To find the present worth of an annuity by means of this table, we must take from the table the present worth of $1, for the given time and rate per cent., and multiply it by 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. 13 7. What is the present worth of an annuity of $375, for years, interest being reckoned at 4 per cent.? 8. What is the present worth of an annuity of $875, for 11 years, interest being 6 per cent. ? 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. Or, which amounts to the same thing: Take the difference of the tabular numbers for these two periods, and multiply by the annuity. Examples. 1. What is the present worth of an annuity of $200, to be continued 5 years, but not to commence till 2 years hence, interest being 6 per cent.? By our table, we find the present worth of $1, for 7 years, at 6 per cent., to be $5.592381, the same for 2 years is $1.833393, the difference is $3.748988, which, multiplied by $200, gives $749.798, for the present worth. 2. A father leaves to his son a rent of $310 per annum, for 8 years, and the reversion of the same rent to his daughter for 14 years thereafter. What is the present worth of the leg. acy of each, at 6 per cent.? $1925.036. Ans. {Daughter's, Daughter's, $1807.854. 3. What is the present worth of a reversion of $100 a year, to commence in four years, and to continue for ten years, interest being at 6 per cent.? Ans. $582.988. 4. What is the present worth of a reversion of $800 a year, to continue 7 years, but not to commence until the end of 8 years, interest being 4 per cent.? Ans. $3508.514. When the annuity is to continue for ever, it is obvious that its present worth will be that sum whose interest for 1 year. is equal to the annuity; therefore to find the present worth of an annuity to continue for ever, we must divide the annuity by the interest of $1 for one year, at the given rate per cent. 5. How much must be paid, at present, for the title to an annuity of $1000, to commence in 7 years, and to continue for ever; interest at 6 per cent.? Dividing $1000 by $0.06, we get for the present worth, if entered upon immediately, $16666.661. From table under compound discount, we find the present worth of $1, for 7 years, at 6 per cent., to be $0.665057; this, multiplied by 16666.66, gives $11084.283, for the present worth of $16666. 663, which is evidently the same as the present worth of the annuity. 6. What is the present worth of a reversion of $100 a year, to commence in 4 years, and to continue forever; interest being 6 per cent.? Ans. $1320.157. 72. We will add the following tables more for curiosity than for any view to their utility. The following table gives the time required for a given principal to double itself, at compound interest, the interest being compounded YEARLY. The following table gives the time required for a given principal to double itself, at compound interest, the interest being compounded HALF-YEARLY. Per cent. Years. Per cent. Years. Per cent. Years. 1 69.487 The following table gives the time required for a given principal to double itself, at compound interest, the interest being compounded QUARTER-YEARLY. Per cent. Years. Per cent. Years. Per cent. Years. The following table gives the time required for a given principal to double itself, at compound interest, the interest being compounded EVERY INSTANT. |