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Div. by 4th pow. 1.26247696)60-000000000000(47.525619794231 Subtract the quotient=47.525619794281

Divide by 1.06-1-06)12.474380205719

Divide by 1.06x1.06-1.1236)207-9063367619(185-035899=1851. Os. 8d. the present worth of the annuity in reversion.

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60-47.5256

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What is the present worth of a reversion of a lease of $60 per anoum, to continue 20 years, but not to commence till the end of 8 years, allowing 6 per cent to the purchaser?

Ans. $431-782 (nearly.)

3. An annuity of $1 in reversion is to commence at the end of 20 years, and is to continue 15 years; what is its present worth at 4 per cent.? Ans. $5.0743 nearly.

4. An annuity of $1 in reversion is to commence after 5 years, and to continue forever; what is its present worth at 6 per cent. ? Ans. $12 45c. 43.

An annuity, several times in reversion, and rate being given, to find the several present values.,

Find the present value of £1 or $1 by Table 4, at the given rate, and for the several given times, which, being severally multiplied by the annuity, the products will be the several present values of that annuity, for the several times given; subtract the several present values, the one from the other, and the several remainders will answer the question.

B

5. A has a term of 6 years in an estate at 601. per annum. has a term of 14 years in the same estate, in reversion, after the 6 years are expired; and C has a further term of 16 years, after the expiration of 20 years. I demand the present values of the several terms at 6 per cent. ?

£ s. d.

Pres. value of £1 for 36y.=14·61722×60=877 0 73
Ditto of ditto for 20 years 11.46992×60=688 3 103
Ditto of ditto for 6 years 4.91732×60=295.0 91=A's term.
Therefore, 877 0 73-688 3 103 £188 16 9 C's term, and
688 3 103-295 0 91-£393 3 11=B's term.

6. For a lease of certain profits for 7 years, A offers to pay $300 gratuity, and $300 per annum, B offers $800 gratuity and $250 per annum, C bids $1300 gratuity and $200 per annum, and D bids $2500 for the whole purchase, without any yearly rent; which is the best offer, computing at 6 per cent. ?

By Table 4, the present worth of $300 per annum for 7 years, at 6 per cent is

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$

1674-714

To which add 300.

Value of A's offer 1974-714

Present worth of $250 per annum for 7 years 1395.595

To which add.

800.

Value of B's offer 2195.595

Present worth of $200 per annum for 7 years=1116.476

To which add

1300.

Value of C's offer 2416.476

D's offer 2500·

Hence it appears that D's offer is the best.

The above questions may be answered by the 4th and 2d Tables.

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

Multiply by 60=annuity.

207.9060

In Table 2d we have 889996

1247436

1871154

1871154

1871154

1663248

1663248

185-035508376=pres. worth of the reversion.

CASE II.

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 continuauce, 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.

1.06×1.06 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 1064-1=26247) 15-7488750(60 the annuity required. Or, 185-036x1·1236=207.906

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185.036×1.26247×1·1236×1·06-1
1.26247-1

60.

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?

S s

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 61. per cent. to the purchaser ?

£ £ £

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Or, 106-106)60.00

1000

£1000 Ans.

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?

£ £ £

100 6: 1000

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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 Compoond Interest. Case II. and III. follow directly from the rule for Case I. and their rules are hence manifest.

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