In this question, the sum of the repetends is 2851303, which divided by 999999, gives 2 to carry to the next column 5,3,0, &c. and the remainder is 851305.

2. Let 3275-319+36-45+123-19+5-3173+112·3513+11·131+125

+29.100.53 be added together..

Ans. 3593-00042.

SUBTRACTION OF CIRCULATING DECIMALS:

RULE.

Make the repetends similar and conterminous, and subtract as usual, observing, that if the repetend of the number to be subtracted be greater than the repetend of the number it is to be taken from, then the right hand of the remainder must be less by unity than it would be if the expressions were finite.

MULTIPLICATION OF CIRCULATING DECIMALS.

RULE.

1. Turn both the terms into their equivalent vulgar fractions, and find the product of those fractions as usual

2. Turn the vulgar fraction, expressing the product, into an equiv. alent decimal one, and it will be the product required.

EXAMPLES.

1. Multiply 54 by 15. 54, and 15 = ¦ ¦ =

1. Change both the divisor and dividend into their equivalent vulgar fractions, and find their quotient as usual.

2 Turn the vulgar fraction, expressing the quotient, into its equiv. alent decimal, and it will be the quotient required.

IS the method of mixing two or more simples of different qualities, so that composition may be of a mean or middle quality; It consists of two kinds, viz. Alligation Medial, and Alligation Al

ternate.

ALLIGATION MEDIAL

Is, when the quantities and prices of several things are given, to find the mean price of the mixture compounded of those things.

RULE.

As the sum of the quantities, or the whole composition, is to their total value; so is any part of the composition to its mean price or value.

EXAMPLES.

EXAMPLES.

1. A Tobacconist would mix 60lb. of tobacco, at 6d. per lb. with 50lb. at 1s. 40lb. at Is. 6d. and 30lb. at 2s. per lb.: What is 1lb. of S this mixture worth?

2. A farmer would mix 20 bushels of wheat at D.1 per bushel, 16 bushels of rye at 75c. per bushel, 12 bushels of barley at 50c. per bushel, and 8 bushels of oats at 40c. per bushel: What is the value of one bushel of this mixture? Ans. 73c. 5m.

3. A wine merchant mixes 12 gallons of wine, at 75c. per gallon, with 24 gallons, at 90c. and 16 gallons at D.1 10c.: What is a gallon of this composition worth?

Ans. 92c. 6m.

4. A goldsmith melted together 8oz. of gold of 22 carats fine, 1lb. 8oz. of 21 carats fine, and 10oz. of 18 carats fine: Pray what is the quality, or fineness of the composition?

5. A refiner melts 5lb. of goid of 20 carats fine with 8lb. of 18 carats fine: How much alloy must be put to it, to make it 22 carats fine?

22-5x20+8x18÷5+8=3
3133

3

Answer. It is not fine enough by 3 carats, so that no alloy must 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

Demon. By connecting the lefs rate with the greater, and placing the difference between them and the mean rate alternately, or one after the other in turn, the quantities refulting are fuch, that there is precifely as much gained by one quantity as is loft by the other, and therefore the gain and lofs, upon the whole, are equal, and are exactly the proposed rate.

In

CASE 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 quantities 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 simple, or ingredient, which is less than that of the compound, with one or any number of those, 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 simples, opposite to the rates with which they are connected.

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

EXAMPLES.

some

1. A merchant has spices, some at Is. 6d. per lb. some at 2s. at 4s. and some at 5s. per lb.: How much of each sort must he mix that he may sell the mixture at 3s. 4d. per lb. ?

In like manner, let the number of fimples be what it may, and with how many foever, each one is linked, fince it is always a lefs with a greater than the mean price, there will be an equal balance of lofs and gain between every two, and confequently an equal balance on the whole.

It is obvious from the rule, that questions of this fort admit of a great variety of anfwers; for having found one anfwer, we may find as many more as we please, by only multiplying or dividing each of the quantities, found by 2, 3, 4, &c. the reafon of which is evident; for if two quantities of two fimples make a balance of lofs and gain with refpect to the mean price, fo must also the double or triple, the half or third part, or any other ratio of these quantities, and fo on ad infinitum. If any one of the fimples be of little or no value with refpect to the reft, its rate is fuppofed to be nothing, as water mixed with wine, and alloy with gold and

filver.

2. *A merchant has Canary wine, at 3s. per gallon, Sherry, at 2s. Id. and Claret at Is. 5d. per gallon: How much of each sort must he take, to sell it at 2s. 4d. per gallon?

Mean rate 28d.

{

25.

25

17

3. How much barley at 40c. rye at 60c. and wheat at 80c. per bushel, must be mixed together, that the compound may be worth 62 c. per bushel ?

Ans. 17 bushels of barley, 17 of rye, and 25 of wheat. 4. A goldsmith would mix gold of 19 carats fine, with some of 16, 13, 23 and 24 carats fine, so that the compound may be 21 carats fine: What quantity of each must he take?

Ans. 5oz. of 16 carats fine, 5oz. of 18, 5oz. of 19, 10oz. of 23, and 10oz. of 24 carats fine.

5. It is required to mix several sorts of wine, at 60c. 90c. and D.1 15c. per gallon, with water, that the mixture may be worth 75c. per gallon Of how much of each sort must the composition consist? Ans. 40 galls. of water, 15 galls. of wine, at 60c. 15 galls. do. at 90c. and 75 galls. do. at D.1 15c.

CASE II.

When the rates of all the ingredients, the quantity of but one of them, and the mean rate of the whole mixture are given, to find the several quantities of the rest, in proportion to the quantity given.

RULE.

Take the differences between each price, and the mean rate, and place them alternately, as in Case 1. 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 merchant has 40lb. of tea, at 6s. per lb. which he would mix with some at 5s. 8d. some at 5s. 2d. and some at 4s. 6d. : How much of each sort must he take, to mix with the 40lb. that he may sell the mixture at 5s. 5d. per lb. ?

lb.

*Note, the 2d. and 3d. queftions admit but of one way of linking, and fo but of one anfwer; yet all numbers in the fame proportion between themfelves, as the numbers, which compofe the answer, will likewife fatisfy the condition of thequestion.