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Take question 1st. for Example.

1. Multiply the tabular number in Table 4, corresponding to the rate and the time of continuance, into the annuity, and the product will be the present worth, to commence immediately.

2. Multiply this present worth by the tabular number in Table 2, corresponding to the rate and the time of reversion, and the product will be the present worth of the annuity in reversion.

In Table 4th we have 3.4651

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When the present worth of the reversion, rate and time are given, to find the annuity.

RULE 1. Multiply that power of the ratio signified by the time of reversion, by the present worth, and the product will be the amount of the present worth for the time before the annuity com

mences.

2. Multiply that power of the ratio signified by the time of continuance plus 1, by the last product.

3. Multiply that power of the ratio, signified by the time, by the aforesaid product, and this last product, divided by that power of the ratio denoted by the time minus unity, will give the annuity,

Or, Divide the continual product of the present worth, that power of the ratio denoted by the time of continuance, that power of it denoted by the time of reversion, and the ratio minus 1, by that power of the ratio denoted by the time of continuance minus 1, and the quotient will be the annuity.

EXAMPLES.

1. What annuity, to be entered upon 2 years hence, and then to continue 4 years, may be purchased for $185 035899, at 6 per ct.? First Method.

1.06×105=1·1236=2d power of the ratio.

Multiply by 185 036-present worth.

67416

33708

561800

.89888 11236

207-9064496 amount for the time of reversion.

Brought up.

207 9064496 amount for the time of reversion. =5th power of the ratio.

Multiply by 1.33822

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Divide by 106-1-26247) 15.7488750(60 the annuity required. Or, 185 036x1·1236=207-906

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2. The present worth of a lease of a house is £431 15s. 7d. 2-7819qrs. taken in reversion for 20 years; but not to commence till the end of 8 years, allowing £6 per cent, to the purchaser: What is the yearly rent?

Ans. £60.

PURCHASING ANNUITIES FOREVER, OR FREEHOLD ESTATES, AT COMPOUND INTEREST.

CASE I.

When the annuity, or yearly rent, and the rate are given, to find the present worth or price.

RULE.*

As the rate per cent. is to £100 or $100 so is the yearly rent, to the value required.

Or, Divide the yearly rent by the ratio less 1, and the quotient will be the value required.

EXAMPLES.

1. What is the worth of a freehold estate of £60 per annum allowing 67. per cent. to the purchaser ?

£ £ £

6: 100 :: 60
60

6)6000

£1000 Ans.

Or, 1·06—1=06) 60.00

1000

2. An estate brings in yearly $75: What will it sell for, allowing the purchaser 5 per cent. compound interest? Ans. $1500.

CASE II.

When the price, or present worth, and rate are given, to find the annuity, or yearly rent.

RULE.

As £100 or $100 is to the rate so is the present worth to its rent. Or, Multiply the present worth by the ratio less 1, and the product will be the yearly rent.

EXAMPLES.

1. If a freehold estate be bought for £1000 allowing £6 per cent. to the purchaser: What is the yearly rent?

£ £ £ 1006: 1000

6

100)6000(£ 60 Ans.

600

0

Or, 1000×06 £60.

2. If an estate be sold for $1500 and 5 per cent. allowed to the buyer; what is the yearly rent?

Ans. $75

*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 neither more nor less be made of that price, than of the annuity, whether it be employed at simple or compound interest It has also been proved under Case I. of the Present Worth of Annuities &c. at Compound Interest. Case II. and III, follow directly from the rule for Case I. and their rules are hence manifest.

CASE III.

When the present worth, or price, and yearly rent, are given, to find the rate.

RULE.

As the present worth is to the rent; so is £100 or $100 to the

rate.

Or, Divide the rent by the present worth; add 1 to the quotient, and the sum will be the ratio of the rate per cent.

Or, Divide the sum of the present worth and rent by the present worth, and the quotient will be the ratio.

EXAMPLES.

1. If an estate of £60 per annum be bought for £1000 what rate of interest was allowed the purchaser for his money?

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2. An estate of $75 per annum was purchased for $1500 what rate of interest had the buyer for his money? Ans. 5 per cent.

To find at how many years' purchase an estate may be bought.

CASE I.

When the rate of interest is given, to find the number of years.

RULE.

Divide £100 or $100 by the rate, and the quotient will be the years.

EXAMPLES.

1. How many years' purchase should a gentleman offer for the purchase of an estate, to have 6 per cent. for his money?

6)100

16.666+16 years.

2. How many years' purchase is an estate worth, allowing 5 per cent. to the purchaser ?

Ans. 20 years.

CASE II.

When the number of years purchase, at which an estate is bought, or sold, is given, to find the rate of interest.

RULE.

Divide £100 or $100 by the number of years, and the quotient will be the rate.

EXAMPLES.

1. A gentleman gives 16 years' purchase for a farm; what interest is he allowed? 16-16-666+)100 000(6 per cent. Ans. 2. A gentleman gives 20 years' purchase for an estate; what interest has he? Ans. 5 per cent.

PURCHASING FREEHOLD ESTATES IN REVERSION.

CASE I.

The rate and rent of a freehold estate being given, to find the present worth of reversion.

RULE.*

1. Find the present worth of the annuity or rent, (by Case 1, of purchasing Freehold Estates, page 326,) as though it were to be entered on immediately.

2. Divide the last present worth by that power of the ratio denoted by the time of reversion (by Discount by Compound Interest) and the quotient will be the answer required.

Or, 1. Having found the present value of the estate, supposing it to be immediate: Multiply the annuity, or rent, by the present worth of 17. or $1 corresponding with the time of reversion and rate in Table 4th, and the product will be the present worth of the annuity, or rent, for the time of reversion; or the value of the present possession.

2. Subtract the value of the possession from the value of the estate, and the remainder will be the value of reversion.

EXAMPLES.

1. Suppose a freehold estate of 60l. per annum to commence 2 years hence, be put up to sale; what is its value, allowing the purchaser 67. per cent.?

*

First Method.

1·06-1-06)60·00-rent per annum.

1000-present worth, if entered on immediately.

By the first step, the present worth is found for the present time; but as the estate is not to be entered on for a certain time, discount for that time must be allowed at Compound Interest. This is the second step, and the propriety of the rule is manifest. Case II. needs no illustration.

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