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when they depend on some contingency, as the life or death of a person ? A. The annuity is then said to be contingent. How do you distinguish annuities, when they do not commence till some future period ? A. They are then said to be in reversion. When is an annuity said to be in arrears ? A. When the debtor keeps it beyond the time of payment. What is the present worth of an annuity ? A. It is such a sum as being now put out at interest, would exactly pay the annuity as it becomes due. What is the amount of an annuity? A. The sum of the annuities for the time, with the interest due on each. How do you find the amount of an annuity at simple interest ? A. First find the interest of the given annuity for one year, then for two years, three years, and so on, up to the given number of years, less one; then mul uply the annuity by the given number of years, and add the produet to the whole interest, and the sum will be the amount of the annuity. How do you find the present worth of an annuity at simple interest ? A. First find the present worth of each yearly payment by itself, discounting from the time it becomes due; then the sum of all these wild be the present worth.

ALLIGATION, Teaches how to compound or mix together several simples of difforent qualities or prices, so that the composition may be of some intermediate quality or price. It is commonly distinguished into two kinds, Alligation Medial, and Alligation Altērnātė.

ALLIGATION MEDIAL, Teaches to find the price or quality of the composition, from having the quantities and prices or qualities of the several simples given.

Case I. To find the mean price or quality of any part of the composition, when the several quantities and their prices or qualities are given.

RULE. First, multiply the quantity of each ingredient by its price or quality; then add all the products together, and add also all the quantities together, into another sum; then say—as the whole composition is to the sum of the products, so is any part of the composition to its mean value or quality.

Note.--Alligation means to mingle, tie, or mix together, two or more simples; and Medial means the middle or mean rare between the extremes.

EXAMPLES. 1. A merchant mixes 20 gallons of brandy at 10 shillings per gallon, with 36 gallons of rum at 6 shillings per gallon, and 40 gallons of gin at 4 shillings per gallon; what is a gallon of the mixture worth?

Ans. 6$. R

S.

gal. s.

gal. S.

gal. 20x10=200 as 96 : 576 .. 1 36X 6=216

1 40x 4=160

96)576(6s. Ans.
96 gal. 576s. 576

gal. s. gal.
0
PROOF as 1:6:: 96

6 DEM.--It is evident, as the whole

1)576 quantity of liquor is to the money

576s. which it cost, so one gallon is to the cost of one gallon.

Note. The student will now perceive, that Alligation is nothing more than the Rule of Three applied to mixing different ingredients.

2. A grocer mixes 60lb. of sugar at 8d. per pound with 2016. worth 12d. per pound; what is the value of 1 pound of the mixture ?

Ans. 9d. 3. A farmer mixes ten bushels of wheat at 5s. a bushel, with 18 bushels of rye at 3 shillings a bushel, and 20 bushels of barley at 2 shillings per bushel; how much is a bushel of the mixture worth?

Ans. 3s. 4. A refiner melted together 8 ounces of gold, of 22 carats fine, 10 ounces of 20 carats fine, 12 ounces of 16 carats fine, 8 ounces of 18 carats fine; will you make out the fineness of the composition ?

Ans. 1814 carats fine. 5. Of what.fineness is that composition, which is made by mixing 4lb. of silver of 8oz. fine, with 21b. 4oz. of 9oz. fine, and 802. of alloy?

Ans. 7oz. fine. | NOTE.-An ounce of pure gold being reduced into 24 equal parts, these parts are called carats; but when gold is mixed with baser metal, the mixture is said to be so many carats fine; thus; if 22 carats of pure gold be mixed with 2 of alloy, it is said to be 22 carats fine; and if 20 carats of pure gold be mixed with 4 of alloy, it is said to bé 20 carats fine. A pound of pure silver, losing nothing in trial, is said to be 12oz.

but if it lose loz. by the fire, or be mixed with loz. of alloy, it is said to be lloz. fine, &c.

ALLIGATION ALTERNATE, Teaches to find what quantity of any number of simples, 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. This is called Alligation Alternate, because the same question frequently admits of different answers.

CASE I.-- When the prices of the several simples are given,

fine;

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gal.

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to find how much of each, at their respective rates, must be taken, to make a compound or mixture, at any proposed price

RULE.-1. Place the rates of the simples in a column under each other, with the mean price at the left hand. 2. Connect each rate which is less than the mean rate, with one or more that is greater. 3. Take the difference between each rate and the mean price, and place it directly opposite to that rate with which it is connected.' 4. If only one difference stand against either rate, it will be the quantity required at that rate; but if there be two or more, their sum will be the quantity.

NOTE.-When all the given prices are greater, or less, thạn the mean rate, they must be linked to a cipher.

EXAMPLES. 1. A grocer has several sorts of wine, some at 60 cents per gallon, some at 50, some at 70, and some at 65; how much of each sort must he mix, that he may sell the mixture at 62 cents per gallon? cts.

DEM.-- It is plain, by con60

3 at 60 cts. necting a rate which is less Mean

than the mean rate, with one 50rate,

Ans.

