CASE II. Rate per cent. debt, and present worth, given to find the time. RULE. Divide the debt by the present worth-the quotient will be the amount of 1 for the time required, (which seek in table 1st) or divide this amount of 1 continually by the ratio-the number of divisions will be the time. EXAMPLE. 4. Given £521 4 11, debt; present worth, £320; rate per cent. 5, to find the time. 521.24687320=1.62889, tab. number, or amount of £1, found opposite 10 years in table 1st. CASE III. Time, debt and present worth, given to find the ratio : RULE. Having found the tab. number, or amount of 1 for the given time, seek the number in table 1st, and at the head of the column you will find the ratio : Otherwise, Extract that root of the number found, denoted by the number of years the root will be the ratio. EXAMPLE. Last question. 1.62889, tab. number for 10 years, at 5 per ct. ΙΟ Or thus, 1.62889.05, Ans. Present worth of Annuities. TABLE IV. Showing the present worth rebate of £1 or $1, annuity to continue Construction of the foregoing Table. This table is constructed from table 3d, thus, to the amount of 1 as due one year hence, viz. .934579 being first term at 7 per cent. Add .873438 second term of table 3d. 1.808017-present value of 1. annuity for 2 years. Add .816297 third number of table 3d. 2.624314 present value of 1. annuity for 3 years, rebate at 7 per cent. per annum. USE. Multiply one year's annuity by the tabular number opposite the number of years the annuity has to continue, and under the rate per cent. the product will be the present value of the annuity, &c. Otherwise, Find the present worth of the first and last year's annuity, which are the greatest and least terms of a geometrical progression; then find the sum of that progression, which is the present worth of the annuity. EXAMPLES. 1. What is £30 yearly rent worth, in ready money, to continue 7 years, at 6 per cent. per annum, compound interest? Ans. £167 9 5. 2. An annuity of £20, to continue 7 years, is to be sold at 5 per cent. compound interest: What is it worth in ready money? Ans. £115 14. 6. 3. A rent of £365, paid yearly, is to be sold for 12 years, at 5 per cent. per annum, compound interest: What is the present worth in ready money ? Ans. £3235 1 9. CASE II. Present worth, rate per cent. and time, given to find the annuity. RULE. Divide the present worth, by the present worth of 1, for the given time and rate; the quotient will be the annuity. EXAMPLE. 4. Present worth, £3235 1 9, time 12 years, and rate per cent. 5: Required the annuity the above would purchase. 3235.087248-8.863251 present worth of 1, at 5 per cent. Table 4th 365 annuity. Ans. CASE III AND IV. Annuity, present worth and ratio, given to find the time. RULE. Divide the present worth by the annuity, the quotient will be the present worth of 1; which seek for in table 4, under the ratio, opposite to which you find the time: Otherwise, divide the present worth of I continually by the ratio, till nothing remainsthe number of divisions will be equal to the time. EXAMPLE. 5. Annuity £365, is sold for £3235 1 9, at 5 per cent.: Required the time of its continuance. 3235.087248-365-8.863252, present worth of 1. which is found under 5 per cent. table 4, answering to 12 years. Ans. Annuities in Reversion. When an annuity or yearly rent does not commence till after the expiration of some time, or when an annuity belongs to several, each of whom shall enjoy it in succession; then the annuity or rent is said to be in reversion. RULE. Find the present worth of the annuity, as if commencing immediately. Second-find what ready money ought to be paid for that sum, rebate at compound interest, being allowed for the term of years, till the commencement of the annuity or lease, &c. EXAMPLE. 1. Required the present value of $100 yearly rent, to continue 14 years, and not to commence till 6 years hence; discount computed at 7 per cent. per annum, compound interest. 100 1.07 amount of 1 for 1 year,—93.457, first term or value at 1 year's end. 100 1.07,14(2.57851 for 14 y.)—38.781, last term. 1.07-1.07) 54.676, difference. 