3. Multiply the sum thus found by the given annuity, and the product will be the amount sought.

RULE II.

Or, multiply the amount of £1 or $1 for 1 year into itself so many times as there are years less by 1; then multiply this product by the annuity; and subtract the annuity therefrom. Lastly, divide the remainder by the ratio less 1, and the quotient will be the amount.

EXAMPLES.

1. What will an annuity of 60l. per annum, payable yearly, amount to in 4 years, at 61. per cent. ?

3

Or, 1+1.06+1.06)2+1·0613×60=£262 9s. 61d.

II. The second rule is derived from the expression, 1-064 X60-1x60

also,

.06

the above amount, and is the rule.

Because the amounts of annuities, at the same rate and for the same time, are as the annuities, if the amount be divided by the amount of £1 or $1 for the same time and rate, the quotient will be the annuity. This is the 2d Rule under Case II. And the 2d Rule of Case III. is readily inferred from the same principle.

Brought up.

Divide by 1.06-106)15-7482(262.47=2621. 9s. 42d. Ans.

Multiply the tabular number under the rate, and opposite to the time, by the annuity, and the product will be the amount.

2. What will an annuity of 601 per annum amount to in 20 years, allowing 61. per cent. compound interest?

Under 61. per cent. and opposite 20, in table 3d, you will find, Tabular number=36.78559

Multiply by 60=annuity.

2207.13540=22071. 2s. 84d. Ans. 3. What will a pension of $75 per annum, payable yearly, amount to in 9 years at 5 per cent. compound interest?

Ans. $826 99 2,3m. 4. If a salary of 1001. per annum, to be paid yearly, be forborne 5 years, at 61. per cent. What is the amount? Ans. 5631. 14s. 2d. 5. What will wages of $25 per month, amount to in a year, at per cent. per month? Ans. $308 38c. 9m.

CASE II.

When the amount, rate per cent. and time are given, to find the annuity,

pension, &c.
RULE I.

Multiply the whole amount by the amount of 11. or $1 for a year, from which subtract the whole amount, divide the remainder by that power of the amount of 11. or $1 for a year, signified by the number of years, made less by unity, and the quotient will be the

answer.

Table 3 is calculated thus: Take the first year's amount, which is 11. multiply it by 1.06+1-206-second year's amount, which also multiply by 1.06-+1=3.1836 third year's amount, &c. and in this manner proceed in calculating tables at any other rates.

RULE II.

Or, find the amount of an annuity of 11. or $1 for the given time and rate (by Case 1;) divide the given sum by this amount; and the quotient will be the annuity required.

1.

EXAMPLES.

What annuity, being forborne 4 years, will amount to £ 262-47696, at 61. per cent. compound interest?

262.47696=amount.

Multiply by 1.06 amount of 11. for 1 year.

Amount of an annuity of 11. for 4 years at 6 per cent. per annum

4.374616 (by Case 1;) and

262.47696

Or, by Table III. the amount of 11. is found to be 4:374616; the answer is found, as before.

and

2. What annuity, being forborne 20 years, will amount to $2207-1354, at 6 per cent. compound interest?

Amount of an annuity of $1 for 20 years at 6 per cent. per annum=36.78559. And,

36-78559)2207-13540($60, Ans.

2207.1354

0

CASE III.

When the annuity, amount and ratio are given, to find the time.

RULE I.

Multiply the amount by the ratio, to this product add the annuity, and from the sum subtract the amount; this remainder being

divided by the annuity, the quotient will be that power of the ratio signified by the time, which being divided by the amount of 11. for 1 year, and this quotient by the same, till nothing remain, the number of those divisions will be equal to the time. Or, look for this number under the given rate in table 1, and in a line with it, you will see the time. Or,

RULE II.

Divide the amount by the annuity; from the quotient subtract 1; from the remainder subtract the ratio; from successive remainders subtract the square, cube, &c. of the ratio, till nothing remain; and the whole number of the subtractions will be the answer. Or, find the quotient in Table III. under the rate, and in a line with it stands the answer.

EXAMPLES.

1. In what time will 601. per annum, payable yearly, amount to £262.47696, allowing 61. per cent. compound interest, for the forbearance of payment?

262-47696 amount.

der the rate, 6, the quotient Or, looking into Table III. un4-374616, stands against 4 years, Ans. as before.

Or, in Table I. under the given rate, you will find 1.262476, and in a line under years, you will find 4.

2. In what time will an annuity of $60 payable yearly, amount to $2207-1354, allowing 6 per cent. for the forbearance of payAns. 20 years.

ment?

PRESENT WORTH OF ANNUITIES, &c. AT COMPOUND INTEREST.

CASE I.

When the annuity, &c. rate and time are given to find the present

worth. RULE 1.*

1. Divide the annuity by the amount of $1 or £1 for 1 year, and the quotient will be the present worth of 1 year's annuity.

* This rule depends on the rule for finding the present worth in Discount at Compound Interest. For each year the present worth is to be found by that rule. Then, the sum of the present worth for the several years, must evidently be the present worth of the whole, and is the rule.

Or, suppose the annuity to be 11. or $1 at 6 per cent. then

1 1.06 for three years;

is the present

1

1

1

1

Then the sum, or +

will be

1.06 1.062

four years, and so on. + 1.063 1.064 the whole present worth. Let any annuity be substituted for the numerator of these several fractions, and you have the rule in the text.

By Note 2, Prob. I. of Geometrical Progression, the sum of the series,

1

1

is 1

1:06

Now if the

1 1 1 1 1 + + + 1.062 1.063 1.064 1.064 ·06' .06 .06 1-064 annuity were to continue forever, or the number of years were infinite, then the index of the denominator of the last expression would be infinite, and the value

1 of the fraction would be infinitely diminished or become nothing, and Would .06

be the present worth of an annuity of 11. or $1 to continue forever at 6 per cent. Hence, if an annuity is a perpetuity, or is to continue forever, its present worth is found by dividing the annuity by the ratio, or the interest of 11. or $1 for a year at the given rate.

·05

The present worth of an annuity of $1 to continue forever at 5 per cent. is $20, and an annuity of $100 at 5 per cent. to continue forever, would

100
5

now be worth $2000, and at 7 per cent. $14284.

1

1

X when the annuity is 1.064 06'

Rule II. is derived from the expression 1— $1 or 11. and the rate 6 per cent. That is when the annuity is $1 or 11. divide the annuity by that power of the ratio indicated by the number of years, and subtract the quotient from the annuity; the remainder divided by the ratio of the series less 1, will be the present worth. But the present worth of aanuities varies as the annuity. Hence the rule is manifest.

Note.

Another rule for obtaining the present worth may be derived from the

preceding. Thus, the sum of the series,

rears at 6 per cent. That is, divide the difference between unity and that power