3. Find, in like manner, the present worth of each year by itself, and the sum of all these will be the value of the annuity sought.

EXAMPLES.

1. What is the present worth of an annuity of 401. to continue 5 years, discounting at 5 per cent. per annum, compound interest?

[year. 1.05)40-00000(38.095= present worth for first

ratio =

2. What is the present worth of an annuity of 211. 10s. 94d. to continue 7 years, at 6 per cent. per annum, compound interest? Ans. 1201. 5s,

3. What is 701. per annum, to continue 59 years, worth in present money, at the rate of 5 per cent. per annum? Ans. 1321-30211.

To find the present worth of a Freehold Estate, or an Annuity to continue forever, at Compound Interest.

RULE.*

As the rate per cent. is to 100l. so is the yearly rent to the value required.

* The reason of this rule is obvious; for since a year's interest of the price, which is given for it, is the annuity, there can

EXAMPLES.

1. An estate brings in yearly 791. 4s. what would it sell for, allowing the purchaser 41 per cent. compound interest for his money?

neither more nor less be made of that price than of the annuity, whether it be employed at simple or compound interest.

The same thing may be shown thus: the present worth of an annuity to continue forever is +++, &c. ad infini

n

n

n

3 r

n

tum, as has been shown before; but the sum of this series, by

n

the rules of Geometrical Progression, is; therefore r—1 :

The following theorems show all the varieties of this rule, I. •=p. II. r—Txp=n. III. +1=r, or r-1.

n

r-l

n

n

The price of a freehold estate, or an annuity to continue forever, at simple interest, would be expressed by +

1

1 +r 1+2r

&c. ad infinitum; but the sum of this se

ries is infinite, or greater than any assignable number, which sufficiently shows the absurdity of using simple interest in these cases.

2. What is the price of a perpetual annuity of 401. discounting at 5 per cent. compound interest? Ans. 8001.

3. What is a freehold estate of 751. a year worth, allowing the buyer 6 per cent. compound interest for his money? Ans. 12501,

To find the present worth of an Annuity, or Freehold Estate, in Reversion, at Compound Interest.

RULE.*

1. Find the present worth of the annuity, as if it were to be entered on immediately.

2. Find the present worth of the last present worth, discounting for the time between the purchase and commencement of the annuity, and it will be the answer required.

EXAMPLES.

1. The reversion of a freehold estate of 791. 4s. per an num to commence 7 years hence, is to be sold: what is it worth in ready money, allowing the purchaser 41 per cent. for his money?

* This rule is sufficiently evident without a demonstration.

Those, who wish to be acquainted with the manner of computing the values of annuities on lives, may consult the writings of Mr. DEMOIVRE, Mr. SIMPSON, and Dr. PRICE, all of whom have handled this subject in a very skilful and masterly man

ner.

Dr. PRICE'S Treatise on Annuities and Revolutionary Payments is an excellent performance, and will be found a very valuable acquisition to those, whose inclinations lead them to studies of this nature.

and 1·045] = 1·360862) 1760·000(1293.297 = 12931. 5s. 114d. present worth of 17601. for 7 years, or the whole present worth required.

2. Which is most advantageous, a term of 15 years in an estate of 1001. per annum, or the reversion of such an estate forever, after the expiration of the said 15 years, computing at the rate of 5 per cent. per annum, compound interest? Ans. The first term of 15 years is better than the reversion forever afterward, by 751. 18s. 74d.

3. Suppose I would add 5 years to a running lease of 15 years to come, the improved rent being 1861. 7s. 6d. per annum; what ought I to lay down for this favour, discounting at 4 per cent. per annum, compound interest? Ans. 4601. 14s. 12d.

POSITION.

POSITION is a method of performing such questions, as cannot be resolved by the common direct rules, and is of two kinds, called single and double.

SINGLE POSITION.

Single Position teaches to resolve those questions, whose results are proportional to their suppositions.

RULE.*

1. Take any number and perform the same operations with it, as are described to be performed in the question.

2. Then say, as the result of the operation is to the position, so is the result in the question to the number required.

* Such questions properly belong to this rule, as require the multiplication or division of the number sought by any proposed number; of when it is to be increased or diminished by itself, or any parts of itself, a certain proposed number of times. For in this case the reason of the rule is obvious; it being then evident, that the results are proportional to the suppositions.

NOTE. 1

may be made a constant supposition in all ques

tions; and in most cases it is better than any other number.