(5.) What annuity, or yearly rent, to be entered upon 3 years hence, and then to continue 5 years, may be bought for 1871. Os. 02.4176d. ready money, at 5 per

cent.?

(6.) The reversion of a lease, to be entered on 5 years hence, and thence to continue 20 years, was sold for 97641. 9s. 4088d., allowing the purchaser 5 per cent., what ought the yearly rent to be?

PURCHASING FREEHOLD ESTATES, OR PERPETUAL ANNUITIES TO BE ENTERED ON IMMEDIATELY.

Proposition 1. Given the annual rent of any perpetual annuity, or freehold estate, to find the value thereof, allowing the purchaser any assigned rate per cent. for his money.

Rule. Divide the rent by the ratio less 1, and the quotient will be the present worth of the estate.

n

Or, Theo. I. p, when n and r are given. If the rents are to be paid either yearly or quarterly, as is generally the case, then the ratio, or r, must represent the amount of £1 for that time, and the annuity, or n, must be divided by 2, 4, &c. to represent the , 4, &c. rent. Here we may observe, that though there be no such thing as a limited time considered in the purchase of perpetual annuities, yet a due regard ought to be had to the times the annuities or rents are paid; for, it is evident the less the intervals between the payments of the rents are, the purchase is more valuable, and vice versa.

Prop. 2. When any sum of money is proposed to be laid out in a perpetual annuity, or freehold estate, to find what annual rent that sum will purchase at any given rate per cent.

Rule. Multiply the proposed sum to be laid out by the ratio less 1, and the product will be the yearly rent. Theo. II. pxr-1—”, when p and r are given.

Prop. 3. The annual rent of any perpetual annuity, or freehold estate, and the sum paid down for it, being given, to find what rate of interest per cent. is paid to the purchaser.

Rule. Divide the annual rent by the sum that is paid for the purchase, the quotient, increased by an unit, will be the ratio, whence the rate per cent. may be found.

Theo. III.

n

-+1=r, when p and n are given.

*

Examples to Prop. 1.

(1.) Au estate brings in 257. yearly rent; required the present worth thereof, allowing the purchaser 4 per cent. compound interest for his money.

First, 104-104, the ratio less 1.

Then 2504 £625, the present worth required.

(2.) Suppose a freehold estate of 2507. yearly rent is to be sold; what is it worth, allowing the buyer 6 per cent. compound interest for his money ?

(3.) What is the present worth of a freehold estate of 250l. per annum, the rent payable half yearly*, allowing the purchaser 4 per cent. for his money?

(4.) What is the present worth of a perpetual annuity of 20001. payable quarterly, (viz. 500l. per quarter,) allowing the buyer 4 per cent. compound interest for his money?

Examples to Prop. 2.

(5.) I propose to lay out 6251. in the purchase of a perpetual annuity, and to make 4 per cent. compound interest for my money; what ought the annuity to be?

104-104, the ratio less 1.

Then, 04×625 £25, the annuity or annual rent required.

(6.) A freehold estate was bought for 41667. 13s. 4d.; what ought the yearly rent to be, allowing the buyer 6 per cent. compound interest for ready money?

(7.) A person is desirous of laying out 17607. in the purchase of a freehold estate, so as to get 4 per cent. compound interest for his money; what must be the annual income of such an estate?

It may not be improper to observe in this place, that, if the ratio be taken according to Table I. p. 242, it will make no difference whether the rents are payable yearly, half-yearly, or quarterly, but, if it be taken according to Table II. page 242, the difference, in this example, will be 61l. 17s. 14d.: this shews, that the second method or table is more accurate than the first; for it is certainly more advantageous to receive the rents half-yearly than yearly.

Examples to Prop. 3.

(8.) Suppose 6251. to be paid for a freehold estate which yields 25l. per annum, what rate of interest has the purchaser for his money?

625)25.00( 04

1.

1.04 the ratio; hence the rate per cent. is 41.

(9.) Suppose a freehold estate of 2501. per annum, costs 4166. 13s. 4d., what rate of interest per cent. is allowed to the purchaser?

(10.) A freehold estate of 60%. a year rent was sold for 12007., what was the rate per cent. (compound interest) allowed the purchaser for the ready money which he paid for the estate?'

