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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 tine 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 muk iply the annuity by the given number of years, and add the product 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 wiki be the present worth.
ALLIGATION, Teaches how to compound or mix together several simples of different 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 Altērnaté.
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 tħe 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 compo sition 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 rate 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. 6s. R
gal. s. gal. S. gai.
gal. s. gal.
6 DEM.-It is evident, as the whole
19576 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 201b. 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 58. 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. 18 carats fine. 5. Of what fineness is that composition, which is made by mixing 4lb. of silver of 8oz. fine, with 21h. 4oz. of 9oz. fine, and 8oz. 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, il 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 be 20 carats fine. A pound of pure silver, losing nothing in trial, is said to be 12oz. fine; 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,
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, than 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 con560
3 at 60 cts. necting a rate which is less Mean 50
than the mean rate, with one rate,
8 at 50 cts. that is greater than the mean Ans.
12 at 70 cts. rate, and setting down the 65
2 at 65 cts. difference between them and Or,
the mean rate alternately,
or one after the other in cts. gal. gal
turn, that the quantities re160. 3+ 8=11
sulting are such, that there 8 8
is precisely as much gained 62 cts.
by one quantity as is lost by
the other, and consequently 65. 2 - 2
the gain and loss, upon the
whole, are equal; and the cts.
whole quantity, at the mean gal. $ cts.
rate, amounts to a sum equal 607 8X60=4,80 to the price of all the sim50
3X50=1,50 62 cts. Ans.
ples, at the rates given. No 2x70=1,40
matter whether the number 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
greater, there will be an 150
equal balance of loss and
gain between every two; and $15,50
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 adnit
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 or treble, or the hall, or the third, and so on, to any degree whatever. This demonstration perhaps will appear more plain after noticing the following example, where only I wo 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, 2810=20 16
it is evident, that the loss on 2x 6=12
the one is exactly balanced by 4 ......4)32(8s.
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.carats finė, 3 af 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 quantil ies required.
EXAMPLES. 1. A grocer wishes to mix teas at 11s., 9s., and 7s. per pound, with 20 pounds at 5s. per pound; how much of each sort must he take to make the composition worth 8s. për pound?
lb. 5s. 63 lb. at 9s.
6 lb. at 78
16. 11- 3 NOTE.—This Case may be proved by
9 1 the one preceding, 8s.
3 Stands against the given quantity 16. 1b.
DEM.--It is evident that the quan26. 18, 3: 20 tities obtained by linking the several
rates, together, would form a comAs 3 : 20 :: 1: Ans. 63 pound equal in value to the mean 1: 6} 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., and 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 48- 2 2
2: 4 at 4s. 60
10 : 20 at 5s.
20 : 40 at 5s. 6d. 72 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