5) 30 5 112 6 1 24 24 20)504 amount of 1st year. 25 4 20)529 4 ditto of 2d. 26 9 24 20)555 13 21 ditto of 4th. 27 15 2 )583 8 10 L.612 12 31 do. of 5th. Ans. Note. The same may be done in federal money, but the first method £.642 6 10 do. of 5th. Ans. is generally more easy. COMPOUND INTEREST BY DECIMALS. A Table of the Amount of £.1 or 1D. at per cent. per month, as practised at the Banks. A Table of the Amount of £.1 or D.1 from 1 Day to 31 Doys, at 6 per CASE I.* When the principal, the rate of interest, and time, are given, to find either the amount or interest. RULE. 1. Find the amount of £.1 or D11 for one year at the given rate per cent. 2. Involve the amount, thus found, to such power, as is denoted by the number of years; or, in Table 1, at the end of Annuities, under * Let r = amount of 11. for 1 year, and p = principal, or given fum; then, 2 fincer is the amount of 11. for 1 year, r will be its amount for 2 years, r 3 for years, and so on; therefore, it will be as 1 : r :: rir 2 3 = amount for the second 2 3 : r amount for the year, or principal for the third: Again, as 1: r⠀r third year, or principal for the fourth, &c. to any number of years. And, if the time or number of years be denoted by t, the amount of 11. for t years, will ber; from hence it will appear that the amount of any other principal sum p t for t years, is pr; for, as 1 ::: p: pr', the fame as in the rule. If the rate of intereft be determined to any other time than a year, as 4,, &c. the rule is the fame, and then t will represent that stated time. r amount of 11. for 1 year, at the given rate per cent. P principal, or fum put out at interest. Leti intereft. t = time. mamount for the time t. Then the following Theorems will exhibit the folutions of all the cafes in compound interest. , The most convenient way of giving the Theorems, especially that for the time, will be by Logarithms, as follows: 1 L.m-L.p. I. tx Log. r+Log.p=Log, m. II. Log. m―tXL.r=L.p. III. =t. L.m-L.p Lar IV. = L.r. t a If the compound intereft, or amount of any fum, be required for the parts of year, it may be determined as follows: I. W ben the time is an aliquot part of a year. RULE.-1. Find the amount of 11. for 1 year, as before, and that root of it, which is denoted by the aliquot part, will be the amount of 11. for the time fought. 2. Multiply the amount, thus found, by the principal, and it will be the amount of the given fum required. II. When the time is not an aliquot part of a year. RULE.-1. Reduce the time into days, and the $65th root of the amount of 11. for 1 year is the amount for 1 day. 2. Raife this amount to that power, whose index is equal to the number of days, and it will be the amount of 11. for the given time. 3. Multiply this amount by the principal, and it will be the amount of the given fum required. To avoid extracting very high roots, the fame may be done by logarithms, thus:' Divide the logarithm of the rate, or amount of 11. for 1 year, by the denominator of the given aliquot part, and the quotient will be the logarithm of the root fought. under the rate, and against the given number of years, you will find the power.* 3. Multiply this power by the principal, or given sum, and the product will be the amount required, from which, if you subtrac the principal, the remainder will be the interest. EXAMPLES. 1. What is the compound interest of 6001. for 4 years, at 6 per cent. per annum ? amount of 11. for 1 year, at 6 per cent. 1.06= Multiply by 1.06 { per annum. 157-486176.157 9s. 8d.