5. What is due on a pension of $1000, which has been forborne 27 years, at 3 per cent., compound interest?

From the preceding table, we find the amount of an annuity of $1 for 27 years, at 3 per cent., to be $40 709634, which, multiplied by 1000, gives $40709‍634 for the amount due.

6. What is the amount of an annuity of $50, which has been forborne 30 years, at 6 per cent., compound interest? Ans. $3952-909. 7. What is the amount of a pension of $300, which has been forborne 19 years, at 5 per cent., compound interest? Ans. $9161·701. 8. What is the amount of a pension of $900, which has been forborne 17 years, at 4 per cent., compound interest? Ans. $21327.761. 9. What is the amount of an annuity of $75, which has been forborne 13 years, at 5 per cent., compound interest? Ans. $1328.474.

Case II.

To find the present worth of an annuity which is to terminate in a given number of years.

The present worth of an annuity is obviously such a sum of money as will, at compound interest, produce an amount equal to the amount of the annuity. Therefore, if we find the amount of the annuity by Case I., we may consider it as the amount of a certain principal, which principal is the same as the present worth. We have already been taught how to find the present worth, by rule under Compound Discount. Hence, we have this

RULE.

First find the amount of the annuity, as if it were in arrears for the whole time, by the aid of the table under Case I. of ANNUITIES.

Then find the present worth of this amount for the given time and rate per cent., by the use of the table under COMPOUND DISCOUNT.

EXAMPLES.

1. What is the present worth of an annuity of $500, to continue 10 years, interest being 6 per cent.?

By the table under Case I., of Annuities, we find the amount of an annuity of $1 for 10 years, at 6 per cent., to be $13 180795; this, multiplied by 500, gives $6590-3975 for the amount of the annuity.

Now, by the table under Compound Discount, we find the present worth of $1 for 10 years, at 6 per cent., to be $0.558395; which, multiplied by 6590 3975, gives $3680-045 for the present worth required.

2. What is the present worth of an annuity of $100, to continue 20 years, at 5 per cent. interest?

Ans. $1246.222.

The work under this rule may be very much simplified by the use of the following table, which gives the present worth of an annuity of $1, or £1, for any number of years not exceeding 30, at 3, 4, 5, and 6 per cent.

To find the present worth of an annuity by means of this table, we must take from it the present worth of $1 for the given time and rate per cent., and multiply it by the number of dollars in the given annuity.

3. What is the present worth of an annuity of $27 for 9 years, at 4 per cent.?

From the table, we find the present worth of $1 for 9 years, at 4 per cent., to be $7.435332; this, multiplied by 27, gives $200-754 for the present worth.

4. What is the present worth of a pension of $75 for 15 years, at 5 per cent.? Ans. $778.474.

5. A young man purchases a farm for $924, and agrees to pay for it in the course of 7 years, paying part of the price at the end of each year. Allowing interest to be 6 per cent., how much cash in advance will pay the debt? Ans. $736 874.

6. Allowing interest to be 6 per cent., how much shall I gain by paying $15 a year for 10 years, in order to cancel a debt of $160, now due? Ans. $49.599.

7. What is the present worth of an annuity of $375 for 13 years, interest being reckoned at 4 per cent.?

Since 375 is of 1000, we may multiply the tabular number by 3, divide by 8, and remove the decimal point three places to the right.

Operation.

9.985648

3

8)29-956944

Ans. $3744.618

8. What is the present worth of an annuity of $875

for 11 years, interest being 6 per cent.?

Since 875 is of 1000, we may multiply by 7, divide by 8, and remove the decimal point three places to the right.

Operation. 7.886875

7

8)55.208125

Ans. $6901.015625

Or, we might have subtracted of the tabular number from itself, and then have removed the decimal point three places to the right, as in this second

Operation.
8)7.886875

985859375

Ans. $6901 015625

The student ought to exercise himself in seeking short and expeditious methods whenever the nature of the operation will admit of contractions.

NOTE.-When an annuity does not commence until a given time has elapsed, or some particular event has happened, it is called a

REVERSION.

Case III.

To find the present worth of an annuity in reversion.

RULE.

Find, by the use of the table under last Case, the present worth of the annuity from the present time up to the end of its continuance; find, also, by the same table, its value for the time before it commences; the difference of these results will be the present worth.