PROBLEMS IN INTEREST. ART. 205. A PROBLEM in arithmetic is a question, or proposition, which requires some unknown truth to be investigated. ART. 206. In the preceding questions in interest, five terms or things have been mentioned; viz. the Interest, Amount, Rate per cent., Time, and Principal. The investigation of these involves five problems: I. to find the interest; II. to find the amount; III. to find the rate per cent.; IV. to find the time; V. to find the principal. With one exception, any three of the preceding terms being given, a fourth may be found by the rules deduced from the solution of the problems. But if the rate per cent., time, and amount are given, an additional rule is necessary to find the principal, which will form a sixth problem; but from its connection with Discount, its solution will be deferred until that subject is considered. The Problems I. and II. have already been examined, and we now proceed to an examination of those remaining. ART. 207. Problem III. To find the rate per cent., principal, interest, and time being given. the Ex. 1. The interest of $300 for 2 years is $48; what is the rate per cent. ? OPERATION. $300 .02 $6.00) 4 8.00 (8 per cent. Ans. 8 per cent. We find the interest on the principal for 2 years at 1 per cent., and divide the given interest by it. Since the interest of $1 at 1 per cent. for 2 years is 2 cents, the interest of $300 will be 300 times as much, equal to $6. Now if $6 is 1 per cent., $48 will be as many per cent. as $6 is contained times in $48, which gives 8 per cent. for the answer. RULE. Divide the given interest by the interest of the given sum at 1 per cent. for the given time, and the quotient will be the rate per cent, required. QUESTIONS. Art. 205. What is a problem in arithmetic?-Art. 206. How many terms or things have been given in the preceding questions in interest? Name them. What does an investigation of these terms involve? Name them. How many terms are given in each problem in order to find a fourth? What two problems have been examined?-Art. 207. What is Problem III. ? Explain the operation. What is the rule for finding the rate per cent., the principal. interest. and time being given? EXAMPLES FOR PRACTICE. 2. The interest of $250 for 1 year, 3 months, is $28.125; what is the rate per cent.? Ans. 9 per cent. 3. If I pay $8.82 for the use of $72 for 1 year, 9 months, what is the rate per cent. ? Ans. 7 per cent. 4. A note of $500, being on interest 2 years, 6 months, amounted to $550; what was the rate per cent.? Ans. 4 per cent. ART. 208. Problem IV. To find the time, the principal, interest, and rate per cent. being given. Ex. 1. For how long a time must $300 be on interest at 6 per cent. to gain $ 36? OPERATION. $300 $18.00) 3 6.00 (2 years. 36.00 Ans. 2 years. We find the interest on the given principal for 1 year, by which we divide the given in terest. Since the interest of $1 for 1 year is 6 cents, the interest of $300 will be 300 times as much, equal to $18. Now, if it require 1 year for the given principal to gain $18, it will require as many years to gain $36 as $18 is contained times in $36. Thus, $36 $182 years for the answer. RULE. Divide the given interest by the interest of the given principal for 1 year, and the quotient is the time. EXAMPLES FOR PRACTICE. 2. If the interest of $140 at 6 per cent. is $42, for how long a time was it on interest? Ans. 5 years. cent. per 3. How long a time must $ 165 be on interest at 6 to gain $ 14.85? Ans. 1 year, 6 months. 4. How long must $98 be on interest at 8 per cent. to gain $25.48 ? Ans. 3 years, 3 months. 5. A note of $ 680 being on interest at 4 per cent. amounted to $727.60; how long was it on interest? Ans. 1 year, 9 months. QUESTIONS. Art. 208. What is Problem IV.? Explain the operation. What is the rule for finding the time, the principal, interest, and rate per cent. being given? ART. 209. Problem V. To find the principal, the interest, time, and rate per cent. being given. Ex. 1. What principal at 6 per cent. will gain $ 36 in 2 years? OPERATION. .06 int. of $1 for ly. 2 .12)3 6.0 0($ 3 0 0 principal. it will require a principal of as are contained times in $36. answer. RULE. Ans. $300. We find the interest of $1 for 2 years, by which we divide the given interest. principal of $ 1 to gain 12 cents, Since it requires 2 years for a many dollars to gain $ 36 as 12 cents Thus, $36.00.12 $300 for the - Divide the given interest or amount by the interest or amount of $1 for the given rate and time, and the quotient is the principal. EXAMPLES FOR PRACTICE. 