1. The Principal, Ratio, and Time given, to find the Interest. RULE.-Multiply together-the decimal expressing the rate yearly; the time, in years and the decimal of a year; and the principal; the product will be the interest. NOTE.-In the preceding rule consider 365 days the year, and 30 days the month; precision in casting interest requiring it. Different, though incorrect rules, have been adopted by differ ent authors, in the preceding case, as also is the case with a great portion of the rules in a common work on arithmetic, by those authors, the subdivisions of this division more especially: such matter has been excluded from this work entirely. Many of those subdivisions, which follow, such as Commission, Brokerage, Insurance, &c., you have simply to multiply the principal by the ratio for the rule. EXAMPLES. 1. What is the interest of $345. 50 cts. for 3 years, 6 months, and 15 days, at 6 per cent. ? OPERATION. 345.50, principal. .06, ratio. 20.7300 EXPLANATION. The principal is multiplied by the rate, .06 per cent., which gives the interest for 1 year; then that product by Time, 3.743 (3 yrs. 6 mo. 15 d.) the 3 years, and the 6 months $77.59239, the interest. with the days, both reduced to the decimal of a year. 2. What is the amount of $284 for 2 years, 2 months, and 2 days; simple interest, 8 per cent.? (int. +$284.) 3. What is the interest of $284 for 5 years, 11 months, 7 days, at 7 per cent.? NOTE.-At 1 per cent., any sum of money will double in 100 years, or 36525 days. At 2 per cent., any sum of money will double itself in half of 100 years, or 18262.5 days, and thus of any rate per cent. Then if you should divide the number of days thus found, by the given sum, the quotient will be the interest for one day, which multiplied by the number of days of a given time in a question, the product will be the interest as before. Or, divide the number of days in 100 years by the ratio, the quotient is the sum of money which will gain $1 a day at that rate. Therefore, if any sum of money, upon which interest is to be computed is less than the quotient, divide the given number of days by that number which shows how many times less the given sum is, than the quotient, and this quotient will be the interest. But multiply the given number of days, if the given sum is more than the required sum at the given rate, by that number which shows how many times larger the given sum is, than the required sum for the known rate, and the product will be the required interest. 4. What is the interest of $67.58 for 3 years, 8 months, and 10 days, at 6 per cent.? Thus, 67.58-6000=.011263}, int. for 1 day, because in 6000 days, 360 days for a year, any sum of money will double, at 6 per cent. Then, 3 yrs. 8 mo. 10 d. 1330 d., X.011263=$14.9802331, Ans. 5. What is the interest of $60 for 150 days, at 6 per cent.? Thus, 150-100-$1.50 cts., Ans. Reason.-$6000 gain $1 a day; in 150 days, it would gain $150. But $60 are 100 times less than the $6000; therefore, $60 would gain, in 150 days, 100 times less than $6000 would gain in that time, viz. $1. 50 cts. However, this is not the fact, if we aim at precision, for in the two preceding and the following questions, if we call 36525 days the 100 years, and then proceed by the foregoing principle, we then have the true result by the same method. It is also believed to be the best. 6. What is the interest of $9675.75 for 120 days, at 6 per cent.? First, 9675.75-6000+3000+600+75+.75, for the sum of the The interest of $6000 whole is equal to all its parts. Another method is given by this operation. $120. NOTE 1.-In the preceding questions, had the rate been more or less, than the 6 per cent., you would have first calculated the interest as for 6 per cent., and then had it been, say for 9 per cent., you would have added half the interest thus found to itself, or would have subtracted it, had it been half less than 6 per cent., &c. &c. NOTE 2.-The preceding operations for casting interest are entirely new and different from any others, and is thought to be far superior to any yet offered, for by these methods we combine brevity with perspicuity, and begin at the foundation of the subject. NOTE 3.-Examples may be supplied to the foregoing on interest, if thought necessary. 2. Principal, Time, and Amount given, to find the Ratio. Q. At what rate per cent. will $950.75 amount to $1235.975 in 5 years? RULE.-The amount, $1235.975-950.75-$285.225, the interest for the 5 years. Then, the principal, $950.75X5 yrs. 4753.75. 3. Ratio, Amount, and Principal given, to find the Time. Q. In what time will $248.39 amount to $270.7451, at 6 per cent.? RULE. From the amount, $270.7451 Take the principal, $248.39 248.39X.06 14.9034) $22.3551 ( 1.5 yr., Ans. 4. The U.S. Courts and the Courts of the several States, with the exception of Connecticut, Vermont, and New Jersey, have adopted the following rule for estimating interest on notes, bonds, &c., when partial payments have been made: RULE.