A Table of the Amount of £1 or $1, from 1 Day to 31 Days, at 6 per cent. per annum. These tables are formed by adding the interest of £1 or $1, to £1 or $1, for the given rate and time. Interest, the interest of £1 or $1 for 1 the amount is 1.00016438+. CASE I.* Thus, by rule for Simple day, is, 00016439+, and 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 $1 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, -3 2 --2 3 --2 * The reason of the rule may be seen by the following process. If the rate be 6 per cent. the amount of £1 or $1 for 1 year, is, by the rule for Simple Interest by Decimals, 106. This is the principal for the second year, and its amount is by the same rule, 1·06+1.06×·06=1+·06 × 1·06—1·06 × 1·06—1·06. That is, the amount of £1 or $1 for two years is equal to the square of the amount of £1 or $1 for one year. This is the principal for the third year, and its amount is, 1.06+1:06 ×·06=1+·06X1·06 =1·06×106 106, that is the amount for three years is the cube of the amount for 1 year. In the same way it may be shown, that the amount for four years is the fourth power of the amount of £1 or $1 for 1 year; for five years, is the fifth power, and so on. The same would be true, whatever be the rate per cent. Now, whatever he the principal, the amount must be so much greater than the amount of £1 or $1 for the same time and rate. Therefore, the amount for any principal will be found by multiplying the amount of £1 or $1, at the given rate and time by the principal and is the rule. Let the principal be $100 or £100, the rate 5 per cent. and the time 5 years. Then, 105 X100 the amount. 100-100-the interest. 5 -- 5 And 1.05 x If the rate of interest be determined to any other time than a year, as 1, 1, &c. the rule is the same. If the compound interest, or amount of any sum, be required for the parts of year it may be determined as follows: 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 subtract the principal, the remainder will be the interest. EXAMPLES. 1. What is the compound interest of £600 for 4 years, at 6 per cent. per annum? amount of £1 for 1 year, at 6 per 1.06= cent, per annum. 157-486176 757-48617600=amount. £157 9s. 84d. interest required. Tabular amnt. of £1 for 4 years, at 6 per cent. per ann. 1.2624769 Multiply by the principal= Amount 600 757-4861400 2. What is the amount of $1500 for 12 years, at 3 per cent. per annum ? $1 035 amount of $1 for 1 year at 31 per cent. per annum. And, 103512×1500=$2266 60c. nearly, Ans. Another method of working compound interest for years, months, and days, which is much more concise than the preceding method. I. When the time is an aliquot part of a year. RULE 1. Find the amount of £1 for 1 year, as before, and that root of it, which is denoted by the aliquot part, will be the amount of £1 for the time sought. 2. Multiply the amount, thus found, by the principal, and it will be the amount of the given sum required. RULE 1. II. When the time is not an aliquot part of a year. Reduce the time into days, and the 365th root of the amount of £1 for 1 year is the amount for 1 day. 2. Raise this amount to that power, whose index is equal to the number of days, and it will be the amount of £1 for the given time. 3. Multiply this amount by the principal, and it will be the amount of the given sum required. *The amounts of £1 or $1 in this table, are so many powers of the amount of £1 or $1 for 1 year; whose indices are denoted by the number of years. Note. When the given time consists of years and months, or years, months, and days; first seek the amount of £1 or $1 in the table of years, then in the table of months, &c. multiply these several 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=£1.33622, ditte for 6 months-£1:02956. Now, 1-33822 × 1·02956 × 480 £661·2341 Answer. 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.* LOGARITHMICK TABLES, AT SIX PER CENT. PER ANNUM, FOR YEARS, MONTHS AND DAYS. Years. Dec. pts. Y. Dec. pts. Y. Dec. pts. | Y. Dec. pts. Months Dec. pts. What is the amount of 1321. 10s. at 6 per cent. per annum, for years, 8 months, and 15 days? Remains 2-3680302, the nearest to which, in the table of logarithms, is 2368101, and the natural number answering thereto is 233·4== £233 8s. Ans. * Although there is a small error in the logarithm for days, yet they are exant enough for common use, And if after the first month you deduet per cent. for each month past (that is, per cent. after 1 month, 1 per cent. after 3 months, &c.) from the logarithm of the number of days, it will give the true an Note, That, after 1 month,per cent. on the logarithm of 1 day is -0C0CC0355, on 2 days, is 00000071a: After 2 months, 1 per cent. on the logarithm of 1 day, 160000071, on 2 days, 00000143: After 10 months, 5 per cent. on the logarithin for 1 day, is 00000355, en 6 days, is. 00004145, &c. CASE II. When the amount, rate and time, are given, to find the principal. RULE. Divide the amount by the amount of £1 or $1 for the given time, and the quotient will be the principal.* Or, If you multiply the present value of £1 or $1 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. 1. What is the present worth of 7571. 9s. 8d. due 4 years hence, dsscounting at the rate of 61. per cent. per annum?. By Table I. Ans. 599-999923582704+=£600, 2. What principal must be put to interest 6 years, at 5 per ct. per annum, to amount to $689-4214033809453125 ? CASE III. Ans. $500. 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 $1 for 1 year, this quotient by the same, till nothing remain, and the number of the divisions will show the time.1 Or, Divide the amount by the principal, and the quotient will be the amount of £1 or $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. * By Case I. the amount is equal to the principal multiplied by that power of the amount of £1 or $1 for 1 year at the given rate, which is indicated by the number of years: therefore, if the amount be divided by this power of the amount of £1 or $1 for 1 year, the quotient must be the principal. Thus, in the exam1·055 × 100, plc in the proof of Case I. 1055 × 100=the amount; therefore, 100, the principal. 1.055 + See Table II. shewing the present value of £1, discounting at the rates of 4, 42, &c. per cent. the construction of which is thus: Amount. Pres. worth. Amount. Pres. worth. As 1:06 : 1 : 1: cent. and time. 9433962, and so on, for any other rate per the amount; divide this This quotient divided by By the example in the proof of Case I. 1-055 × 100 by the principal, 100, and the quotient will be 1055. the ratio, and this quotient by the ratio, and so on, will be exhausted by five divisions, which shows the number of years. EXAMPLE. In what time will $500 amount to $689 42c. 1m+, at 5 per cent. per annum? 5001689-421+ When the principal, amount and time, are given, to find the rate per et. RULE. Divide the amount by the principal, and the quotient will be the amount of 11. or $1 for the given time; then, extract such root as the time denotes, and that root will be the amount of 11. or $1 for 1 year, from which subtract unity, and the remainder will be the ratio.* Or, Having found the amount of 11. or $1 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. At what rate per cent. per annum will $500 amount to $689-421403+ in 6 years? 689-421403+ 500 =1378843—; and ✓1-378843-1055. Then 1·055-1055-ratio. Hence the rate is 51 per cent. per anAnswer. num, DISCOUNT BY COMPOUND INTEREST.† The sum, or debt to be discounted, the time and rate, given, to find the present worth. RULE. Divide the debt by that power of the amount of 11. or $1 for 1 year, denoted by the time, and the quotient will be the present worth, which, subtracted from the debt, will leave the discount. *Proceeding as in the preceding demonstration, and extracting that root of the quotient, which is shown by the number of years, we have the amount of £1 or $1 for 1 year. From this subtract 1, and the remainder is the ratio. Thus in the preceding example,√ 105=1·05, and 1·05-105, the ratio. 5 As the present worth is such a principal, as at the given rate and time, would amount to the debt, this rule must be the same as that of Case II. of Compound Interest, the principal being in this case the present worth, and the amount the sum or debt.ˆ Or, By Case I. of Compound Interest by Decimals, the amount of |