2. What principal will gain $24.225 in 4 years, 3 months, at 6 per cent.? Ans. $95. 3. What principal will gain $5.11 in 3 years, 6 months, at 8 per cent. ? Ans. $18.25. 4. The interest on a certain note at 9 per cent. in 1 year and 8 months amounted to $42; what was the full amount of the note? Ans. $280. § XXIV. COMPOUND INTEREST. ART. 210. Compound InteREST is interest on the principal and interest, when the interest is not paid at the end of the year, or when it becomes due. The law specifies, that the borrower of money shall pay the lender a certain sum for the use of $100 for a year. Now, if he does not pay this sum at the end of the year, it is no more than just that he should pay interest for the use of it as long as he shall keep it in his possession. The computation of compound interest is based upon this principle. QUESTIONS. Art. 209. What is Problem V.? Explain the operation. What is the rule for finding the principal, the interest, time, and rate per cent. being given ? -Art. 210. What is compound interest? On what principle is it based? ART. 211. To find the compound interest of any sum of money at any rate per cent. for any given time. Ex. 1. What is the compound interest of $500 for 3 years, 7 months, and 12 days, at 6 per cent.? Ans. $117.54,1. $500 3 0.0 0 500 We first multiply the principal by the interest of $1 for 1 year, and add the interest thus found to the principal for the amount, or new principal. We then find the interest on this amount for 1 year, and proceed as before; and so also with the third year. For the months and days we find the interest on the amount for the last year, and, adding it as before, we subtract the original principal from the last amount for the answer. RULE. - Find the interest of the given sum for one year, and add it to the principal; then find the amount of this amount for the next year; and so continue, until the time of settlement. If there are months and days in the given time, find the amount for them on the amount for the last year. Subtract the principal from the last amount, and the remainder is the compound interest. QUESTIONS.- -Art. 211. Explain the operation in computing compound inWhat is the rule? terest. NOTE.-1. If the interest is to be paid semiannually, quarterly, month. ly, or daily, it must be computed for the half-year, quarter-year, month, or day, and added to the principal, and then the interest computed on this, and each succeeding amount thus obtained, up to the time of settlement. 2. When partial payments have been made on notes at compound in terest, the same rule is adopted as given in Art. 204. EXAMPLES FOR PRACTICE. 2. What is the compound interest of $761.75 for 4 years? Ans. $199.94,1. 3. What is the amount of $ 67.25 for 3 years, at compound interest? Ans. $80.09,5. 4. What is the amount of $78.69 for 5 years at 7 per cent. ? Ans. $110.36,4. 5. What is the amount of $128 for 3 years, 5 months, and 18 days, at compound interest? Ans. $156.71,7. 6. What is the compound interest of $76.18 for 2 years, 8 months, 9 days? Ans. $12.96,7. ART. 212. There is a more expeditious method of computing compound interest than the preceding, by means of the following TABLE, Showing the amount of $1, or £1, for any number of years not exceeding 20, at 3, 4, 5, 6, and 7 per cent., compound interest. 1 092727 5 6 10 11 12 13 14 15 16 17 18 19 20 1.124864 1.157625 1.191016 1.225043 123456789 10 1.169858 1.215506 1.262476 1.310795 1.159274 1.216652 1.276281 1.338225 1.402552 1.194052 1.265319 1.340095 1.418519 1.500730 1.229873 1.315931 1.407100 1.503630 1.605781 1.266770 1.368569 1.477455 1.593848 1.718186 1.304773 1.423311 1.551328 1.689478 1.838459 1.343916 1.430284 1.628894 1.790847 1.967151 1.384233 1.539454 1.710339 1.898298 2.104852 11 1.425760 1.601032 1.795856 2.012196 2.252191 12 1.468533 1.665073 1.885649 2.132928 2.409845 13 1.512589 1.731676 1.979931 2.260903 2.578534 14 1.557967 1.800943 2.078928 2.396558 2.750032 15 1.604706 1.872981 2.182874 2.540351 2.952164 16 1.652847 1.947900 2.292018 2.692772 3.158815 17 1.702433 2.025816 2.406619 2.854339 3.379932 18 1.753506 2.106849 2.526950 3.025599 3.616528 19 1.806111 2.191123 2.653297 3.207135 3.869685 20 QUESTIONS.If the interest is to be paid semianually, quarterly, &c., how is it computed? How, when partial payments have been made? - Art. 212. Explain the method of computing compound interest by means of the table. |