they are to be multiplied together. Thus PR. implies, that the principal is to be multiplied by the ratio. When one letter is placed cbove another, like the index of a power, the first is to be raised to a power, whose index is denoted by the second. Thus RT implies, that the ratio is to be raised to a power, whose index shall be equal to the time, that is, the number of years. 2. What is the amount of 40 dollars for 11 years, at 5 per cent. compound interest? RT. × P. = A.; therefore, 1'0511 × 40 = 68'4. Ans. $68'40. 3. What is the amount of $6 for 4 years, at 10 per cent. compound interest? Ans. $8784. 4. If the amount of a certain sum for 5 years, at 6 per cent. compound interest, be $5'352, what is that sum, or principal? If the number of terms be 6, the ratio 1'06, and the last term 5'352, what is the first term? This question is the reverse of the last; therefore, 5. What principal, at 10 per cent. compound interest, will amount, in 4 years, to $87846 ? Ans. $6. 6. What is the present worth of $68'40, due 11 years nence, discounting at the rate of 5 per cent. compound interest? Ans. $40. 7. At what rate per cent. will $6 amount to $8'7846 in 4 years? If the first term be 6, the last term 8'7846, and the num ber of terms 5, what is the ratio? the ratio; and then, by extracting the 1'10 for the ratio. 8. In what time will $6 amount to cent. compound interest? A. =RT, that is, 8'7846 the 4th power of 4th root, we obtain Ans. 10 per cent. $87846, at 10 per = 1'4641 = 1'10T.; therefore, P. if we divide 1'4641 by 1'10, and then divide the quotient thence arising by 1'10, and so on, till we obtain a quotient that will not contain 1'10, the number of these divisions will be the number of years. Ans. 4 years. 9. At 5 per cent. compound interest, in what time will 40 amount to $68'40? Having found the power of the ratio 1'05, as before, which is 1'71, you may look for this number in the table, under the given rate, 5 per cent., and against it you will find the number of Ans. 11 years. years. 10. At 6 per cent. compound interest, in what time will $4 amount to $5'352? Ans. 5 years. Annuities at Compound Interest. 115. It may not be amiss, in this place, briefly to show the application of compound interest, in computing the amount and present worth of annuities. An ANNUITY is a sum payable at regular periods, of one year each, either for a certain number of years, or during the life of the pensioner, or forever. When annuities, rents, &c. are not paid at the time they become due, they are said to be in arrears. The sum of all the annuities, rents, &c. remaining unpaid, together with the interest on each, for the time they have remained due, is called the amount. 1. What is the amount of an annual pension of $100, which has remained unpaid 4 years, allowing 6 per cent. compound interest? The last year's pension will be $100, without interest; the last but one will be the amount of $100 for 1 year; the last but two the amount (compound interest) of $100 for 2 years, and so on; and the sum of these several amounts will be the answer. We have then a series of amounts, that is, a geometrical series, (T 114,) to find the sum of all the terms. If the first term be 100, the number of terms 4, and the ratio 1'06, what is the sum of all the terms? Consult the rule, under ¶ 113, ex. 11. 1'064 - 1 '06 X 100 437'45. Ans. $437'45. Hence, when the annuity, the time, and rate per cent. are given, to find the amount,-RAISE the ratio (the amount of $1, &c. for 1 year) to a power denoted by the number of years; from this power subtract 1; then divide the remainder by the ratio, less 1, and the quotient, multiplied by the annuity, will be the amount. Note. The powers of the amounts, at 5 and 6 per cent. up to the 24th, may be taken from the table, under ¶ 91. 2. What is the amount of an annuity of $50, it being in arreans 20 years, allowing 5 per cent. compound interest? Ans. $1653'29. 3. If the annual rent of a house, which is $150, be in arrears 4 years, what is the amount, allowing 10 per cent. compound interest? Ans. $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 22 years? years? in 24 years? in 20 Ans. to the last, $25407'75. ¶ 116. If the annuity is paid in advance, or if it be bought at the beginning of the first year, the sum which ought to be given for it is called the present worth. 5. 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 6 per cent. compound interest, would, in 4 years, produce an amount equal to the amount of the annuity in arrears the same time. By the last rule, we find the amount = $437'45, and by the directions under ¶ 114, ex. 4, we find the present worth = $346'51. Ans. $346'51. Hence, to find the present worth of any annuity,-First find its amount in arrears for the whole time; this amount, divided by that power of the ratio denoted by the number of years, will give the present worth. 6. What is the present worth of an annual salary of $100 to continue 20 years, allowing 5 per cent.? Ans. $1246′22. The operations under this rule being somewhat tedious, we subjoin a 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. What ready money will purchase an annuity of $150, to continue 30 years, at 5 per cent. compound interest? The present worth of $1 annuity, by the table, for 30 years, is $15'37245; therefore, 15'37245 × 150 = $2305'867, Ans. 8. What is the present worth of a yearly pension of $40, to continue 10 years, at 6 per cent. compound interest? at 5 per cent. ? to continue 15 years? years? 25 years? U⭑ 20 34 years? When annuities do not commence till a certain period of time has elapsed, or till some particular event has taken place, they are said to be in reversion. 9. 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? The present worth is evidently a sum which, at 6 per cent. compound interest, would in 2 years produce an amount equal to the present worth of the annuity, were it to commence immediately. By the last rule, we find the present worth of the annuity, to commence immediately, to be $346'51, and, by directions under TT 114, ex. 4, we find the present worth of $346'51 for 2 years, to be $308'393. Ans. $308'393. Hence, to find the present worth of any annuity taken in reversion, at compound interest,-First, find the present worth, to commence immediately, and this sum, divided by the power of the ratio, denoted by the time in reversion, will give the answer. 10. What ready money will purchase the reversion of a ease of $60 per annum, to continue 6 years, but not to commence till the end of 3 years, allowing 6 per cent. compound interest to the purchaser ? The present worth, to commence immediately, we find to 295'039 be, $295'039, and 1'063 =247'72. Ans. $24772. It is plain, the same result will be obtained by finding the present worth of the annuity, to commence immediately, and to continue to the end of the time, that is, 3 + 6 = 9 years, and then subtracting from this sum the present worth of the annuity, continuing for the time of reversion, 3 years. Or, we may find the present worth of $1 for the two times by the table, and multiply their difference by the given annuity. Thus, by the table, The whole time, 9 years, 6'80169 The time in reversion, 3 years, = 2'67301 11. What is the present worth of a lease of $100 to continue 20 years, but not to commence till the end of 4 years, |