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be equal to the number of decimal figures in both the factors of that product.
RULE. Multiply as in whole numbers; and in the product, point off as many figures for decimals, as there are decimal places in both factors. If the number of figures in the product be less than the number of decimal places in both factors, prefix ciphers to supply the deficiency.
18. Find the product of 658 by .249.' 7.06 by 3.65. .593 by 5.62. .146 by .244. 658 7.06 593
.146 .249 3.65 5.62
..244 5922 3530 1186
584 2632 4236 3558
584 1316 2113 2965
292 173.842 25.7690 3.33266 .035624
19. Multiply 423 by .27; that is, find .27 of 428 20. What is the product of 3.067 by 8.2? 21. What is the product of .6247 by 23 ? 22. What is the product of .099 by .04 ? 23. What is the product of 113 by .0647 ? 24. What is 7.03 X.9 X 31.6 x 23.758=? 25. Multiply 9 dolls. 7 cts. 6 mills (9.076] by 46. 26. What cost 28 yards of cloth, at $7.515 per yd.? 27. What cost 15.9 yd. of cloth, at $9.427 per yd.? 28. What cost 275 lemons, at 9 mills apiece?
29. At 7 cents and 3 mills per yard, what is the value of 18704 yards of satin ribbon?
30. What is the value of a township containing 30519.75 acres of land, at 4 dolls. 8 cts. and 5 mills
acre ? 31. What is .06 of 1532 dollars? Or, what is the product of 1532 multiplied by .06?
32. What is 03 of 476 dollars and 78 cents ?
33. If an insurance office charge .015 of the value of a house for insuring it against fire, what will be the expense of insuring a house, valued at $437.25 ?
34. Multiply 26.000375 by .00007.
The learner will perceive, that any decimal number is multiplied by 10, 100, 1000, &c., by merely removing the decimal point as many places to the right hand as there are ciphers in the multiplier. Thus, 6.25 X 10=62.5. 6.25 X 1000=6250.
DIVISION OF DECIMALS.
It has been shown, in multiplication of decimals, that there must be as many decimal places in a product as there are in both its factors; and it follows, that, in division of decimals, there must be as many decimal places in the divisor and quotient together, as there are in the dividend. Therefore, the number of decimal places in the quotient must be equal to the difference between the number of decimal places in the dividend, and the number of decimal places in the divisor.
RULE Divide as in whole numbers; and in the quo tient, point off as many figures for decimals, as the decimal places in the dividend exceed those in the divisor; that is, make the decimal places in the divisor and quotient counted together, equal to the decimal places in the dividend,
If there be not figures enough in the quotient to point off, prefix ciphers to supply the deficiency.
When there are more decimal places in the divisor, than in the dividend, render the places equal, by annexing ciphers to the dividend, before dividing:
After dividing all the figures in the dividend, if there be a remainder, ciphers may be annexed to it, and the division continued. The ciphers thus annexed, must be counted with the decimal places of the dividend.
36. How many times is 57.2 contained in 2406.976 ?
57.2)2406.976(42.08 37. What is the quotient of 11.7348 by 254 ?
254)11.7348.0462 38. What is the quotient of 4066.2 by .648?
39. What is the quotient of 3.672 by .81 ?
The sign of addition, of .81)3.672(4.5333+
more, here shows, that the true 324 quotient is more than the
ceding figures express. We 405
might continue the division,
but we should never arrive at 270 243
a complete quotient. For the
purposes of business, it is sel270
dom necessary to extend the 243
quotient below thousandths; 270 but, in the following exercises, 243
those quotients that do not 27
terminate, may be extended to
millionths. 40 How many times is 4.72 contained in 637.5317 41. What is the quotient of 2.739.5 by 74? 42. What is the quotient of 409.967 by .5906 ? 43. What is the quotient of 125 by .1045 ? 44. What is the quotient of 709 by 3.574 ? 45. What is the quotient of 7382.54 by 6.4252? 46. What is the quotient of 715 by .3075 ? 47. What is the quotient of 267.15975 by 13.25 ? 48. What is the quotient of .0351648 by 423? 49. What is the quotient of .009 by .00016 ?
