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.f EXAMPLES. 1. What is the present worth of 7571. 9s. 8d. due 4 years hence, dsзcounting at the rate of 61. per cent. per annum? By Table 1. By Table II. Divide by the tabular 1-2624769)757-4861400(£600 Ans. Mult. by the present worth of 11. for 4 years, at 6 per cent per ann. § Amount 757-48614 .79-20936 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.t 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 examT-655 × 100, 5 ple in the proof of Case I. 105 x 100 the amount; therefore, 100, the principal. 1·055 + See Table II. shewing the present value of £1, discounting at the rates of 4, 4, &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. 1055 x 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 51 per cent. per annum ? 500/689-421+ When the principal, amount and time, are given, to find the rate per ct. 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? 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,✔ 1·05=105, and 1.05—1—05, the ratio. 51 + 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. Or, By Case I, of Compound Interest by Decimals, the amount of EXAMPLES. 1. What is the present worth, and discount, of £600 due 3 years hence, at £6 per cent. per annum, compound interest? Divide by 106=1.19101)600-00000(503-7741= £503 15s. 53d. present worth, and £600-£ 503 15 53 £96 4s. 6d. discount. In this Table, corresponding to the time and rate, we have 839619 Multiply by 600 present worth of £1 for the time and rate. debt, or principal. 503-771400 present worth of the debt. 2. What is the present worth of £312 10s. due 2 years hence, at 4 per cent. per annum, compound interest? Ans. £236 3s. 3d. 2.97qrs. 3. What ready money will discharge a debt of $1000 due 4 years hence, at $5 per cent. per annum, compound interest? Ans. $822 70c. 2m. 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 has 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 laid 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. 1 11. or $1 for any number of years, is equal to that power of the amount of 11. or $1 indicated by the number of years. Hence if the amount be 11. or $1 the principal will be the reciprocal of the power of the amount of 11. or $1 adicated by the number of years; thus, if 1=-amount at 6 per cent. for 4 years, then, the principal which will produce the amount at the rate and time. 1.064 Therefore, if the sum to be discounted at that rate and time, then present worth, and is the rule. 1 its 1:064 Note. The present worth of 11. or $1 for any time and rate, is the reciprocal of the amount of 11. or $1 for the same time. The amount is the sum of the annuities for the time it has been forborne, with the interest due on each. CASE I. To find the amount of an annuity at Simple Interest. RULE. 4, Multiply the sum of the natural series of numbers, 1, 2, 3, &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.* EXAMPLES. 1. What is the amount of an annuity of £100 for four years, computing interest at 6 per cent. ? 1+2+3=6, sum of the natural series to the number of years less 1. 61. interest of annuity for 1 year. 6×6=361. the whole interest. 100x44001. product of annuity and time. Ans. 4361. amount. 2. If a pension of $20 be continued unpaid for six years, what is its amount at 6 and 7 per cent.? Ans. At 6 per cent. $138. At 7 per cent. $141. 3. If an annuity of $20 to be paid half each half year is forborne for six years; what is its amount at 6 per cent.? Ans. $159 60c. 4. If a pension of £33 is forborne for 12 years, at 7 per cent. what is the amount ? Ans. CASE II. To find the present worth of an annuity at Simple Interest. RULE. Let the present worth of each year be found by itself, discounting from the time it is due; then, the sum of all these will be the present worth.t *It is plain that upon the first year's annuity there will be due so many year's interest, as the given number of years less one, and gradually one year less upon each 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. This is the rule. + This rule depends on the principles of discount. The annuity may be considered for each year, as a debt, due 1, 2, 3, &c. years hence, of which the pres EXAMPLES. 1. Find the present worth of an annuity of $100 continued five years at six per cent. $ As 106 100: 100: 94-3396, the present worth for 1 year. 112: 100 :: 100: 89-2857, 118: 100: 100: 84-7457, 124 100 100 80 6451, : 130: 100: 100: 76-9230, $425-9391, present worth required. 2. Find the present worth of an annuity of 751. continued for 4 years at 7 per cent. ? Ans. 3. What is the present worth of a pension of $20 to be continued for 6 years, at 6 per cent.? Ans. ANNUITIES OR PENSIONS, IN ARREARS, AT COMPOUND INTEREST. CASE I. When the annuity, or pension, the time it continues, and the rate per cent. are given, to find the amount. RULE 1.* 1. Make 1 the first term of a Geometrical Progression, and the amount of £1 or $1 for 1 year at the given rate per cent. the ratio. 2. Carry the series to so many terms as the number of years, and find its sum. ent worth is to be found. Hence the sum of the present worth for the several years, must be the present worth for the whole. This rule is very absurd in practice. It is obvious on inspecting the operation of Ex. 1. that the difference between the present worth of the several years is continually diminishing. Whence, after a certain number of years, the present worth of an annuity of $100 would produce more than $100 interest in one year, which is greater than the annuity to be purchased. * I. From the nature of an annuity, as explained in the proof of the rule, Case I. of Annuities at Simple Interest, there is due one year's interest less than the number of years the annuity has been continued. Now, by Case I. of Compound Interest, the amount of £1 or $1 at the given rate, is equal to that power of the amount for one year, which is indicated by the number of years. This amount is obtained for one less than the number of years, by forming the geometrical series as directed in the Rule, or beginning with unity. Thus in Ex. 1, the series is, 1, 106, 1·062, 1·063, and the last term is the amount of £1 or $1 for one less than four, the number of years. The sum of this series is the amount at Compound Interest, of an annuity of £1 or $1 for four years. The amount of any other annuity for the same time and rate, will be as much greater or less, as the annuity is greater or less than £1 or $1, that is, the amount of the annuity of £1 or $1 must be multiplied by the annuity to obtain its amount. Hence, the rule is manifestly correct. In Ex. 1, the above series amounts, by 1.064-1 Prob. III. of Geometrical Progression, to and this multiplied by the |