OPERATION. We find the compound in$ 2370 • .790 = $ 3000 Ans. terest of $ 1 for the given time, and at the given rate; and proceed as in like cases in simple interest (Art. 359). RULE. - Divide the given compound interest by the compound interest of $ 1 for the given time at the given rate. EXAMPLES. 2. What principal, at 7 per cent. compound interest, will produce $ 205.90 in 6 years and 6 months ? Ans. $372.16. 3. What sum of money, at compound interest, will produce $ 1026.54 in 3 years 2 months and 12 days? Ans. $ 5000. 4. What sum of money must be invested at compound interest at a semiannual rate of 31 per cent. to produce $ 857.25 in 154 years ? Ans. $ 450. 379.° To find the RATE PER CENT., the principal, the interest, or the amount, and the time being given. Ex. 1. At what rate per cent. must $ 500 be at compound interest to become $ 703.55 in 7 years ? Ans. 5 per cent. OPERATION $ 703.55 ; 500 $ 1.4071, which for 7 years, in the table, denotes a rate of 5 per cent. Since $ 500 becomes $ 703.55 in 7 years at the required rate, $1 in the same time at the same rate will amount to do as much, or $ 1.4071. Corresponding to this amount of $ 1 for the given time, we find in the table (Art. 377) 5 per cent., the rate required. RULE. Divide the amount by the principal, and the quotient will be the amount of $1 for the given time and the required rate; and in the table, over this amount, may be found the rate per cent. required. Note. — If the given time contains a part over an exact number of periods, look in the table, against the number denoting the whole periods or years in the given time, for that amount of $1 which comes the nearest to the one found by dividing. Then see if the approximate amount, increased by its rate of interest for the fractional period, will equal the other amount; if so, the rate corresponding to the approximate amount will be the rate per cent. required; if not, the rate of the approximate amount will be as much greater or smaller than the required rate, as the interest added to the approximate amount is greater or smaller than that required to produce the amount found by dividing. The rule can only be well applied when the rate per cent. sought is within the limits of the table. EXAMPLES. 2. At what rate per cent. will $ 400 amount to $ 640.405, at compound interest, in 12 years? Ans. 4 per cent. 3. At what rate per cent. must $ 2500 be loaned, to produce $ 2096.147 compound interest, in 9 years? Ans. 7 per cent. 4. At what rate per cent. will any sum of money double itself at compound interest in 11 years? Ans. 6 per cent. 5. At what rate per cent. will $10,000 amount to $ 31479.70 in 19 years and 8 months ? Ans. 6 per cent. 380.° To find the TIME, the principal, the compound interest, and the rate per cent. being given. Ex. 1. In what time will $ 500, at 7 per cent. compound interest, amount to $ 655.398 ? Ans. 4 years. OPERATION. 655.398 • 500 $ 1.310796, which, at the given rate in the table, denotes 4 years' time. Since $500 amounts to $ 655.398 at 7 per cent. in the required time, $ 1 at the same rate, in the same time, will amount to do as much, or $ 1.310796. Corresponding to this amount of $1 at the given rate, we find in the table (Art. 377) 4 years, the time required. RULE. Divide the amount by the principal, and the quotient will be the amount of $ 1 at the given rate for the required time; and in the table, against this amount, may be found the time required. NOTE. - If the required time cannot be found exactly in the table, the number against that amount of $1 which under the given rate is next less than the amount found by dividing, will denote the whole periods or years. Then, find the fractional period or part of a year, by dividing 1 whole period or year by the ratio of the difference between the amount corresponding to the whole periods or years and that found by dividing to the difference between the former of the amounts and that next larger in the table; and the value of the fraction obtained as the result may be expressed in months or days, or both. EXAMPLES. Ans. 12 years. 2. In what time will $ 400 amount to $ 640.405 at 4 per cent. compound interest ? 3. In what time will $ 6000 amount to $ 9021.78 at 7 per cent. compound interest ? 4. In what time will any sum double itself at 5 per cent. compound interest ? Ans. 14y. 2mo. 13d. 5. In what time will any sum double itself at 6 per cent. compound interest ? Ans. Ily. 10mo. 20+d. 6. A gentleman has deposited $ 450, for the benefit of his son, in a savings bank, at compound interest at a semiannual rate of 31 per cent. He is to receive the amount as soon as it becomes $ 1781.664. Allowing that the deposit was made when the son was 1 year old, what will be his age when he can come in possession of the money ? Ans. 21 years. DISCOUNT AND PRESENT WORTH. 