per cent. ? per cent. ? per cent. ? per cent. ? 4. What is the compound interest of D500 for 4 years ar o Ans. D131.23.8. 5. What is the compound interest of D700 for 7 years at 5 per cent. ? Ans. D284.97. 6. What is the compound interest of D850 for 10 years at 6 Ans. 1672.22.08. 7. What is the compound interest of D1100 for 15 years at 5 Ans. D1186.82.08. 8. What is the compound interest of D4400 for 20 years at 6 Ans. 19711.39.6. 9. What is the compound interest of D20000 for 13 years at 6 Ans. D22658.56 10. What is the amount of D768 for 3 years at 6 per cent compound interest ? Ans. For 3 years, 1.1910160x1.014674 for =12084929687840 amount of 100 for 31; then x 768D. princ. =D928.12.2. Ans. 11. What is the difference between the simple interest of D200 for 3 years, and the compound interest for the same time? Ans. D2.203 % per cent. ? When the amount, rate, and time, are given, to find the principal RULE III. Divide the amount by the amount of 1 or D100 for the given time, and the quotient will be the principal. Or if you multiply the present value of 1 or D100-for the given number of years, at the given kate per cent. by the amount the product will be the principal or present worth. 12. What principal must be put to interest 6 years, at 5} pe cent. per annum, to amount to D689.4214033809453125 ? Thus, 1.3788426)689.4214033809453125(500D. Ans. Note.---We have a variety of cases in Compound Interes, and Annuities, which, for want of room, are necessarily omitted a sufficient number will be given for all practical purposes. REVIEW. What is Compound Interest ? How can you compute interest by the use of the tables ? What is rule 2? How can you find the principal ? In what time will any sum double at 6 per cent. simple interest ? Ans. 16 years, 8 months. In what time at compound interest ?. Ans. 11 years, 8 months, 22 days When is it lawful to compute compound interest ? 13. What is the compound interest of the following sums D1200 for 5 years, and D480 for 7 years, at 7 per cent. ? ANNUITIES. An Annuity is a sum of money payable at regular periods, for a certain time, or for ever. Annuities sometimes depend on some contingency, as the life or death of a person, and the annuity is then said to be contingent. Sometimes annuities are not to commence till a certain number of years have elapsed, and the annuities are then said to be in reversion. The annuity is said to be in arrears when the debtor keeps it beyond the time of payment. The present worth of an annuity is such a sum, as being now put out at interest would exactly pay the annuity as it becomes due, and is the sum which must be given for the annuity if it be paid at its commencement. The amount is the sum of the annuities for the time it had been forborne, with the interest due on each. To find the amount of an annuity, at simple interest. RULE I. Multiply the natural series of numbers, 1, 2, 3, 4, &c., to the number of years, less 1, by the interest of the annuity for one year, and the product will be the interest which is due on the annuity. Multiply the annuity by the time, and the sum of the two products will be the amount. 1. What is the amount of an annuity of D100, for 4 years, computing interest at 6 per cent. ? Thus: 1+2+3=6, sum of the natural series to the number of years, less 1. D6 interest of annuity for one year; then 6x6=D36, the whole interest 100 X4=D400+36=436, amount. Ans. (See note at close of annuities.) 2. If a pension of D20 be continued unpaid for 6 years, what is its amount at 6 and 7 per cent. ? At 6 per cent., D138 ; at 7 per cent., D141. Ans. REMARKS. It is plain that upon the first year's annuity there will be due so many years' interest as the given number of years, less 1, and gradually one year less upon each such succeeding year, to that preceding the last, which has but one year's interest, and the last bears none. There is, therefore, due in the whole as many years' interest of the annuity as the sum of the series, 1, 2, 3, &c., to the number of years, diminished one. It is evident then, that the whole interest due must equal this sum of the natural series, multiplied by the interest for one year, and that the amount will be all the annuities, or the product of the annuity and time added to the whole interest ; and this is the rule. The annuity, time, and ratio given, to find the amount at com. pound interest. RULE II. 1. Make 1 the first term of a geometrical progression, and the amount of Di, and ide given rate per cent. the ratio. 