-interest required. BY TABLE I. Tabular amount of 11. for 4 years, at 6 per cent. per ann. 1.2624769 Multiply by the principal= 600 Amount = 757.4861400 2. What is the amount of D.1500 for 12 years, at 3 per cent. per annum ? A = D.1:035 amount of D.1 for 1 year at 3 per cent. per annum. nd, 103512 x 1500 D.2266 60c. nearly, Ans. Another method of working compound interest for years, months, and days, which is much more concise than the preceding method. RULE. To the logarithm of the principal, found in any Table of logarithms, add the several logarithms, answering to the number of years, months and days found in the following tables, and their sum will be the logarithm of the amount for the given time, which being found in any table of logarithms, the natural number corresponding thereto will be the answer.t Logarithmick * The amounts of £.1 or D.1 in this table, are so many powers of the amount of £.1 or D.1 for 1 year; whofe indices are denoted by the number of years. Note. When the given time confifts of years and months, or years, months, and days; firft feek the amount of £.1 or D.1 in the table of years, then in the table of months, &c. multiply these feveral amounts and the principal continually together, and the last product will be the amount required. Thus, if the amount of £.480 in 5 years, at 6 per cent per annum, were required; the amount of £.1 for 5 years=£.1.33822, ditto for 6 months=£.1·02956 Now, 1.33822X1-02956 × 480=£.661.2541 Answer. + Although there is a small errour in the logarithm for days, yet they are exact enough for common ufe. And if after the first month you deduct per cent. for each Logarithmick TABLES, at 6 per Cent. per Annum, for Years, Months and Days. What is the amount of 1321. 10s. at 6 per cent. per annum, for 9 years, 8 months, and 15 days? To the log. of £.132.52.122216 Remains 2.3680302, the nearest to which, in the table of logarithms, is 2.368101, and the natural number answering thereto is 233-4-.233 8s. Ans. CASE II. When the amount, rate and time, are given, to find the principal. RULE. Divide the amount by the amount of £.1 or D.1 for the given time, and the quotient will be the principal. Or, If you multiply the present value of £.1 or D1 for the given number of years, at the given rate per cent by the amount, the product will be the principal, or present worth.* EXAMPLES. each month past (that is, per cent. after 1 month, 14 per cent. after 3 months, &c.) from the logarithm of the number of days, it will give the true answer. Note, That, after 1 month, per cent. on the logarithm of 1 day is 000000355, on 2 days, is 000000715: After 2 months, 1 per cent. on the logarithm of 1 day, is 00000071, on 2 days, ·00000143 : After 10 months, 5 per cent. on the logarithm for 1 day, is '00000355, on 6 days, is, '00002145, &c. * See Table II. fhewing the prefent value of 11. difcounting at the rafes of 4, 41, c. per cent. the conftruction of which is thus: Amount. Pres. worth. Amount. Pres. worth. As 1.06 : cent. and time. 1 : 1 : 9433962, and so on, for any other rate per EXAMPLES. 1. What is the present worth of 7571 9s. 8d. due 4 years hence, discounting at the rate of 61. per cent. per annum? Divide by the tabular BY TALE I. amount of 11. for 4 years, =1-2624769) 757-4861400 (£.600 Ans. BY TABLE II. Mult. by the present worth of 11. Į Amount=757.48614 for 4 years, at 6 per. cent. per ann. = ⚫79 20936 Ans. 599-999923582704+.600 2. What principal must be put to interest 6 years, at 54 per cent. per annum, to amount to D.689-4214035809453125? Ans. D.500. CASE III. When the principal, rate and amount, are given, to find the time. RULE. Divide the amount by the principal; then divide this quotient by the amount of £.1 or D.1 for 1 year, this quotient by the same, till nothing remain, and the number of the divisions will show the time. Or, Divide the amount by the principal, and the quotient will be the amount of £.1 or D.1 for the given time, which seek under the given rate in table 1, and, in a line with it, you will see the time. EXAMPLE. In what time will D.500 amount to D.689 42c. 1m.+, at 51 per t. per annum? cent. When the principal, amount and time, are given, to find the rate per cent. RULE. Divide the amount by the principal, and the quotient will be the amount of 11. or ID. for the given time; then, extract such root as the time denotes, and that root will be the amount of 11. or D. for 1 year, from which subtract unity, and the remainder will be the ratio. Or, Having found the amount of 11. or D. for the time as above directed, look for it in Table 1st. even with the given time, and directly over the amount you will find the ratio. EXAMPLE, |