381. DISCOUNT is an allowance or deduction made for paying money before it is due. 382. The PRESENT WORTH is the amount of ready money that will satisfy a debt before it is due. It is equivalent to a principal which, being put at interest, will amount to the debt at the time of its becoming payable. Thus, $ 100 is the present worth of $ 106 due one year hence at 6 per cent.; for $ 100 at 6 per cent. will amount to $ 106 in that time; and $ 6 is the discount. 383. In discount, the rate per cent., the time, and the sum on which the discount is made, are given, to find the present worth, which corresponds precisely to the rate per cent., the time and the amount being given, in either simple or compound interest, to find the principal. 384. The interest or percentage of any sum cannot properly be taken for the true discount; for we have seen (Art. 382) that the interest for one year is the fractional part of the sum at interest, denoted by the rate for the numerator, and 100 for the denominator ; and the discount for one year is the fractional part of the sum on which discount is to be made, denoted by the rate for the numerator, and 100 plus the rate for the de nominator. Thus, if the rate of interest is 6 per cent., the interest for one year is 187 of the sum at interest; but if the rate per cent. of discount is 6, the discount for one year is 16 of the sum on which discount is made. 385. Business men, however, often deduct, or “take off," from the face of a bill or note due at some future time, a greater percentage than the interest would be for the given time at the given rate. Therefore, the true present worth and discount are not obtained by that method, but only a NOMINAL PRESENT WORTH and a NOMINAL DISCOUNT, The true discount is equal to the interest on the true present worth of the debt, while the nominal discount is equal to the interest on the face of the debt. 386. To find the true present worth of any sum, and the discount, for any time, at any rate per cent. Ex. 1. What is the present worth of $ 12.72, due one year hence, discounting at 6 per cent.? What is the discount ? Ans. $ 12 present worth ; $ 0.72 discount. OPERATION. $ 0.7 2, Amount of $ 1, 1.0 6 ) $1 2.7 2 ( $12, Present worth. 106 212 $1 2.7 2, Given sum. Discount. Since $1 is the present worth of $ 1.06 due one year hence, at 6 per cent., it is evident that the present worth of $ 12 72 must be as many dollars as $ 1.06 is contained times in $ 12.72. We thus find the present worth to be $ 12, which, subtracted from the given sum, gives $ 0.72 as the discount. RULE. — Divide the given sum by the amount of $1 for the given time and rate, and the quotient will be the PRESENT WORTH. From the given sum subtract the present worth, and the remainder will be the DISCOUNT, EXAMPLES. year hence ? 2. What is the discount on $ 802.50, at 7 per cent., due one Ans. $ 52.50. 3. What is the present worth of $117.60, due one year hence, at 12 per cent. ? 4. What is the present worth of $ 769.60, due 3 years and 5 months hence ? Ans. $ 638.672. 5. What is the present worth of $ 678.75, due 3 years 7 months hence, at 74 per cent. ? Ans. $ 534.975. 6. What is the discount on $ 600, due 5 years hence, at 5 per cent. ? 7. A merchant has given two notes; the first for $79.87, to be paid January 21, 1836; the second for $87.75, to be paid December 17, 1856. How much ready money will discharge both notes February 10, 1855 ? Ans. $ 154.545. 8. C. Gardner owes Samuel Hall as follows: $ 365.87, to be paid December 19, 1855; $ 161.15, to be paid July 16, 1856; $112.50, to be paid June 23, 1854 ; $ 96.81, to be paid April 19, 1858. What should Hall receive as an equivalent, January 1, 1854 ? Ans. $ 653.40. 9. What is the present worth of $ 67.25 due 3 years hence ? 10. What is the present worth of $ 80.095, due 3 years hence at compound interest ? Ans. $ 67.25. 11. What is the discount on $ 110.364 due 5 years hence, at 7 per cent. compound interest ? Ans. $ 31.677. 387. To find a nominal present worth of any sum due at some future time, and the discount on the same at a given rate, reckon the interest on the face of the debt for the given time and rate, and the same will be the nominal discount; and this discount subtracted from the face of the note will give the nominal present worth. EXAMPLES. 1. I have bought of Paine and Woodard a bill of goods amounting to $960, on six months, but for ready money they take off from the face of the bill, for the time, 5 per cent. What was the amount paid ? Ans. $ 912. 2. How much more is the nominal than the true discount on $ 5000 due one year hence, at 7 per cent. ? 3. When money is worth 6 per cent. a year, how much may be gained by biring money to pay $ 4440 due 6 months hence, allowing the present worth of this debt to be reckoned by deducting the nominal discount? Ans. $ 3.996. |