-Cast the interest to the time when the payments shall at least be equal to the interest, then discharge the interest from the payment, subtract the excess, if any, from the principal, and cast the interest on the new principal as before, and so on. Questions.-What is Simple Interest?-Give an Ex. Give the proportion. What is the lawful interest of New York?-Of Louisiana?-How do you write the reward?-Why?-What do you understand by per cent.? What is the principal?-The ratio?-The amount?-Repeat the rule to the first case. When is it you use this rule?-Why?-What do you consider for sake of precision?-Are there other rules by different authors for the first case? Are those rules to be considered correct?-How respecting such rules in this work?-Are the rules in most works on arithmetic generally correct?-For Commission, Brokerage, &c., how should you proceed?-How would you find the required days in which any sum of money will double?-Then what should you do?-Or, you may do what?-Give the next step. Give the last step. Give the reason. aim at precision, what should you call 100 years? Then how should you proceed? What is it?-What do you understand by the sixth example?Why?-How do you proceed when the interest by that method is anore than 6 per cent.?-If less, how?-Why are these methods best?-What is the 2d case?-What is the rule?-The 3d case?-The rule?-What is the 4th case?-The rule?-Lastly, what should you observe? If we Example. A had a note against B for $1166.666, dated May 1, A. D. 1796, upon which were the following payments, viz.: 1. Dec. 25, 1796, $166.666. What is due Aug. 3, 1801? 4. June 14, 1799, $333.333. 5. Apr. 15, 1800, $620. Set down the sum upon which the interest is to be cast, with the time, interest, payments, and the excess of payments above the interest of the sum, in columns, as in the following: Principal. TABLE OF OPERATION. Time. Interest. Payments. Excess. $1166.666 7 mo. 24d. $45.499 $166.666 $121.167 1. What sum of money will produce as much interest in 3 years, as $210.15 can produce in 5 years? First, 34 years 39 months, and 5 yrs. 65 months. 2. In what time will the interest of $72.60 equal that of $15.25 for 64 days, at any rate of interest? Int. of $15.25 for 64 days, at 8 per cent., is $0.21399, and the int. of $72.60 for 365 days, at 8 per cent., is $5.808. Then, as 5.808: 365:: 21399: 13,5135 days, Ans. 3. The interest of a certain sum, at simple interest, for 16 yrs.,. at 5 per cent. yearly, remaining unpaid, wanted but $9.32 of the principal. What is the sum? $100 on interest for 16 years, at per cent., is equal to $180, and 200-180-$20. Then, as 20: 100 :: 9.32: $46.60, Ans. 4. If $100 in 5 years be allowed to gain $20.50, in what time will any sum double, at the same rate of interest? Int. of $100 for 5 years, at 6 per cent., is $30, simple interest. 30: 6:20.50:41 per cent. 6:16:41:241 years, Ans. Then, as COMPOUND INTEREST. It is the interest on the interest and the sum of money which has become due, and remains unpaid. RULE.-Make the amount at the time interest is due, a new principal, on which cast the interest to the time when interest is again due, and so on, and finally subtract the first principal from the last amount. Or, thus; The required amount of the several amounts for the several years, is the last term of an increasing series of Continual Proportionals, whose first term is the principal, whose ratio is the amount of $1 for 1 year, and whose number of terms is the number of years plus 1. See Continual Proportionals. Questions.-What is Compound Interest?-How do you find it? Example.-If one mill had been put at interest at the commencement of the Christian era, what would it amount to, Jan. 1, 1837, at compound interest, supposing the principal to double every 12 years? 7153-1 *Thus, 2×2 091061972.992, Ans. $11417981541647679048466287755595961 NOTE. It would take the present inhabitants of our globe more than a million of years to count it; and its value in pure gold, formed into a globe, would be many million times larger than all the bodies that compose the solar system, however incredible it may appear. DISCOUNT. It is an allowance made for the payment of a sum of money before it becomes due, and is the difference between that sum due some time hence, and its present worth. Or, it is an allowance for advancing money on notes, bills, &c., payable at a future day. The present worth of any sum or debt due some time hence, is such a sum as, if put to interest, would, in that time and at that rate per cent. for which the discount is to be made, amount to the sum or debt then due. * Respecting the 153-1, see Involution. |