403. Ex. 1. What is the interest of $30 for 1 year, at 6 per cent.? 6 Analysis. We have seen that 6 per cent. is ; that is, $6 for $100, 6 cents for 100 cents, &c. (Art. 386.) Since therefore the interest of $1 (100 cents) for 1 year is 6 cents, the interest of $30 for the same time must be 30 times as much; and $30×.06 Operation. $30 Prin. .06 Rate. $1.80 Int. 1 yr. We first multiply the principal by the given rate per cent. expressed in decimals, as in percentage, and point off as many decimals in the product as there are decimal places in both factors. Ex. 2. What is the interest of $140.25 for 1 year, 1 month, and 10 days, at 7 per cent.? What is the amount? Operation. .07 Rate. 12)$9.8175 Int. 1 yr. 3) 8181 66 1 mo. 2727 10 d. 66 $10.9083 Interest. $140.25 Prin. added. $151.1583 Amount. 1 month is of a year; therefore the interest for 1 month is of 1 year's interest. 10 days are of 1 month, consequently the interest for 10 days, is of 1 month's interest. The amount is found by adding the principal and interest together. Note.-1. In adding the principal and interest, care must be taken to add dollars to dollars, cents to cents, &c. (Art. 374.) 2. When the rate per cent. is less than 10, a cipher must always be prefixed to the figure denoting it. (Art. 387. Obs. 1.) It is highly important that the principal and the rate should both be written correctly, in order to prevent mistakes in pointing off the product. Ex. 3. What is the interest of $250.80 for 4 years, at 5 per cent.? What is the amount ? Solution.-$250.80X.05 $12.54, the interest for 1 year. Now $12.54X 4=$50.16, 66 66 4 years. And $250.80+$50.16=$300.96, the amount required. 40. From the foregoing illustrations and principles we deduce the following general RULE FOR COMPUTING INTEREST. 1. FOR ONE YEAR. Multiply the principal by the given rate, and from the product point off as many figures for decimals, as there are decimal places in both factors. (Art. 324.) II. FOR TWO OR MORE YEARS. by the given number of years. III. FOR MONTHS. Multiply the interest of 1 year Take such a fractional part of 1 year's in terest, as is denoted by the given number of months. IV. FOR DAYS. Take such a fractional part of one month's interest, as is denoted by the given number of days. The amount is found by adding the principal and interest together. OBS. 1. The reason of this rule is evident from the consideration that the: given rate per cent. per annum denotes hundredths. (Arts. 386, 398.) Now when the rate is 6 per cent. we multiply by .06, when 7 per cent. by .07, &c., and point off two figures in the product; consequently the result will be the same as to multiply by T, TU, &c. 6 100 2. In calculating interest, a month, whether it contains 30 or 31 days, or even but 28 or 29, as in the case of February, is assumed to be one twelfth of a year. Therefore, for 1 month we take of 1 year's interest; for 2 months, ; for 3 months, ; for 4 months, ; for 6 months, ; for 8 months, , &c. 1 1 2 30 Again, 30 days are commonly considered a month; consequently the interest for 1 day, or any number of days under 30, is so many thirtieths of a month's interest. (Art. 303. Obs. 2.) Therefore, for 1 day we take of 1 month's interest; for 2 days, 5; for 3 days, ; for 5 days, ; for 10 days, 1, &c. This practice seems to have been originally adopted on account of its convenience. Though not strictly accurate, it is sanctioned by general usage. 0 1 6 3. Allowing 30 days to a month, and 12 months to a year, a year would contain only 360 days, which in point of fact is, or less than an ordinary year. Hence, To find the interest for any number of days with entire accuracy, we must take so many 365ths of 1 year's interest, as is denoted by the given number of days; or, find the interest for the days as above from this subtract of 1 73 QUEST.-404. How is interest computed for a year? iw for any number of years ? How for months? How for days? How find the amount? Obs. In reckoning interest. what part of a year is a month considered? How many days are commonly considered a month? Is this practice accurate? itself, and the remainder will be the exact interest. The laws of New York, and several other states, require this deduction to be made. In business, when the mills in the result are 5, or over, it is customary to add 1 to the cents; if under 5, to disregard them, EXAMPLES. 1. What is the interest of $423 for 1 yr., at 7 per cent.? 2. What is the interest of $240.31 for 3 yrs., at 6 per cent. ? 3. What is the interest of $403.67 for 2 yrs., at 5 per cent. ? 4. What is the interest of $640 for 1 yr., at 8 per cent.? 5. What is the interest of $430.45 for 2 yrs., at 7 per cent.? 6. What is the interest of $185.06 for 4 yrs., at 6 per cent.? 7. What is the interest of $864.80 for 5 yrs., at 4 per cent.? 8. What is the interest of $763 for 4 months, at 7 per cent.? 9. What is the interest of $940.20 for 6 mo., at 6 per cent.? 10. What is the interest of $243.10 for 5 mo., at 8 per cent.? 11. What is the interest of $195.82 for 7 mo., at 6 per cent.? 12. What is the interest of $425.35 for 9 mo., at 6 per cent. ? 