months, we must add the interest on 1 year's interest for the sum of the above intervals, 7 years. Exercise XLII 1. A note of $350 drawing 5% interest, payable annually, runs 4 years, 6 months. Find the amount due at maturity, if no payments have been made. 2. A man bought $500 worth of goods on 9 months' credit; he paid for them at the end of 3 years, 3 months. Allowing 6% interest, payable annually, how much was due? 3. A gentleman holds six $1000 railroad bonds, due in 3 years, interest 6% payable semi-annually: no interest having been paid, what amount is owing him when the bonds mature?-R. N. H., p. 260. NOTE. Frequently annual or periodic interest is payable semiannually or quarterly. 4. Find the annual interest on $800 for 1 year, 9 months, at 5% per annum, payable quarterly. 5. The annual interest on a sum of money for 4 years, 6 months, at 5%, paid annually, was $85.75. What was the sum? SUGGESTION. Find the annual interest on $1. 6. Find the amount due at the end of 5 years, 9 months, of a note of $1200, bearing interest at 6%, payable annually, if no payments have been made. REMARK.-In some states, the law provides that unpaid annual interest shall bear interest at the legal rate. In the following problem, the unpaid interest bears the legal rate of interest. 7. In a state whose legal rate is 6%, H makes a note of $1250 for 3 yr. 3 mo., with interest at 8%, payable semiannually; he pays no interest; find the amount due at maturity.-S. & K., p. 197. IV. COMPOUND INTEREST 351. Compound interest is the interest that accrues by making the interest, due at the close of any interval, a part of the interest-bearing principal for the next succeeding interval. EXAMPLE Find the compound interest on $1 for 3 years at 5%, payable annually. SOLUTION: $1.00 principal (P). .05 $0.05 1.00 $1.05 .05 = = = = rate (R). int. for 1st yr. = (1 + R) = amount for 1st yr. (A). (1 + R)2 = amount for 2d yr. int. for 3d yr. (1 + R)3 = amount for 3d yr. the compound int. for 3 years. A careful study of this example will show that the compound amount of $1, at any rate, for any number of years, equals (1+R) raised to the power indicated by the number of years. Then to get the amount of n dollars, take n times the amount of $1. Using the initial letters, we have we have the following formulas: = Write the rule corresponding to each formula. 352. The interest interval is generally a year, half-year, or quarter-year. The length of the interval is indicated by inserting annually, semi-annually, or quarterly. - √1 + interval rate 1. The half-interval rate Thus, if the annual rate is 21%, the semi-annual rate would √1.21 11.10 1.10, or 10%. = This fact may be illustrated thus: 1. Am't of $100 for 1 yr. at 21% annually = 1.21 × $100 = $121. 2. Am't at 21% semi-annually = √1.21× √1.21 × $100 = $121. 3... the semi-annual multiplier = = V1.21, and the rate = √1.21-1. Similarly, it can be shown that the quarterly multiplier 1.21, and the rate √1.21 — 1. = TABLE Showing the amount of $1 at compound interest from 1 year to 20 years. Yr. 2 per cent. 3 per cent. 34 per cent. 4 per cent. 5 per cent. 6 per cent. 1 1.025 1.04 2 1.06 1.03 1.05 1.124864 1.191016 1.215506 1.262477 1.338226 2345 3 1.312087 1.384234 1.45997 .34488 1.425761 1.511069 13 1.378511 1.468534 1.563956 14 1.412974 1.51259 1.448298 1.159693 1.194052 1.229255 1.265319 Yr. 1 1.539454 1.710339 1.898299 1.601032 1.795856 2.012197 1.665074 1.885649 2.132928 1.618695 1.731676 1.979932 2.260904 1.557967 1.675349 1.800944 2.078928 | 2.396558 16 1.484506 1.604706 1.733986 1.872981 2.182875 2.540352 18 17 1.521618 1.652848 1.794676 1.947901 2.292018 1.559659 1.702433 1.857489 2.025817 2.406619 1.753506 1.922501 2.106849 2.52695 2.191123 2.653298 2.692773 2.854339 3.0256 3.207136 19 1.59865 20 1.638616 1.806111 1.989789 1.340096 1.418519 1.50363 1.593848 1.4071 1.477455 1.423312 1.551328 1.689479 1.480244 1.628895 1.790848 7 per cent. 8 per cent. 1.07 1.08 1.225043 1.259712 1.310796 1.360489 1.402552 1.469328 9 per cent. 10 per cent. 11 per cent. 12 per cent. 1.10 1.11 1.12 1.09 1.21 1.2544 1.2321 1.295029 1.331 1.404908 1.411582 1.573519 1.4641 1.51807 1.538624 1.762342 1.50073 1.586874 1.718186 1.85093 9 1.838459 1.999005 10 1.967151 2.158925 2.367364 1.6771 1.771561 2.593742 11 2.104852 2.331639 2.580426 2.853117 12 2.252192 2.51817 2.812665 3.138428 2.409845 2.719624 3.065805 13 3.452271 4.177248 14 2.578534 2.937194 3.341727 3.797498 4.71712 1.870414 1.973822 2.07616 2.210681 2.304537 2.475963 2.558036 2.773078 2.83942 3.105848 3.151757 3.478549 3.49845 3.895975 3.883279 4.363492 4.31044 4.887111 4.784588 5.473565 4.594973 5.310893 5.05447 5.895091 5.559917 6.543551 6.115909 7.263342 6.7275 8.062309 6.130392 6.86604 7.689964 8.61276 9.646291 Exercise XLIII Find the compound interest on: 1. $500 for 4 years at 5%, payable annually. 2. $1700 for 2 years at 4%, payable semi-annually. 3. $450 for 5 yrs., 4 mos., at 6%, payable semi-annually. Find the compound amount on: 4. $800 for 2 years at 6%, compounded semi-annually. 5. $1848 for 5 years, 4 months, 15 days, at 7%, compounded annually. 6. Find the difference between the simple and the compound interest on $180 for 2 years at 8%, if the interest is compounded quarterly. 7. What principal will amount to $763.205 in 2 years, 9 months, at 6%, compounded annually? 8. Find the compound interest on $500 for 3 years, 2 months, at 6%, compounded annually. = SOLUTION: By referring to the table (page 172), it will be seen that the amount of $1 for 3 yr. at 6% $1.191016. Computing the interest on this amount for 2 mo., the amount of $1 for 3 yr. 2 mo. is $1.202926. Therefore, the amount of $500 500 $1.202926 $601.463. $601.463 $500 = $101.463, compound interest. = = Making use of the table, find the compound interest on: 9. $535 for 3 yr. 5 mo. at 7%. 10. $750.80 for 6 yr. 7 mo. at 6%. 11. $672.28 for 2 yr. 3 mo. 18 da. at 6%. 12. What rate per cent per annum, compounded semiannually, is equivalent to 44%, compounded annually? |