the At 1 per cent. for the given time, 30 dollars will gain 22 cents 5 mills; rate, therefore, is so many times 1 per cent. as 22 cents 5 mills is contained in the whole gain, which is $1.35, i. e. $1.35 $0.225—.06, or 6 per cent. Ans. Hence, III. The principal, interest and time being given, to find the rate. RULE. Divide the given interest by the interest on the given principal, at one per cent. for the given time, and the quotient will be the rate per ceut. 2. If the interest on 575 dollars for 8 months be 23 dollars, what is the rate per cent.? Ans. 6 per cent. 3. If the interest of 1600 dollars for 1 year and 3 months, be 100 dollars, what is the rate? Ans. 5 per cent. 175. 1. If the interest on 30 dollars at 6 per cent. per annum, be 1 dollar and 35 cents, what is the time? The interest on 30 dollars for 1 year at 6 per cent. is 1 dollar and 80 cents. Now, if the given interest be divided by the interest on the given principal for one year, the quotient will evidently be the number of years that principal was on interest-$1.35-$1.80 0.75yr. 4 months (145), the answer. Therefore, IV. The principal, rate and interest being given, to find the time. RULE.-Divide the given interest by the interest of the given principal for 1 year at the given rate, and the quotient will be the time in years and decimal parts. 2. If the interest on 575 dollars at 6 per cent. be 23 dollars, what is the time? Ans. 8 months. 3. If the interest of 1600 dollars at 5 per cent. be 100 dollars, what is the time? Ans. 1.25yr. lyr. 3mo. 2. Commission and Ensurance. DEFINITIONS. 176. Commission is an allowance of so much per cent. to an agent for transacting business for another. Insurance is a contract by which certain persons, or companies, agree to make good losses of property by fire, storms, &c. in consideration of the payment to the insurer of so much per cent. on the value of the property insured. Premium is the sum paid by the owner of the property for the insurance. The written contract of insurance is called a policy. The policy should always cover a sum equal to the estimated value of the property insured, together with the premium; that is, a policy to secure the payment of 100 dollars at 2 per cent. must be made out for 102 dollars, RULE. 177. Multiply the sum on commission, or insurance, by the rate per cent., and the product will be the commission, or premium (162). 178. The methods of computing interest on notes and bonds differ in different places. Those in most general use are the following: I. Find the amount of the principal up to the time of payment, and also the amount of the endorsements from the time they were made up to the time of payment; deduct the latter from the former, and the remainder will be the sum due. This method is evidently erroneous; for suppose a note be given for 100 dollars with interest, and 6 dollars be paid at the end of each year for four years, which is endorsed on the note. Now the interest of the principal for this time is 24 dollars, just equal to the sum of the payments; but by this method the several payments all draw interest from the times they are made, the first 3 years, the second 2, and the third 1.1.08+72+36= $2.16, which goes towards paying the principal, and in this way any debt would in time be extinguished by the payment of the interest annually. II. Compute the interest up to the time of the first payment, and if the payment exceed the interest, deduct the excess from the principal, and cast the interest on the remainder up to the second payment, and so on. If the payment be less than the interest, place it by itself, and cast the interest up to the next payment, and so on till the payments exceed the interests, then deduct the excess from the principal, and proceed as before. By this method the interest is supposed to be always due whenever a payment is made; and although, on that account, it is not always perfectly correct, it is perhaps sufficiently so for common use. This method is extonsively used, and is established by law in Massachusetts. III. If the contract be for the payment of interest annually, the interest becomes due at the end of each year, and if it be not extinguished by payment, interest is to be cast upon that interest, from the time it becomes due up to the time of payment. If the contract be for a sum payable at a specified time, no interest is due till the time of payment arrives, and endorsements made before that time, are to be applied exclusively to the principal. After the debt falls due, the interest is to be extinguished an nually, if the payments are sufficient for that purpose. These last are the principles upon which interest is allowed by the courts of law in Vermont, and upon these are founded the two following rules: RULE I. When the contract is for the payment of interest annually, and no payments have been made, find the interest