TABLE, Showing the Present Worth of $1, or £1, for any number of years, from 1 to 30, at 3, 4, 5, and 6 per cent., compound discount. The present worth of a given sum of money, discounting at compound interest, is easily obtained by the preceding table. 2. How much money must be placed out at compound interest to amount to $1000 in 20 years, the interest being 5 per cent. ? Ans. $376.889. 3. What is the present worth of $1000, due 27 years hence, discounting at 3 per cent., compound interest ? From the preceding table, we find the present worth of $1 for 27 years, at 3 per cent., to be $0:450189; this, multiplied by 1000, gives $450-189 for the present worth required. 4. What is the present worth of $3525, due in 3 years, discounting at 6 per cent., compound interest ? Ans. $2959.657. 5. What is the present worth of $350, due 5 years hence, discounting at 6 per cent., compound interest ? Ans. $261:54. 6. What is the present worth of $375, due 17 years hence, discounting at 4 per cent., compound interest ? Ans. $192:515. 7. What is the present worth of $672, due 13 years hence, discounting at 5 per cent., compound interest ? Ans. $356:376. 8. What is the present worth of $400, due 19 years hence, discounting at 6 per cent., compound interest ? Ans. $132.205. 9. What is the present worth of $111, due 29 years hence, discounting at 3 per cent., compound interest ? Ans. $47.102. ANNUITIES. 70. AN ANNUITY is a fixed sum of money, which is paid periodically for a certain length of time. Case 1. To find the amount of an annuity which has been forborne for a given time. It is obvious that the last year's payment will be simply the annuity without any interest; the last but one will be the amount of the annuity for one year; the last but two will be the amount of the annuity for two years, and so on; and the sum of all these partial amounts will give the total amount due. Now we discover that these partial amounts, or payments, form a geometrical progression, whose first term is the annuity, the ratio is the amount of $1 for 1 year, and the number of terms is equal to the number of years; therefore, the amount of an annuity is found by summing the terms of a geometrical progression, when the first term, the number of terms, and the ratio, are given. This may be done by the following: RULE From the amount of $1 for 1 year, raised to a power whose exponent is equal to the number of years, subtract $1; divide the remainder by the interest of $1 for 1 year; then multiply the annuity by this quotient. Note.—The different powers of the amount of $1 for one ycar, may be taken from the table under Art. 68. EXAMPLES. 1. What is the amount of an annuity of $200, which has been forborne 14 years, at 6 per cent., compound interest? From table under Art. 68, we find the 14th power of the amount of $1 for one year, at 6 per cent., to be $2.260904; subtracting $1, and dividing the remainder by $0.06, the interest of $1 for one year, we get 21•01506; then multiplying $200, the annuity, by 21•01506, we find $4203:012 for the amount required. 2. Suppose a person, who has a salary of $700 a year, payable quarterly, to allow it to remain unpaid for 4 years; how much would be due him, allowing quarterly compound interest, at 12 per cent. per annum ? Ans. $3527-453. 3. What is due on a pension of $150 a year, payable half-yearly, but forborne 2 years, allowing half-yearly compound interest, at 6 per cent. per annum ? Ans. $313.772. 4. What is due on a pension of $350 a year, payable quarterly, but forborne 21 years, allowing quarterly compound interest, at 12 per cent. ? Ans. $1003.088. Questions under this rule may be easily wrought by the following table, which shows the amount of an annuity of $1, or £1, forborne for any number of years not exceeding 30, at 3, 4, 5, and 6 per cent., compound interest. |