2 Involve the amount thus found to such a power, as is denoted by the number of years. 3. Multiply this power by the principal, or given sum, and the product will be the amount required. 4. Subtract The most convenient way of giving the theorem for the time, as well as for all the other cases, will be by logarithms, as follows: If the compound interest, or amount of any sum, be required for the parts of a year, it may be determined as follows: I. When the time is any aliquot part of a year. RULE. 1. Find the amount of 11. for one year, as before, and that root of it, which is denoted by the aliquot part, will be the amount sought. 2. Multiply the amount thus found by the principal, and it will be the amount of the given sum required. II. When the time is not an aliquot part of a year. RULE. 1. Reduce the time into days, and the 365th root of the amount of 11. for one year is the amount for one day. 2. Raise 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 sum required. To avoid extracting very high roots, the same may be done by logarithms thus divide the logarithm of the rate, or amount of 11. for one year, by the denominator of the given aliquot part, and the quotient will be the logarithm of the root sought. 4. Subtract the principal from the amount, and the remainder will be the interest. EXAMPLES. 1. What is the compound interest of 500l. for 4 years, at 5 per cent. per annum ? amount of 11. for one year at 5 105 3. What is the amount of 7211. for 21 years, at 4 per cent. per annum ? Ans. 1642l. 19s. 10d. 4. What is the amount of 2171. forborn 24 years, at 5 per cent. per annum, supposing the interest payable quarterly? Ans. 2421. 138. 4žd. ANNUITIES. ANNUITIES. AN ANNUITY is a sum of money payable every year, for a certain number of years, or for ever. When the debtor keeps the annuity in his own hands, beyond the time of payment, it is said to be in arrears. The sum of all the annuities for the time they have been forborn, together with the interest due upon each, is called the amount. If an annuity be to be bought off, or paid all at once, at the beginning of the first year, the price, which ought to be given for it, is called the present worth. To find the Amount of an Annuity at Simple Interest. RULE.* 1. Find the sum of the natural series of numbers 1, 2, 3, &c. to the number of years less one. 2. Multiply * DEMONSTRATION. Whatever the time is, there is due upon the first year's annuity, as many years' interest as the whole number of years less one; and gradually one less upon every succeeding year to the last but one; upon which there is due only one year's interest, and none upon the last; therefore in the whole there is due as many years' interest of the annuity, as the sum of the series 1, 2, 3, 4, &c. to the number of years less Consequently one year's interest, multiplied by this sum, must be the whole interest due; to which if all the annuities be added, the sum is plainly the amount. Q. E. D. one. Let r be the ratio, n the annuity, t the time, and a the amount. Then will the following theorems give the solutions of all the different cases. 2. Multiply this sum by one year's interest of the annuity, and the product will be the whole interest due upon the annuity, 3. To this product add the product of the annuity and time, and the sum will be the amount sought. NOTE. When the annuity is to be paid half-yearly or quarterly; then take, in the former case, the ratio, half the annuity, and twice the number of years; and, in the latter case, the ratio, the annuity, and 4 times the number of years, and proceed as before. I EXAMPLES. 1. What is the amount of an annuity of 5ol. for 7' years, allowing simple interest at 5 per cent. ? 1+2+3+4+5+6=21=3X7 21. 10s. I year's interest of gol. cannot be found equal to the amount, the problem is impossible in whole years. NOTE. Some writers look upon this method of finding the amount of an annuity as a species of compound interest; the annuity itself, they say, being properly the simple interest, and the capital, whence it arises, the principal. 2. If a pension of 600l. per annum be forborn 5 years, what will it amount to, allowing 4 per cent. simple interest ? Ans. 32401. 3.. What will an annuity of 250l. amount to in 7 years, to be paid by half-yearly payments, at 6 per cent. per annum, simple interest ? Ans. 20911. 5s. To find the present Worth of an Annuity at Simple Interest. RULE.* Find the present worth of each year by itself, discountIng from the time it becomes due, and the sum of all these will be the present worth required. EXAMPLES. * The reason of this rule is manifest from the nature of discount, for all the annuities may be considered separately, as so many sin gle and independent debts, due after 1, 2, 3, &c. years; so that the present worth of each being found, their sum must be the present worth of the whole. The estimation, however, of annuities at simple interest is highly unreasonable and absurd. One instance only will be sufficientto shew the truth of this assertion. The price of an annuity of 50l. to continue 40 years, discounting at 5 per cent. will, by either of the rules, amount to a 'sum, of which one year's interest only exceeds the annuity. Would it not therefore be highly ridiculous to give, for an annuity to continue only 40 years, a sum, which would yield a greater yearly interest for ever? It is most equitable to allow compound interest. present worth, and the other letters as before. |