50. If 17 boxes of oranges cost $ 93.29, what is the cost of a single box!
51. If $ 550.725 be divided equally among 15 men, what will be each man's share ?
52. If 37.5 barrels of flour be divided equally among 25 men, how much will each man have ?
53. If 46.75 yards of cloth cost $ 251.702, what is the cost of 1 yard of the cloth ?
51. Divide 3712 by 42; annexing ciphers to the remainders, until eight decimal figures are obtained in the quotient.
55. What is the quotient of 9 divided by 266 ?
In this exainple it will be necessary to annex a sufficient number of decimal ciphers to the dividend, before the operation of dividing can be commenced.
56. What is the quotient 1 divided by 8?
10 by 12.
3 by 16. 2 by 13. 6 by 26. 14 by 15. 40 by 72. 7 by 599.
Any decimal number is divided by 10, 100, 1000, &c. by merely removing the decimal point as many places to the left hand as there are ciphers in the divisor. Thus 14.8:10=1.48 14.8:1000=.0148.
REDUCTION OF DECIMALS.
CASE I. To reduce a vulgar fraction to a decimal.
RULE, Divide the numerator by the denominator, and the quotient will be the decimal. 59. Reduce to a decimal. 8)7.000
Decimal ciphers are here annexed
to the dividend as directed in the .875 Ans.
rule for division of decimals. 60. Reduce the fractions 1, 3; }, }, 1, H4, , azd n'as to decimals.
61. Simplify of %, and reduce it to a decimal.
The learner will discover, that the above fractions, , tí , and it cannot be reduced to exact decimal expressions. The quotient of 2 by 3 is .6666, &c., continually. The quotient of 2 by 11 is .181818, &c.; the same two figures being repeated continually. The quotient of 1 by 27 is .037037, &c.; the same three figures being repeated continually. Decimals of this kind are treated in the next Article, under the head of Infinite Decimals. For most purposes, however, three or four decimal places will express any fraction with sufficient accuracy, unless the integer of the fraction is of very high value. CASE II. To reduce a decimal to a vulgar fraction.
Write the decimal denominator under the decimal, and erase the decimal point: view the expression as a vulgar fraction, and reduce it to its lowest terms.
65. Reduce .4375 to a vulgar fraction.
.4375=10066 ; and to reduce this fraction to its lowest terms, we divide the terms by their greatest common measure, which is 265. The result is, 16.
66. Reduce .375 to a vulgar fraction.
CASE III. To reduce the lower denominations of a compound number to the decimal of a higher denomination.
RULE. Reduce the given quantity to a vulgar fraction, (as taught in page 40), then reduce the vulgar fraction to a decimal.
The decimal quotients which do not terminate, may, in the examples of this case, be extended as low as the seventh place.
71. Reduce 12 s. 6 d. 3qr. to the decimal of a £. 72. Reduce 2qr. 14 lb. to the decimal of a cwt. 73. Reduce 1 R. 14 rods to the decimal of an acre.
74. Reduce 13 dwt. 16 gr. to the decimal of a pound, Troy weight.
75. Reduce 1 pk. 1 pt. to the decimal of a bushel. 76. Reduce i bl. to the decimal of a tun of wine. 77 Reduce 4 yd. 6 in. to the decimal of a mile. 78. Reduce 5 square yards to the decimal of an acre. 79. Reduce 14 cubic feet to the decimal of a cord. 80. Reduce 21 h. 50 m. 31 s. to the decimal of a year.
81. Express £19. 13s. 9 d. decimally; making the £ the unit, and the s. and d. a decimal.
82. Reduce 17 hhd, 9 gal. 3 qt. 1 pt. to a decimal expression; the hogshead being the unit.
83. Reduce 15 tons, 1 qr. 14 oz. to a decimal expreso sion; the ton being the unit.
34. Reduce 4 miles, 7 fur. 9r. 3yd. 6 in. to a decima) expression; the mile being the unit.
85. Reduce 25 rods,19yd.7 ft. 115 in., square measure to a decimal expression; the rod being the unit.