THE BUYING AND SELLING FREEHOLD ETSATES TO BE ENTERED ON IMMEDIATELY, ACCORDING TO A NUMBER OF YEARS' RENT, OR INCOME, FOR THE PURCHASE-MONEY.

Proposition 1. The purchase-money, or present worth, of a freehold estate being given, to find at what rent it must be let to clear itself in a given time.

Rule. Divide the present worth by the proposed time, and the quotient will be the annual rent.

Prop. 2, Given the purchase, or present worth, of a freehold estate, and the annual rent it lets for, to find in what time it will clear itself, or bring in the purchasemoney.

Rule. Divide the present worth by the annual rent, and the quotient will be the time required.

Prop. 3. Given the annual rent of a freehold estate, and the time in which it will clear itself, to find the purchase, or present worth, of such an estate.

Rule. Multiply the rent by the time.

Prop. 4. Given the time in which a freehold estate brings in the purchase-money, or clears itself, to find what rate per cent. the purchaser has for his money.

Rule. Divide the time more I by the time, and the quotient will be the ratio, whence the rate may be found.

Examples to Proposition 1.

(1.) The reversion of a freehold estate of 5007. per annum, to commence 5 years hence, is to be sold; what is it worth in ready money, allowing the purchaser 4 per cent. for his money?

500÷04-125007. value of the estate, if entered on immediately. 1-04×1-04×1-04×1·04×104=1·2166529024, amount of 14. for

3 years.

1-2166529024 : 11.:: 12500 : 10274-088834 £10274 1-91-281, present worth of the reversion.

Or thus by Theorem I.

Here n=500, r=1·04, and T÷5.

1-04 × 1·04 x 104 × 1.04 × 1·04 =1•2166529024➡T, and 1.2166529024×04-048666116096—r—1 XrT.

Hence 500÷048€66116096=10274-088834=£10274 1 91-281, as before.

(2.) If a freehold estate of 607. 10s. per annum, to commence 10 years hence, is to be sold; what is it worth, allowing the purchaser 5 per cent. for present payment?

(3.) A freehold estate of 2907. per annum, to commence 4 years hence, is to be sold; what is it worth, allowing the purchaser 4 per cent.?

Examples to Prop. 2.

(4.) A freehold estate, to commence 5 years hence, is sold for 102747. Is. 94d-281, allowing the purchaser 4 per cent. for his money; what is the yearly rent?

First, £10274 1 91.281=10274-088834. 1.04 1.04 × 1·04 × 1·04 × 1·04=1·2166529024. .1.2166529024×10274 088834=12500 (nearly) the amount of the purchase-money to the time the reversion begins. Then, 12500 ×·04=£500, the yearly rent.

By Theo. II.

04 × 1.045 ×10274·088834—r—1xrTxp=04x

1 216652904×10274·088834=£500, (nearly,) the annuity required.

(5.) If a freehold estate, to commence 10 years hence, is sold for 7421. 16s. 84d-8, allowing the purchaser 5 per cent; what is the yearly rent?

(6.) If a freehold estate which commences 4 years hence, be sold for 61977. 6s. 5d., allowing the purchaser 4 per cent. for his money, what ought the yearly rent to be?

Note. The preceding rules and examples include all kinds of annuities which do not depend upon chance.

SIMPLE INTEREST BY DECIMALS.

Put p the principal, or sum put to interest.

r=the ratio, being the rate per cent. divided by 100.
t=the time, or years, the money is at interest.
i=the interest for the time t.

a the amount.

Then, at 24 per cent. r = 025 | At 4 per cent. r = '04

Hence the decimal parts of a year, for any number of days, weeks, months, &c. may be readily found.

Proposition 1. Given the principal, time, and rate per cent., to find the interest or the amount.

Rule. Multiply the principal, time, and ratio, together, the last product will be the interest; to which add the principal to find the amount.

Theorem, ptr=i, and ptr+p=a. When p, t, and r, are given.

Prop. 2. Given the amount, (or the interest,) time, and rate, to find the principal.

Rule. Multiply the time by the ratio, and add an unit to the product; by this sum divide the amount, and the quotient will be the principal.-Or, divide the interest by the product of the time and ratio, and the quotient will be the principal.

Theo.

a

fr

=p. When a, (or i,) t, and r, are given.

A a