9. Of $1,237.50 for 7 yr. 2 mo. 26 da. at 23%. 10. Of $1,875.60 for 12 yr. 3 mo. 10 da. at 33%. 11. Of $784.15 for 57 da. 12. Of $4,225 for 126 da. Answers.-(1) $35.695; (2) $55.47-; (3) $682.13-; (4) $1,565.20; (5) $22.63+; (6) $137.56-; (7) $229.26+; (8) $258.76+; (9) $246.35—; (10) $844.37—; (11) $7.45—; (12) $88.73—. SIXTY-DAY METHOD. 20. For 1 year, at 6%, the interest of any principal is .06 of the principal itself, and for 2 months, or 60 days, the interest is .01 of the principal. Hence, If the decimal point of any sum be moved two places to the left, it will give the interest of that sum for 60 days at 6%. Thus, the interest of $3,472.75 for 60 days at 6% is $34.73—, and of $692 it is $6.92. Having the interest for 60 days, it is easy, by operations that will suggest themselves, to find the interest for any other number of days. EXAMPLE 1.-Find the interest of $8,368 for 99 days at 6%. SOLUTION. Interest for 3 days of 6 days. Ans. $138.072 Interest for 99 days. EXAMPLE 2.-What is the interest at 9% of $1,264.76 for 49 days? SOLUTION. 21. Rule.-Take .01 of the principal for the interest at 6% for 60 days, and then, by the method of aliquot parts, find the interest for the given time at the rate specified. EXAMPLES FOR PRACTICE. 22. By the sixty-day method, find the interest 1. Of $8,000 for 87 days at 6%. Answers. (1) $116; (2) $48.40; (3) $6.55+; (4) $20.07+; (5) $320.625; (6) $765.26—; (7) $225; (8) $7.23—; (9) $248.83—; (10) $226.42—. EXACT INTEREST. 23. When interest is to be computed for one or more entire years at a specified rate per year, the fact that 12 months of 30 days each are usually regarded as a year does not affect the result-it is only when months and days, or days alone, become an element of the given time, that the interest is greater than it should be. The average length of a month in an ordinary year is 30,5 days, and in a leap year it is 30 days. A day is not of a year, but of a common year, and of a leap year. Hence, 360 days, or 73, of a common year, and 368, or 89, of a leap year. By the ordinary method of finding interest, the result is either or greater than it should be. 3669 60 Thus, the interest of $7,300 for 60 days at 6%, as found by the usual method, is $73. In equity it is $7,300.06 X 38% = $72. That is, each $73 interest should be $72. 24. The method practised by the government and by most banks is to compute the interest for the number of entire years in a period, and then treat the remaining days as so many 365ths of a year. EXAMPLE. Find the exact interest of $8,000 from Jan. 5, 1873, to July 23, 1880, at 6%. SOLUTION. From Jan. 5, 1873, to Jan. 5, 1880 = From Jan. 5, 1880, to July 23, 1880 = Jan. Feb. Mar. Apr. May June July 7 yr. 100 = 388 = 188 yr. 26+29 +31 +30 +31 + 30+ 23 = 200 days 200 $8,000.067198 = $3,622.30. Ans. 25. The same result is obtained by adding to the interest for 7 years the interest for 200 days less of itself, found by the usual method. Thus, Exact interest for 7 yr. 200 da. = $3,622.30 Ans. EXAMPLE. What is the exact interest of $4,800 for 198 days of an ordinary year, at 6% ? SOLUTION. 73) $158.40 = Interest for 198 days, counting 360 2.1 7 days of the year. $156.23 Exact interest for 198 days. Ans. EXPLANATION. The interest is first found by the usual method, and of the result deducted. 26. Rule.-Find the interest by the ordinary method for the whole number of years included in the period. Count the number of days that remain, and by the same method find the interest for the days. If the days are part of an ordinary year, diminish the interest for the days by of itself; if they are part of a leap year, diminish the interest by of itself. Add the interest for the years to that for the days, and the result will be the exact interest. 27. EXAMPLES FOR PRACTICE. Find the exact interest 1. Of $10,000 for 123 days at 6%. 2. Of $12,800 for 168 days at 6%. 3. Of $6,400 for 213 days at 5%. 4. Of $22,800 for 2 yr. 73 da. at 4%. 5. Of $960 for 5 yr. 300 da. at 3%. 6. Of $484.80 for 6 yr. 202 da. at 51%. 7. Of $13,000 from Jan. 17, 1897, to Nov. 29, 1897, at 41%. 8. Of $968.40 from Apr. 19, 1865, to July 1, 1896, at 3%. 9. Of $1,234.60 from Dec. 23, 1888, to Mar. 17, 1890, at 6%. 10. Of $43,000 from May 29, 1891, to Nov. 3, 1895, at 7%. Answers.-(1) $202.19+; (2) $353.49; (3) $186.74; (4) $2,006.40; (5) $195.62; (6) $174.74; (7) $506.47-; (8) $906.41-; (9) $91.12+; (10) $13,342.96-. ANNUAL INTEREST. 28. Unless otherwise specified, interest upon debts is understood to be payable annually. In case it is not so paid, it is permitted in some States to charge interest upon overdue interest. In some other States this practice is illegal. Where it is intended to charge "annual interest," the written obligation should contain the words "interest payable annually." EXAMPLE. What is the interest at 6% of $2,400 for 6 years 6 months, interest payable annually, if no interest is paid until the end of the time? SOLUTION. = $936.00 Interest for 1 year = $2,400.06 = $144.00 1 5 5.5 2 $1,09 1.5 2 Ans. EXPLANATION.-The first year's interest remains unpaid for 5 yr., the second for 41 yr., etc. The sum of these periods is 18 yr. One year's interest of the principal is $144, and the interest of this for 18 yr. is $155.52. The sum of $936 and $155.52 is the entire interest due. 29. Rule. Find the interest of the principal for one year, and for the entire time the debt runs. Find the sum of the several periods that the annual interest remains unpaid, and for this time find the interest of one year's interest of the debt. The sum of the interest of the main debt and that of the unpaid annual interest will be the result required. EXAMPLES FOR PRACTICE. 30. In the following examples, assume that the annual interest is payable but unpaid, and find the entire interest due at the end of the given time: 1. Of $240 for 2 yr. 6 mo. at 6%. 2. Of $380 for 3 yr. 8 mo. at 5%. 3. Of $1,000 for 4 yr. 9 mo. at 7%. 4. Of $1,200 for 3 yr. 3 mo. 15 da. at 6%. 5. Of $387.50 for 5 yr. 6 mo. 20 da. at 4%. 6. Of $7,625 for 8 yr. 7 mo. 18 da. at 5%. Answers.—(1) $37.73-; (2) $74.42; (3) $376.60; (4) $253.74; (5) $94.03+; (6) $3,921.79+. PROBLEMS IN INTEREST. 31. Given the interest, rate, and time, to find the principal. We know that I = Prin. X rate X time (in years); or, more briefly, Prt 100 = I, That is, the principal is equal to 100 times the interest divided by the product of the rate per cent. and the time. 126D-14 |