Operation.

$12.041582-present worth of $1 for 22 yrs.

Explanation. The present worth of an annuity of $1 for 22 years must be equal to its present worth for 12 years, plus its present worth for the 10 succeeding years. Hence the present worth of an annuity of $1 for 10 years deferred 12 years must equal its present worth for 22 years, minus its present worth for 12 years. The present worth of $250 is evidently 250 times the present worth of $1.

RULE.-Find from the table the present value of an annuity of $1, commencing at once and continuing till the TERMINATION of the annuity, and also till the reversion COMMENCES. Multiply the differences of these present values by the given annuity.

Note. If the annuity is perpetual, the present worth of $1, commencing at once, is found according to the last article.

Ex. 2. What is the present value of a leasehold of $1800, deferred 10 years and to run 20 years, at 5 per cent, compound interest? Ans. $13771.2888.

Ex. 3. A lease, whose rental is $1000 a year, is left to two sons. The elder is to receive the rent for 9 years and the youngest for the 12 years succeeding. What is the present value of each son's interest, allowing 6 per cent, compound interest ? Ans. to last, $4962.385. Ex. 4. What is the present value of a perpetuity of $900, to commence in 30 years, allowing 4 per cent. compound interest ? Ans. $6937.17.

ART. 154. To find the annuity, the present or final value, time and rate being given.

Ex. 1. An annuity running 20 years, at 7 per cent. compound interest, is worth $15,000; what is the annuity? Ans. $1415.89.

Explanation.-Since $10.593997, at 7

per cent. compound interest for 20 years, yields an annuity of $1, $15000 will yield an annuity equal to $15000÷10.593997.

RULE.-Divide the present or final value of the given annuity by the present or final value of an annuity of $1, for the given rate and time.

Ex. 2. An annuity in arrears for 8 years, at 5 per cent. compound interest, amounts to $47745.545; what is the annuity? Ans. $500. Ex. 3. A yearly pension, unpaid for 12 years, at 6 per cent. compound interest, amounted to $1591.7127; what was the pension ? Ans. $100.

Ex. 4. The present value of a lease, running 25 years, at 6 per cent. compound interest, is $15340.037; what is the annual income? Ans. $1200.

CONTINGENT ANNUITIES.

ART. 155. When the annuity is to cease with the life of a certain person or persons, it becomes necessary to ascertain the probability of the person or persons, upon the continuance of whose life the annuity depends, surviving a given period. The measure of this probability is called Expectation of Life, and has already been noticed under Life Insurance.

In computing contingent annuities, the expectation of life of the person or persons named, as shown in Bills of Mortality, is taken as the time of the annuity. It can then be computed as an annuity certain.

to

A table, showing the present value of an annuity of $1, continue during the life of an individual, is called a Table of Life Annuities.

ART. 156. To find the present value of a life annuity.

Ex. 1. What is the present value of a life pension of $150, the age of the pensionary being 75 years; interest, 5 per cent.? Ans. $748.35.

RULE.-Multiply the present value of a life annuity of $1,

as shown in the table, by the given annuity.

Ex. 2. Suppose a person 60 years of age is to receive an annual salary of $600 during life. What is the present value of such income, at 6 per cent., compound interest?

Ans. $4982.40. Ex. 3. What must be paid for a life annuity of $560 a year, by a person aged 55 years, at 5 per cent., compound interest? Ans. $5794.32.

ART. 157. To find how large a life annuity can be bought for a given sum, by a person of a given age.

Ex. 1. How large a life annuity can be purchased for $2400, by a person aged $65 years, at 7 per cent. Ans. $350.51.

Operation.

2400.000 6.847 350.51.

RULE.-Divide the given sum by the present value of an annuity of $1 for the given age and rate.

Note. This is the reverse of the preceding article.

Ex. 2. How large a pension, at 6 per cent., compound interest, can be bought for $1600, the age of the pensionary being 50 years?

Ex. 3. How large an annuity can be bought for $3000, by a person aged 25; interest, 5 per cent. ?

Ans. 190.15.

ALLIGATION.

ART. 158. In various kinds of business, it is sometimes convenient or necessary to mix articles of different values or qualities, thus forming a compound whose value or quality differs from that of its ingredients. This process is called ALLIGATION (L. ad, to, and ligatus, bound); a name suggested by the method of solving some of its problems by joining or binding together the terms.

The various problems in Alligation may be divided into two classes, commonly called Alligation Medial and Alligation Alternate.

ALLIGATION MEDIAL.

ART. 169. Alligation Medial teaches the method of finding the average value or quality of a mixture, the value or quality, and also the quantity, of whose ingredients are known.

Ex. 1. A farmer mixed together 50 bushels of oats, at 40 cents per bushel; 30 bushels of barley, at 50 cents per bushel; and 25 bushels of corn, at 60 cents per bushel. What was a bushel of the mixture worth?

Cts.

OPERATION.

Cts.

40 x 50-2000 50 x 30 1500 60 x 25-1500

=

105 )5000

4711

Explanation.-Since the value of 50 bushels of oats, at 40 cents a bushel, is 2000 cents; of 30 bushels of barley, at 50 cents a bushel, 1500 cents; and 25 bushels of corn, at 60 cents a bushel, 1500 cents; the value of the mixture is 2000 cents +1500 cents +1500 cents= 5000 cents. But the mixture contains

50 bushels + 30 bushels +25 bushels=105 bushels. Hence, the value of 1 bushel of the mixture is of 5000 cents=4713 cents.

Ex. 2. A goldsmith melted together 12 oz. of gold, 20

carats fine; 6 oz., 18 carats fine; and 10 oz., 16 carats fine. What was the quality of the mixture? Ans. 18 carats. Explanation.-Since a carat of gold

OPERATION.

Carats.

Carats.

20 × 12 = 240
18 ×
6 = 108
16 × 10 160

28 )508
184

is the twenty-fourth part of the mass regarded as a unit (here an oz.), 12 oz. of gold, each oz. containing 20 carats of pure gold, contain 12 times 20 carats =240 carats; 6 oz. of gold, each containing 18 carats, contain 108 carats; 10 oz. of gold, each containing 16 carats, Hence, 12 oz. +6 oz. +10 oz. 28 oz. of mixture contain 240 carats + 108 carats + 160 carats =508 carats, and 1 oz. of the mixture must, therefore, contain of 508 carats 18 carats.

contain 160 carats.

Note. The regarding of a carat as a unit of measure of the pure gold in a given mass is not essential to the explanation of the above solution. For, suppose the comparative qualities of the above varieties of gold, represented respectively by the numbers 20, 18, and 16. Now, as these numbers represent the comparative qualities of the three varieties of gold, it is clear they must contain a common unit of quality. The number 20 denotes that the quality of the first variety contains this common union of quality 20 times; and, hence, 20 is the measure of its quality. But the effect of 12 oz. in determining the quality of a mixture is 12 times as great as the effect of 1 oz.; hence, 12×20 or 240, is the effective quality of 12 oz. of gold, if the quality of 1 oz. is 20.

RULE.

Multiply the value or quality of each article by the number of articles, and divide the sum of the products by the sum of the articles. The quotient will be the average value or quality of the mixture.

Ex. 3. A grocer mixed 15 lbs. of coffee, at 18 cents a pound; 35 lbs., at 16 cents a pound; and 40 lbs., at 14 cents a pound. What is a pound of the mixture worth? Ans. 15 cents.

Ex. 4. A grocer mixed 25 gallons of wine, at 90 cents a