8 at 50 cts. that is greater than the mean 7062cts.

12 at 70 cts. rate, and setting down the 65

2 at 65 cts. difference between them and

the mean rate alternately,

or one after the other in cts. gal. gal.

turn, that the quantities re60 3+ 8=117 sulting are such, that there 50 8

8

is precisely as much gained 62 cts.

Ans. 2+12=14

by one quantity as is lost by

the other, and consequently 65. 2 = 2

the gain and loss, upon the Or,

whole, are equal; and the cis.

whole quantity, at the mean gal. $ cts.

rate, amounts to a sum equal 8X60=4,80 to the price of all the sim50

3X50=1,50 ples, at the rates given. No 432 cts.

Ans.
2x70=1,40

matter whether the number
65-
12X65=7,80

of simples be great or small,

or with how many a simple 25 25)15,50(62 is yoked, since one, that is 62 150

less than the mean rate, is al

ways linked with one that is 50

50 greater, there will be an 150

equal balance of loss and

gain between every two; and $15,50

0

consequently an equal' baA sum equal to the amount of all the lance on the whole. It is simples at the given nrices.

also plain that it will admit

Or,

701_

607

70J

Proof.

50

8s.

a

of different answers; because having one answer, we nay find as many more as we please, by multiplying or dividing each of the quantities found, by 2, 3, 4, &c. The reason of which is plain; for if two quantities of two simples make a balance of loss or gain with respect to the mean price, so must double of treble, or the hall, or the third, and so on, to any degree whatever. This demonstration perhaps will appear inore plain after noticing the following example, where only two simple quantities are given :

2. A grocer ha's brandy worth 10s. per gallon, and rum worth 6s.; how much of each must he mix, that he may sell the mixture at 8s. per gallon?

Ans. 2 gallons of each, or iqt. Ipt. &c. of each. S.; gal. Proof ĐẾM.-From this example 10, 2X10=20

it is evident, that the loss on 2X 6=12

the one is exactly balanced by 4.

4)3278s. the gain on the other; for on 32

the sale of the 2gal. of brandy

there is a loss of 4s., but it is 0

balanced by the gain on the rum, which is 4s.; and it is also plain, that 1 gallon of each, or 2 quarts of each, would bear the same proportion.

3. A goldsmith has gold of 20 carats fine, of 18, of 22, and of 16; how much must he take of each to make the mixture 19 carats fine ? Ans. Boz. or lb. of 20 caräts finė, 3 of 18 carats fine, 1 of 22 carats fine, and 1 of 16 carats fine; admitting of other answers, as may be seen by referring to example 1, on page 195.

Case II.- When one of the ingredients is limited to a certain quantity, to find the several quantities of the rest, in proportion to the given quantity.

RULE.--Take the difference between each price and the mean rate, and set them down 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 grocer wishes to mix teas at 11s., 9s., and 7s. per pound, with 20 pounds at 58. per pound; how much of each sort must he take to make the composition worth 8s. per pound ?

lb.

63 lb. at 9s. 63 lb. at 7€

Ans. 2016. at 11s.

3

1:

$.

15. 11. 3 NOTE.—This Case may be proved by 9

1 the one preceding. 8s. 7

1

3 Stands against the given quantity 1b. 16.

DEM.--It is evident that the quanlb. , 3 :

20

tities obtained by linking the several

rates together, would form a comAs 3: 20:: 1: Ans. 6 pound equal in value to the mean

63 rate; but since one of the quantities

is given, the others must be increased or diminished in proportion. Then it is plain, that the quantity which stands against the rate of the given quantity, is to the given quantity as the quantity standing against any other rate, is to the quantity of that rate.

2. How much wine at 8s., 5s., arrel at 4s. per gallon must be mixed with 3 gallons at 7s. per gallon, so that the mixture may be worth 6 shillings per gallon?

Ans. 3gal. at 7s., 6 at 8s., 3 at 5s., and 6 at 4s. CASE III.— When the whole composition is limited to a certain quantity.

RULE.-First find an answer as before, by linking; then say, as the sum of the quantities, or differences, thus found, is to the given quantity, so is each ingredient found by linking, to the required quantity of each.

Note.—The student will discover, by noticing the different cases in this rule, that they are only applications of the Rule of Three.

EXAMPLES 1. A grocer has wine at 4s., at 5s., at 5s, 6d., and at 6s. à gallon, and he wishes to make a mixture of 72gal., so that it i nay be afforded at 5s. 4d. per gallon; how much of each sort must he take ? d. gal. gal.

gal. gal. Answer 2 2

2: 4 at 4s. 60.

gal. gal.

8+2=10 64d.

10 : 20 at 5s.
as 36:72 ::
16+4=20

20 : 40 at 5s. 6d. 72 4 = 4

4: 8 at 6s. 36

72 Proof Dem. It is plain, that the sum of the differences or quantities found by linking, bears the same proportion to the given quantity as the differences, severally, bear to a fourth propor

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667

per gallon.

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