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to the given rate per cent. for quarterly or half yearly payments, will be the true amount.
Rate per cent.
The construction of this table is from an Algebraic Theorem given by the learned Mons. De Moivre, in his Treatise
of Annuities on Lives, 3 1,007445 1,011181
which in words is thus: 3 1,008675 1,013031 For half yearly payments 4 1,009902 1,014877 take a unit from the ra4} 1,011126 1,016720
tio, and from the square 5 1,012348 1,018559 54 1,013567 1,020395
root of the ratio, half the 6 1,014781 1,022257 quotient of the first re7 1,017204 1,025880 mainder, divided by the
latter will be the tabular number; for quarterly payments, use the fourth toot, as above, and take one-fourth of the quotient.
1. What will an annuity of $200 amount to in 5 years, to be paid in half yearly payments, at 6 per cent. per annum?
Ans. $1144 8c. 2m. to Agreeably to table first, the tabular number is 5.63709 x 200 = $1127 41c. 8m. x 1.014781, (tabular number answering to 6 per cent. in table second, for half yearly payments) $1144.08.2m. + answer.
2. What will an annuity of $500 amount to in 5 years, at 6 per cent.?
Ans. $2818.54.6 +. 3. What will an annuity of $1000, payable yearly, amount to in 10 years?
Ans. $13180.79.4 +. 4. What will annuity of $30, payable yearly, amount to in 3 years?
Ans. $95.50.8 +.
Table Third, shewing the present worth of an Annuity of $1, from One year to Thirty-seven.
per cent. 6
per cent. | Yrs. 5 per cent. | 6
per cent. Yrs, 5
per cent. 6
14 9,898641 9,294984
EXPLANATION OF THE TABLE. What is the present worth of $1, to continue for 4 years at 6 per cent. per annum? Ans. 3.465106, agreeing with the tabular number opposite to 4 years at 6 per cent. per annum.
First, find the present worth of $1, by discount for 1 year at 6 per ct. per annum, which is $0.943396 2d year the present worth is
$3.465105 Tabular number for 4 years at 6 per. cent. as in table 3d.
1. What is the present worth of $50 per annum for 6 years at 6 per cent. per annum? Ans. $245.86c. 6m.
Annuities in Reversion. The annuity, time, and rate given, to find the present worth as in case 2. Multiply the number, under the rate and opposite th etime in table 3d, by the annuity, the product will be the present worth for yearly payments. If the payments are to be made half-yearly, or quarterly, the present worth so found for yearly payments, must be multiplied by the proper number in table 2d.
Q. What is meant by annuities in reversion?
A. Sums of money, which are paid yearly for a limited period, but which do not commence till after the expiration of a given period, are called annuities in reversion. Given the time of reversion, time of continuance and rate
per cent. to find the present worth of the reversion. RULE.—Take two numbers under the given rate in table 3, corresponding to the different periods of time, viz; time of reversion and time of continuance, and take the difference between the tabular numbers, answering to the times as above mentioned, and multiply that difference by the
annuity, for the present worth annually, if the payments be half yearly or quarterly, we must use table 2 as above stated.
1. The reversion of a freehold is $60 per annum for 4 years, to commence 2 years hence, what is the present worth, allowing 6 per cent. for prompt payment.
Illustration.-Time of continuance 6 years. Tabular number for 6 years at 6 per cent. found in table 3, is
4.917324 Time of reversion; 2 years tabular No. 1.833393
3.083931 3.083931 x 60 $185.03c.5m. x Answer.
2. What is the present worth of a reversion of a lease of $100 per annum, to continue 10 years; but is not to commence till the end of 2
Ans. $655.041. 3. What is the present worth of a reversion of a lease for $120 per annum to continue 9 years, but not to commence till the end of 4 years at 5 per cent to the purchaser?
Ans. $701.71c. 4m.
PERPETUITIES. Annuities which continue for ever, are called perpetuities.
Given the Annuity and rate per cent. to find the present
worth, RULE.—Divide the annuity by the ratio less 1, for the present worth.
Note.--Table 2d must be resorted to, as in temporary annuities, when the payments are half-yearly or quarterly.
1. What is the present worth of an annuity of $150 to continue for ever, allowing 5 per cent. to the purchaser? Operation.-1.05–1 = .05)150.00
2. What is an estate of $260 per annum, to continne for
ever, worth in present money, allowing 6 per cent. to the purchaser?
Ans. $4333.33.3 +. 3. A property in fee simple rents for $120 per annum, what is the present worth, allowing 5 per cent. to the purchaser?
DISCOUNT BY COMPOUND INTEREST. The ratio in compound interest is the amount of $1 for one year, which is found thus: as 100 : 106 : :1 = $1.06 is the amount of $1 for one year.
Example 1.—What is the present worth of 600 for 3 years; hence at 6 per cent. compound interest 1.063 = 1.191016)600( = $503.77 + Ans.
2. What is the amount of $503.77, in 3 years at 6 per cent.?
Ans. $600. 3. What is the present worth of $520, due 5 years hence, at 6 per cent. compound interest. Here 520 1,3312256)520(=390.62 answer.
ALLIGATION, From the Latin (ad. to, and ligo to bind,) it being necessary in sundry cases to link or bind the quantities.
We shall not omit the rule of Alligation, the object of which is to find the value of several things of the same kind of different values. The following examples will sufficiently demonstrate it.
When the quantities and rates of the simples are given to find the rate of a mixture compounded of these simples.
Rule.--Find the value of each quantity, according to their respective costs, then divide the sum of the pro