7. What is the interest of $126 for 5 yr. 6 mo. 18 d., at 5% ? Ans. $34.961. NOTE.-The years, months and days may be reduced to the decimal part of a year. Thus, 5 yr. 6 mo. 18 d. is 5.55 yr. The interest on $126 for 1 yr. at 5% is $6.30; and for 5.55 yr. it is 5.55 times $6.30, or $34.96. This method is often much the shortest. 4 mo. 4 d., Ans. $2095.19. 8. What is the amount of $1785 for 4 yr. at 4% ? 9. What is the interest of $640.20 for 3 yr. 3 mo. 3d., at 61%. 2. SIX PER CENT. METHOD. Ans. $135.59. In calculating interest 30 days are reckoned to the month, and 12 months, or 360 days, to the year. The interest on $1 for 1 year being 6 cents, for one month it is of 6 cents, or a cent; hence, the interest on a dollar is as many cents as half of the number of months. The interest for 30 days being a cent, or 5 mills, for 1 day it is of 5 mills, or of a mill; hence, the interest on $1 is as many mills as one-sixth the number of days. From the foregoing we derive the following rule to find the interest at 6% per annum: RULE. Write one-half the number of months as cents, and one-sixth the number of days as mills, and multiply the sum of these by the number of dollars. SOLUTION. 5 yrs. 3 mo. = WRITTEN PROBLEMS. 1. What is the interest of $955 for 5 yr. 3 mo. 12 d., at 6% ? Ans. $302.735. Analysis.-5 yr. 3 mo. equal 63 mo. Writing of 63 mo., or 311⁄2 mo., as cents, and of 12 days, or 2 days, as mills, we have for the interest of $1 for the given time $.317, and multiplying this by 955, the number of dollars, we have the required interest, $302.735. = 63 mo. = = .317 955 $.317 $302.735. NOTE. To find the interest at any other per cent., divide the interest at 6% by 6, and multiply by the given rate. 2. What is the interest of $644 for 6 yr. 7 mo., at 6% ? Ans. $254.38. 3. What is the interest of $725 for 9 yr. 5 mo., at 5% ? Ans. $341.35. 4. What is the interest of $650.30 for 6 yr. 3 mo., at 8% ? Ans. $325.15. 5. What is the amount of $720.50 for 3 yr. 9 mo., at 10% ? Ans. $990.69. 6. What are the interest and the amount of $85 for 2 yr. 8 mo. 12 d., at 6% ? Ans. Int., $13.77; amt., $98.77. 7. What are the interest and the amount of $1250 for 3 yr. 7 mo. 15 d., at 8% ? Ans. Int., $362.50; amt., $1612.50. 8. What is the interest on $550 from May 8, 1876, to June 13, 1877, at 6% ? Ans. $36.208. 9. What is the interest on $1116 from Aug. 16, 1873, to Mar. 28, 1875, at 7% ? Ans. $126.294. 10. What are the amount and the interest on $125.40 from Sept. 25, 1876, to Mar. 16, 1878, at 6% ? Ans. Amt., $136.50; int., $11.10. 3. METHOD BY ALIQUOT PARTS. 1. What is the interest of $1250 for 3 yr. 5 mo. 18 d., at 8%? SOLUTION. $1250.08 $100, Ans. $346.67. Explanation. The interest of $1250 for 1 yr., at 8% = $100, and for 3 yr. 3 times $100, or $300; the interest for 3 mo. is of $100, or $25; for 1 mo. it is of $25, or $8.33}, and for 2 mo. 2 times $8.333, or $16.67, and for 18 d., or of a month, § of $8.33}, or $5. Hence, the total interest for 3 yr. 5 mo. and 18 d. is $300+$25+$16.67+$5, or $346.67. Total interest = $346.67. 2. What is the interest of $620 for 6 yr. 8 mo., at 6%? Ans. $248. 3. What is the interest of $800 for 3 yr. 3 mo. 15 d., at 7%? Ans. $184.33. 4. What is the interest of $660 for 2 yr. 2 mo. 2 d., at 9% ? Ans. $129.03. P 5. What is the interest of $300 for 1 yr. 1 mo. 1 d., at 10% ? 4. METHOD BY DAYS. 1. Find the interest of $78 for 90 d., at 6%. Ans. $32.58. Ans. $1.17. Analysis. Since the interest on $1 for 1 mo., or 30 d., at 6% is ct., for 60 d. it is 1 ct.; hence, of the number of days equals the number of cents interest. Multiplying this by 78, the number of dollars, gives the total interest, 117 cts., or $1.17. From this we derive the following RULES. 