781.085 93.457, first term. $874.54, present worth of the annuity or rent, as if commencing immediately; but as it does not commence till after 6 years, this present worth of the annuity must be discounted as a sum due 6 years hence: Therefore, 874.54 1.07,6(1.50073)=$582.74, the real value of the an nuity or rent. Ans. 2. Required the present worth of an annuity of £75, which is not to commence till 10 years hence, and then to continue 7 years after that time, at 6 per cent. compound interest. Ans. £233 15 9. 3. An annuity of £24, to begin 7 years hence, and to continue 21 years, is to be sold at 6 per cent. compound interest: What is it worth to the purchaser? Ans. 187 15 5. CASE II. Present worth, ratio, time and reversion, given to find annuity. RULE 1st. Divide the present worth by the present worth of 1, for the time given, (according to table 3); this gives the value of the reversion as commencing immediately. 2nd. Divide this present worth by the present worth of 1 annuity for the number of years, the annuity is in reversion, (by table 4); this quotient will be the annuity. EXAMPLE. A, purchasing the reversion of an annuity, to commence 2 years hence, and to continue 4 years, pays £185.035876, being allowed compound interest at 6 per cent.: Required the annuity. 185.035876.89, present worth of 1 for 2 years,=207.906, present worth of the annuity, as commencing immediately-and 207.906—3.4651, present worth of 1 annuity for 4 years,=£60. Answer. Present worth of Annuities, or Leases for ever. CASE I. To find for how many years purchase an annuity, or rent for ever, may be bought at any given rate per cent.; and then to find the present worth. RULE.-Divide 100 by the rate per cent. the quotient will be the number of years required; this number multiplied by the annuity, gives the present worth. EXAMPLE. What is the present worth of $500, annuity for ever; allowing the purchaser 8 per cent. for his money? 100 8 12.5=121⁄2 years purchase. then 12.5 × 500=$6250. present worth. Ans. What is the present worth of an estate, whose yearly rent is $3000, at 5 per cent.? Ans. $60000. CASE II. When the annuity and rate are given to find the present worth. RULE.-Divide the annuity by the ratio, the quotient is the present worth. EXAMPLE. 1. Prove 1st example of case first. CASE III. When the present worth and rate are given to find the annuity. RULE. Multiply the present worth by the ratio, the product will be the annuity. EXAMPLE. An estate is sold at 8 per cent. it amounts to $6250: Requir6250.x.08 $500. Ans. ed the yearly rent. CASE IV. Present worth, and yearly rent, given to find the ratio. RULE. Divide the annuity by the present worth, the quotient will be the ratio. EXAMPLE. An annuity of $500, is sold for $6250: Required the rate per cent. 500.00 6250.08 ratio or 8 per cent. Ans. The number of years purchase, given to find the rate per cent. RULE.-Divide 100 by the given years, the quotient will be the rate per cent. EXAMPLE. Ans. An annuity is bought at 12 years purchase: Required the rate per cent. 100 12.5 8 per cent. CASE VI. Annuities for ever in reversion. Having the annuity, rate, and time of reversoin, given to find the present worth : RULE. Find the present worth by case 1st or 2d, (as if commencing immediately); divide this present worth by that power of the ratio more 1. denoted by the number of years the reversion continues; this quotient is the present worth. EXAMPLE. What sum must be paid for the reversion of an annuity or rent of $500, to commence after 6 years, at 7 per cent. compound interest? 500-07 7142.85, present worth as commencing immediately; and 7142.85 1.07,°(=1.5)=$4761.90, present worth. Ans. An annuity of $300. for ever, 6 years in reversion, is sold at 5 per cent.: Required the present worth. Ans. $4477.60. CASE VII. Present worth, rate per cent. and time of reversion, given to find the annuity or yearly rent. RULE. Find that power of the ratio more 1. (see table 1st, compound interest) denoted by the number of years in reversion; multiply this power by the ratio, the product multiplied by the present worth, gives the annuity. EXAMPLE. An estate six years in reversion, at 7 per cent. amounts to $4761.90: Required the yearly rent. 1.07,6=1.5 amount for 6 years, ratio .07=.105 and 4761.90 x.105 $500. Ans. An annuity 6 years in reversion, being sold at 5 per cent. amounts to $4477.61: Required the annuity. Ans. $300. |