I demand the price of 34 cwt. of this mixture? Ans. $34 12cts. 5m. 7. A Goldsmith melted together 5 lb. of silver bullion, of 8 oz. fine, 10 lb. of 7 oz. fine, and 15 lb. of 6 oz. fine pray what is the quality, or fineness of this composition Ans. 6oz. 13pwt. 8gr. fine.

8. Suppose 5 lb. of gold of 22 carats fine, 2 lb. of 21 carats fine, and 1 lb. of alloy be melted together; what is the quality, or fineness of this mass ?

Ans. 19 carats fine.

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. 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, being reduced 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 ingredient, 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 one, or any number 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.

1. A merchant has spices, some at 9d. per lb. some at 1s. some at 2s. and some at 2s. 6d. per lb. how much of each sort must he mix, that he may sell the mixture at 1s, 8d. per pound?

2. A grocer would mix the following quantities of sugar; viz. 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 10cts. 2b. at 13cts. and 2lb. at 16 cts. per lb. 3. A grocer has two sorts of tea, viz. at 9s. and at 15s. per lb. how must he mix them so as to afford the composition for 12s. per lb.?

Ans. He must mix an equal quantity of each sort.

4. A goldsmith would mix gold of 17 carats fine, with some of 19, 21, and 24 carats fine, so that the compound may be 22 carats fine; what 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 5s. 7s. and 9s. per gallon, with water at 0 per gallon together, so that the mixture may be worth 6s. per gallon; how much of each sort must the mixture consist of?

Ans. 1 gal. of Rum at 5s. 1 do. at 7s. 6 do at 9s. and 3 gals. water. Or, 3 gals. rum at 5s. 6 do. at 7s. 1 do. at 9s. and 1 gal. water.

6. A grocer hath several sorts of sugar, viz. one sort at 12 cts. per lb. another at 11 cts. a third at 9 cts. and a fourth at 8 cts. per lb.; I demand how much of each sort must he mix together, that the whole quantity may be afforded at 10 cents per pound?

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 50 cts. per bushel, so that a bushel of the composition may be sold for 38 cents; what quantity of each must be taken.

*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 7s. 6d. per gallon, to reduce it to 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 oats 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 gallón, I mixed other rum at 68. 8d. per gallon, and some water; then I found it stood me in 6s. 4d. per gallon; I demand how much rum and how much water I took ?

Ans. 95 gals. rum at 6s. 8d. and 30 gals, water.

CASE III.

When the whole composition is limited to a given quantity.

RULE.

Place the difference between the mean rate, and the several prices alternately, as in CASE ; 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 1s. 3s. 63. 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 ?

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. 334 gals. water, and 663 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 loss; 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 oz. 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. 44 of 5 oz. fine, and 7 of 10 oz. fine.

5. Brandy at 3s. 6d. and 5s. 9d. per gallon, is to be mixed, so that a hhd. of 63 gallons may be sold for 12l. 128.; how many gallons must be taken of each ?

Ans. 14 gals. at 5s. 9d. and 49 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.

So {3, 4, 6, 8, &c. is an ascending arithmetical series: 8, 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 called the extremes.*

PROBLEM I.

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

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

By having any three of these parts given, 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.