1. Multiply the number of dollars by the number of days, and divide by 60; the result is the interest at 6% in cents. 2. When at any other per cent. than 6, multiply the number of dollars by the number of days, and this result by the given rate, and divide by 360. 2. What is the interest of $95 for 20 d., at 8% ? 3. What is the interest of $76 for 30 d., at 6% ? Ans. 38 cts. 4. What is the interest of $1254 for 25 d., at 6% ? Ans. $5.221. 5. What is the interest of $96.50 for 36 d., at 6% ? Ans. $.579. 6. What is the interest of $117.35 for 48 d., at 6% ? Ans. $.9388. 7. What is the interest of $60 for 40 d., at 7% ? Ans. $.46%. 8. What is the interest of $125 for 54 d., at 8% ? Ans. $1.50. 9. What is the interest of $1215 for 18 d., at 51% ? Ans. $3.34. 10. What is the interest of $37.50 for 26 d., at 8% ? Ans. $.213. 11. What is the interest of $87.75 for 50 d., at 10%? Ans. $1.22. 12. What is the interest of $118.63 for 45 d., at 7% ? Ans. $1.038+. 13. What is the interest of $78.45 for 1 mo. 6 d., at 6% ? Ans. $.47 +. 14. What is the interest of $1260.60 for 6 mo. 14 d., at 7%? Ans. $47.55+. 229. Exact Interest may be found by reducing the time to days, dividing the interest for one year by 365, and multiplying by the given number of days. Or, find the interest by any of the common methods, and of the result for the common years, or for leap take years. SECTION XI. GENERAL PROBLEMS IN INTEREST. CASE I. Principal, Interest and Time being given, to find the Rate. ORAL EXERCISE. 1. At what per cent. will $50 in 3 yr. give $12 interest? SOLUTION 1.-The interest of $50 for 1 yr. at 1% is 50 cents, and for 3 yr. 3 times 50 cents, or $1.50; to give $12 interest will therefore require a rate of as many per cent. as $1.50 is contained times in $12, or 8%. SOLUTION 2.-At 1% the interest on $1 for 3 yr. is 3 cents, and on $50 it is 50 times 3 cents, or $1.50; it will therefore require as many per cent. as $1.50 is contained times in $12, or 8%. 2. At what per cent. will $200 gain $40 in 4 yr.? yr. 6 mo.? 8. At what per cent. will $560 gain $182 in 3 yr. 3 mo.? 9. At what per cent. will $105 gain $29.40 in 3 10. At what per cent. will $180 gain $49.50 in 5 yr.? 11. At what per cent. will $250 amount to $287.50 in 3 yr.? 12. At what per cent. will $325 amount to $403 in 4 yr.? 13. At what per cent. will $175 amount to $224 in 3 yr. 6 mo.? 14. At what per cent. will $940 amount to $1236.10 in 5 yr. 3 mo.? 15. At what per cent. will a principal double itself in 2 yr.? 4 yr.? 5 yr.? 6 yr.? 8 yr.? 12 yr.? 20 yr.? 25 yr.? 16. At what per cent. will a principal treble itself in 10 yr.? 12 yr.? 16 yr.? 15 yr.? 30 yr.? 17. At what per cent. will a principal quadruple itself in 5 yr.? 10 yr.? 20 yr.? 25 yr.? 40 yr.? 18. At what rate per cent. will the interest of $650 for 3 yr. 6 mo. amount to $159.25? WRITTEN PROBLEMS. 1. At what rate per cent. will $630 in 3 yr. 8 mo. gain $138.60 interest? Ans. 6. SOLUTION. = 3 yrs. 8 mo. 33 yr. $138.60 $23.10=6 Analysis. The interest of $630 for 3 yr. 8 mo., or 3 yr., at 1% is $23.10. If $23.10 is the interest of $630 at 1%, $138.60 is the interest of the same sum at as many per cent. as $23.10 is contained times in $138.60, or 6%. From the foregoing we derive the rule to find the rate per cent. : RULE. Divide the given interest by the interest of the given principal for the given time, at 1 per cent. 2. The interest on $1255 for 6 yr. 3 mo. is $313.75 : what is the rate per cent.? 3. The interest on $125.50 for 2 the rate per cent.? Ans. 4. yr. 4 mo. is $17.57 : what is Ans. 6. 4. The interest on $640 from Mar. 12, 1873, to Jan. 27, 1875, was $60: what was the rate? Ans. 5%. |