to nothing, the extremes therefore are 16 and 0, and the ratio 4.

ANNUITIES.

An annuity is a sum payable periodically, for a certain length of time, or forever.

An annuity, in the proper sense of the word, is a sum paid annually, yet payments made at different periods, are called annuities. Pensions, rents, salaries, &c. belong to annuities.

When annuities are not paid at the time they become due, they are said to be in arrears.

The sum of all the annuities in arrears, with the interest on each for the time they have remained due, is called the

amount.

The present worth of an annuity, is the sum which should be paid for an annuity yet to come.

When an annuity is to continue forever, its present worth is a sum, whose yearly interest equals the annuity. Now as the principal, multiplied by the rate, will give the interest, the interest, divided by the rate, will give the principal.

Hence to find the present worth of an annuity, continuing forever.

Divide the annuity by the rate per cent.

1. What is the worth of $100 annuity, to continue for. ever, allowing to the purchaser 4 per cent.? per cent.? 8 per cent.? 10 per cent.? 20 per cent. ?

allowing 5 15 per cent. ? Ans. to last, $500.

2. What is an estate worth, which brings in $7,500 a year, allowing 6 per cent.?

A. $125,000.

ANNUITIES AT COMPOUND INTEREST.

It has been shown (page 215) that Compound Interest is that which arises from adding the interest to the principal

What is an annuity? When is an annuity in arrears? What is the amount? What is the present worth of an annuity? What is the rule to find the present worth?

at the close of each year, and making the amount a new principal. The amount of $1 for one year at 6 per cent. is $1.06, and it will be found, that if the principal be multiplied by this, the product will be the amount for 1 year, and this amount multiplied by 1.06, will be the amount for 2 years, and so on. Hence we see that any sum at compound interest, forms a geometrical series, of which the ratio is the amount of $1 at the given rate per

cent.

1. An annuity of $40 was left 5 years unpaid, what was then due upon it, allowing 5 per cent. compound interest? It is evident that for the fifth or last year, the annuity alone is due; for the fourth, the amount of the annuity for 1 year; for the third the amount of the annuity for 2 years, and so on; and the sum of these amounts will be the answer, or what is due in 5 years.

From this we find that the amount of an annuity in arrears, forms a geometrical progression, whose first term is the annuity, the ratio, the amount of $1 at the given rate, and the number of terms, the number of years.

The above example, then, may be resolved into the following question. What is the sum of a geometrical series whose first term is $40, the ratio 1.05, and the number of terms 5? First find the last term, by the first rule in Geometrical progression, and then the sum of the series by the second rule. The answer will be found to be $221.02. Hence, to find the amount of an annuity in arrears, at compound interest,

Find the sum of a Geometrical series, whose first term is the annuity, whose ratio, the amount of $1 at the given rate per cent., and whose number of terms is the number of years.

Note. A table, showing the amount of $1 at 5 and 6 per cent., compound interest, for any number of years not exceeding 24, will be found on page 217.

2. What is the amount of an annuity of $50, it being in arrears 20 years, allowing 5 per cent. compound interest? A. $1653,29. 3. If the annual rent of a house, which is $150, be in

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arrears 4 years, what is the amount, allowing 10 per cent. compound interest? A: $696,15. 4. To how much would a salary of $500 per annum amount in 14 years, the money being improved at 6 per cent., compound interest? in 10 years? in 20 years?" in 22 years? in 24 years?

Ans. to the last, $25,407,75. 5. Find the amount of an annuity of $150, for 3 years, at 6 per cent. A. $477,54.

A rule has been given, for finding the present worth of an annuity, to continue forever; but it is often necessary to find the present worth of an annuity, which is to continue for a limited number of years; thus,

6. What is the present worth of an annual pension of $100 to continue 4 years, allowing 6 per cent. compound interest?

The present worth is evidently a sum, which, at com. pound interest, would in 4 years produce an amount equal to the amount of the annuity, for the same time.

Now to find a given amount, at compound interest, we multiply a sum by the amount of $1 at the given rate per cent. as many times successively at there are years.

Hence, to find a sum, which will produce a given amount in a certain time, we must reverse this process and divide by the amount of $1 for the given time.

Applying this to the above example, we find by the preceding rule, that the amount is $437,46. Dividing this by the amount of $1 for 4 years, we find the present worth, 437,46 1,26247=$346,511, Ans. Hence to find the present worth of an annuity,

Find the amount in arrears for the whole time, and divide it by the amount of $1 at the given rate per cent., for the given number of years.

What is the rule for finding the amount of an annuity? What is the rule for finding the present worth of an annuity?

The operations under this rule, will be facilitated by the following

TABLE,

showing the present worth of $1, or £1 annuity, at 5 and 6 per cent. compound interest, for any number of years from 1 to 34.

It is evident, that the present worth of $2 annuity is 2 times as much as that of $1; the present worth of $3 will be 3 times as much, &c. Hence, to find the present worth of any annuity, at 5 or 6 per cent.,-Find, in this table, the present worth of $1 annuity, and multiply it by the given annuity, and the product will be the present worth.

7. Find the present worth of a $40 annuity, to continue 5 years, at 5 per cent. A. $173.173. 8. Find the present worth of $100 annuity, for 20 years, at 5 per cent. A. $1,246.22. 9. Find the present worth of an annuity of $21,54 for years at 6

7

per cent.

10. Find the present worth of an continue 12 years, at 6 per cent. 11. Find the present worth of an

20 years, at 5 per cent.

A. 120.244+ annuity of $100, to A. $838.384 annuity of $936, for A. $11,664.629

As the present worth of any annuity may be found, by multiplying the annuity by one of the numbers, in the

above table, it is plain that if any present worth be divided by the same number, it will give the annuity itself.

Hence to discover of what annuity any given sum is the present worth, we may use the above, as a table of divisors, instead of multipliers.

What annuity to continue 19 years, will $6,694.866 purchase, when money will bring 6 per cent. ? A. $600. An annuity is said to be in reversion, when it does not commence until some future time.

12. What is the present worth of $60 annuity, to be continued 6 years, but not to commence till 3 years hence, allowing 6 per cent. compound interest?

The present worth is evidently such a sum as would in 3 years, at six per cent., compound interest, produce an amount, equal to the present worth of the annuity, were it to commence immediately.

We must therefore first find the present worth of an annuity of $60 to commence immediately, according to the last rule. This we shall discover to be $295.039.

We now wish to obtain a sum, whose amount in 3 years will equal this present worth. This may be found by dividing the $295.039 by the amount of $1 for 3 years thus, $295.039 1,19101=247.72.

Ans. $247,72. Hence to find the present worth of any annuity taken in reversion, at compound interest,

Find the present worth to commence immediately, and this sum divided by the amount of $1 for the time in reversion, will give the answer.

13. What is the present worth of a lease of $100 to continue 20 years, but not to commence till the end of 4 years, allowing 5 per cent. ? what if it be 6 years in reversion? 8 years? 10 years? 14 years?

Ans. to last, $629,426. 14. What is the present worth of $100 annuity, to be continued 4 years, but not to commence till 2 years hence, allowing 6 per cent. compound interest? A. $308,393.

When is an annuity said to be in reversion? What is the rule for finding the present worth of an annuity taken in reversion?