13. At 7 per cent., what is the int. of $738 for 1 yr. and 2 mo. ? 14. At 6 per cent., what is the int. of $894 for 1 yr. and 8 mo. ? 15. At 7 per cent., what is the amount of $926 for 6 mo.? 16. At 7 per cent., what is the amt. of $648 for 2 mo. 15 d.? 17. At 6 per cent., what is the amt. of $1000 for 1 mo. 11 d.? 18. At 5 per cent., what is the amt. of $1565.45 for 3 mo.? 19. At 6 per cent., what is the amt. of $872 for 4 mo.? 20. What is the int. of $681 for 10 days, at 6 per cent. ? 21. What is the int. of $483.26 for 15 d., at 7 per cent.? 22. What is the int. of $569.40 for 20 d., at 6 per cent.? 23. What is the amt. of $95 for 1 yr. and 6 mo., at 5 per cent.? 24. What is the amt. of $148 for 8 mo. 12 d., at 6 per cent.? 25. What is the amt. of $700 for 30 d., at 7 per cent.? 26. What is the int. of $340 for 60 d., at 54 per cent. ? 27. What is the int. of $4685 for 90 d., at 6 per cent. ? 28. What is the amt. of $3293 for 30 d., at 7 per cent. ? 29. What is the amt. of $5265 for 15 d., at 6 per cent. ? 30. What is the int. of $8310 for 10 d., at 7 per cent. ? 31. What is the int. of $50625 for 21 d., at 7 per cent. ? 32. What is the amt. of $65256 for 4 mo., at 7 per cent.? SECOND METHOD OF COMPUTING INTEREST. 405. There is another method of computing interest, which is very simple and convenient in its application, particularly when the interest is required for months and days, at 6 per cent. 406. We have seen that for 1 year, the interest of $1 at & per cent. is 6 cents., or $.06; (Art. 404;) therefore, Hence, The interest of $1 for 1 month, at 6 per cent., is 5 mills; for every 2 months, it is 1 cent; and for any number of months, it is as many cents, or hundredths of a dollar, as 2 is contained times in the given number of months. 407. Since the interest of $1 for 1 month (30 days) is 5 mills, That is, the interest of $1 for every 6 days, is 1 mill, or $.001; and for any number of days, it is as many mills, or thousandths of a dollar, as 6 is contained times in the given number of days. 408. Hence, to find the interest of $1 for any number of days, at 6 per cent. Divide the given number of days by 6, and set the first quotient figure in thousandths' place, when the days are 6, or more than 6 ;, but in ten thousandths' place, when they are less than 6. OBS. For 60 days (2 mo.) the interest of $1 is 1 cent; (Art. 406,) wnen, therefore, the number of days is 60 or over, the first quotient figure must occupy hundredths' place. QUEST.-408. How find the interest of $1 for any number of days, at 6 per cent. ? Ex. 1. What is the interest of $185 for 1 year, 6 months and 18 days, at 6 per cent.? Analysis. The interest of $1 for 1 year is 3 cents; for 6 months it is 3 cents; and for 18 days it is 3 mills. (Arts. 406, 407.) Now 06+.03+.003 $.093. Since therefore the interest of $1 for the given time is $.093, the interest of $185 must be 185 times as much. 409. From these principles we may derive a Operation. $185 Prin. .093 Int. $1. 555 1665 $17.205 Ans. SECOND RULE FOR COMPUTING INTEREST. I. To compute the interest on any sum, at 6 per cent. Multiply the principal by the interest of $1 for the given time, at 6 per cent., and point off the product as in multiplication of decimals. (Art. 324.) II. To compute int. at any rate, greater or less than 6 per cent. First find the interest on the given sum at 6 per cent.; then add to this interest, or subtract from it, such a fractional part of itself, as the required rate exceeds or falls short of 6 per cent. The amount is found by adding the principal and interest together as in the former method. (Art. 404.) OBS. 1. The amount may also be found by multiplying the given principal by the amount of one dollar for the time. 2. The reason of the first part of this rule, is manifest from the principle that the interest of 2 dollars for any given time and rate, must be twice as much as the interest of 1 dollar for the same time and rate; the interest of 50 dollars, 50 times as much as that of 1 dollar, &c. 6 3. When the required rate is 7 per cent., we first find the interest at 6 per cent., then add of it to itself; if 5 per cent., subtract of it from itself, &c., for the obvious reason, that 7 per cent. is once and 1 sixth, or of 6 per cent.; 5 7 per cent. is only 5 of 6 per cent., &c. 4. When the decimal denoting the int. of $1 for the days, is long, or is a repetend, it is more accurate to retain the common fraction. (Art. 387. Obs. 2.) 2. What is the interest of $746 for 4 months and 18 days, at 6 per cent. ? Ans. $17.158. QUEST.-409. What is the second method of computing interest, at 6 per cent.? When the mte per cent. is greater or less than 6 per cent., how proceed? |