of the principal for each year, separately, up to the time of payment; then find the interest of these interests, severally, from the time they become due up to the time of payment, and the sum of all the interests added to the principal will be the amount: but if payments have been made, find the amount of the principal, and also the amount of the payments to the end of the first year; subtract the latter amount from the former, and the remainder will be the principal for the second year; proceed in the same way from year to year up to the time of payment. NOTE. It will sometimes happen that, when a note has endorsements, there will be years in which no payments are made; for which years the interest is to be found by the former part of the rule; and also when the amount of the payment is less than the interest of the principal, subtract the amount from that interest, and find the amount of the remainder up to the final payment. QUESTIONS FOR PRACTICE. 1. A's note to B for 100 dollars, with interest annually, at 6 per cent. was dated January 1, 1820; what was due, principal and interest, January 1, 1824 ? 2. B's note to C for 50 dollars, with interest annually, was dated Nov. 20, 1822, on the back of which were the following endorsements, viz. May 20, 1823, received 14 dollars, and Feb. 26, 1824, 30 dollars; what was due Jan. 2, 1825? 2 prin. 38.58 3d prin. 9.574 3. D's note to E for $1000, with interest annually, was dated May 5, 1822, on which the following payments were made, viz. Nov. 17, 1822, 300 dollars; April, 23, 1823, 50 dollars, and August 11, 1823, 520 dollars: what was due June 5, 1824 ? Ans. $201.713. 4. C's note to D for 200 dollars, with interest annually, was dated June 15, 1821, on the back of which was endorsed, Sept. 15, 1821, 4 dollars, and Jan. 21, 1823, 15 dollars: what was due June 15, 1824? Ans. $217.224. RULE II. When the contract is for a sum payable at a specified time, with interest, and payments are made before the debt becomes due; find the interest of the principal up to the first payment, and set it aside; subtract the payment from the principal, and find the interest of the remainder up to the next payment, which interest set aside with the former, and so on up to the time the debt becomes due; and the sum of the interests added to the last principal, will be the amount due at that time: after the debt falls due, the interest is to be extinguished annually, if the payments are sufficient for that purpose. QUESTIONS FOR PRACTICE. 1. E's note to F for $75.25, payable in 2 years, with interest, was dated May 1, 1822, on which was endorsed, Jan. 13, 1823, $25.25; what was due May 1, 1824 ? Compound Interest. 179. What will be the interest of $40 for 3 years, at 6 per cent., the interest being added to the principal at the end of each year? The interest of 40 dollars for 1 year is (40X.06-) $2.40, and $2.40+ 40.842.40, the principal for the second year, the interest of which is (42.40.06) $2.544 for the second year, and $2.544+42.40 $44.944, the principal for the third year, the interest of which is (44.944.06) $2.696, and $2.696+44.944 $47.64, the amount of principal and interest at the end of three years, from which subtracting 40 dollars, the first principal, we have (47.64 40.) $7.64 for the interest of 40 dollars for years. Interest computed upon interest, as above, is called Compound Interest. 180. COMPOUND INTEREST is that which arises from making the interest a part of the principal at the end of each year, or stated time for the interest to become due. RULE. Find the amount of the given principal for the first year, or up to the first stated time for the interest to become due, by simple interest, and make the amount the principal for the next year, or stated period; and so on to the last. From the last amount subtract the given principal, and the remainder will be the compound interest required. QUESTIONS FOR PRACTICE. 1. What is the compound interest of $125 for 2 years and 6 months, at 6 per cent.? $125. principal. .06 rate. 7.50 int. for 1st. yr. 125. prin. added, 132.50 am't. for 1 yr. .06 7.9500 int. for 2d yr. 132.50 140.45 .03 4.2135 140.45 144.6635 125. 2. What is the compound interest of $100 for 4 years, at 6 per cent.? Ans. $26.247. 3. What is the compound interest of $200 for 1 year, at 6 per cent., due every four months? Ans. $12.241. 4. What is the amount of $236 at 6 per cent., compound interest, for 3 years, 5 months, and 6 days? Ans. $288.387. 5. What is the amount of $150 at 6 per cent., compound interest, for 2 years, the interam't. for 2d yr. est becoming due at the end of every 6 months? int. for 6 mo. am't. for 2 yrs. $19.663 com. int. required. Ans. $168.826. 6. What is the compound interest of $768 for 4 years, at per cent.? Ans. $201.58. 7. What is the compound interest of $560 for 3 years and 6 months, at 6 per cent.? Ans. $126.977. |