2. Carry.the series to as many terins as the number of years, and find the sum. 3. Multiply the sum thus found by the given annuity, and the product will be the amount sought. Or, multiply the amount of Di for one year, at the given rate per cent., into itself as many times as there are years given; from the product subtract one, then divide the remainder of the interest of D1 for 1 year, at its given rate per cent., and multiply the quotient by the annuity for the amount required. A table showing the amount of A table showing the present Di or D100, from 1 to 15 worth of DI or D100, from 1 years, at 5 and 6 per cent. to 15 years, at 5 and 6 per cent. Years. 15 per cent. 6 per cent 5 per cent. 6 per cent. Years 4 1.009902 1.5 1.011126 5 1.012348 5.5 1.013567 6 1.014781 1.014877 1.016720 1.018550 1.020395 1.022257 The construction of table 3 is from an algebraic theorem, which may be in words, thus : for half-yearly payments, take a unit from the ratio, and from the square root of the ralio; half the quotient of the first remainder, divided by the latter will be the tabular number; for quarterly payments, use the 4th root, and take one quarter of the quotient. Table 1st is calculated thus : take the first year's amount, which is Di, multiply it by 1.06+1=2.06, second year's amount, which also multiply by 1.06+1=3.1836, third year's amount, &c., at 6 per cent., and in this manner calculate the other tables. RULE III. Multiply the tabular number under the rate, and opposite to the time, by the annuity, and the product will be the amount. 3. What will an annuity of D60 per annum amount to in 20 years, allowing 6 per cent., compound interest ? Thus : 36.785592 x D60 ann.=D2207.13.5520. Ans. 4. What will an annuity of D60 per annum, payable yearly, amount to, in 4 years, at 6 per cent. ? Thus : 1+1.06+1.062 +1.063=4.374616, sum, xD60, ann. =D262.47.696. Ans. (Rule 2.) 5. What will a pension of D75 per annum, payable yearly, amount to in 9 years, at 5 per cent., compound interest ? Ans. D826.99.2% The annuity, time, and rate given, to find the present worth. RULE IV. Divide the annuity by the amount of Di for 1 year, and the quotient will be the present worth of one year's annuity. 2. Divide the annuity by the square of the ratio, and the quotient will be the present worth for 2 years. 3. In like manner, find the present worth of each year by itself, and the sum of all these will be the present value of the annuity. 6. What ready money will purchase an annuity of D60, to continue 4 years, at 6 per cent., compound interest ? ratio 1.06)60.00000(56.603=present worth 1 year. ratio 2= 1.1236)60.00000(53.399 2 years. ratio 3= 1.191016)60 00000(50.377 3 years. . ratio 4= 1.26247696)60.00000(47.525 4 years. D207.904 Ans. (See table, compound interest.) OR, BY TABLE 2. the table, by the annuity, the product will be the present worth for yearly payments. When the payments are to be made quarterly, or half-yearly, the present worth, found as above, must be multiplied by the proper number in table 3d. Thus : question 6, tabular No. 3, 46510 x 160=207.906. Ans. 7. What is the present worth of an annuity of D60 per an. num, for 20 years, at 6 per cent. ? Ans. D688.19.50. 8. What is the present worth of 175 per annum, for 7 years, at 5 per cent. ? Thus: 5.78637 x 75=D433.97.775. Ans. Table 2d is thus made : Divide D1 by 1.06=.94339 the present worth of the first year, which, divided by 1.06 is equal to .88999, which added to the first year's present worth is = 1.83339, the second year's present worth ; then .88999 divided by 1.06, and the quotient added to 1.83339 gives 2.67301 for the 3d year's present worth, &c. Annuities in reversion at compound interest. RULE V. Take two numbers under the given rate in table 2d, that stand opposite the sum of the two given times, and the number opposite the time when the annui.y is to commence, or time of reversion, and multiply their difference by the annuity for the present worth. 9. The reversion of a freehold estate of D60 per annum, for 4 years, to commence 2 years hence; what is the present worth, allowing 4 per cent. for present payment ? Thus : tab, no. 5.24214-1.88609=3.35605 x 160=1201.36.300. Ans. 10. What is the present worth of a reversion of a lease for D120 per annum, to continue 9 years, but not to commence till the end of 4 years, at 4 per cent., to the purchaser ? Thus : 9.98565—3.62989=6.35576 x 120=D762.69.1. Ans. PERPETUITIES AT COMPOUND INTEREST. Perpetuities are such annuities as continue for ever. The annuity and rate given, to find the present worth. RULE VI. Multiply the amount of D1 for one year, at the given rate per cent., involved to the time of reversion, by the ratio, for a divi. sor, by which divide the yearly payments; the quotient will be the answer. |