Exercises in Notation and Numeration of Decimals. 156. Write the following numbers in figures : 13. Seven hundred eighteen ten-thousandths. 14. Nine millionths. 15. Eight, and eight hundred four ten-millionths. 16. Fifteen thousand eighty, and four thousand six millionths. 17. One thousand, and one hundred-millionth. 18. Seventy million, and seven millionths. 19. Sixteen thousand, and fourteen hundred-thousandths. 20. Eight million, and eighteen ten-millionths. 157. Though decimals can be added, subtracted, multiplied, and divided as integral quantities, yet as parts of a unit, they are fractions. Thus, 0.3 is ; 0.37 is 0.375 is 375; etc. 1000; 100 158. When the denominators of Decimals are written, the Decimals appear as Common Fractions. Every principle and operation in Common Fractions is equally applicable to Decimals. 159. The denominator of a decimal is 1 with as many ciphers annexed as there are figures in the decimal. 160. For Addition and Subtraction of Decimals, see pages 12-27. Multiply as in whole numbers, and point off as many figures for decimals in the product as there are decimal places in both factors, counted together. NOTE 1. If the number of figures in the product is less than the number of decimal places in the two factors, the deficiency must be supplied by prefixing ciphers to the product, as in Exs. 22 and 24. NOTE 2. The pointing off is in reality the multiplying of the denominators of the factors, or it shows what the product of the denominators is. 31. Multiply 36.874 by 0.5421. 32. Multiply 0.14687 by 0.00054. 33. Multiply 0.17288 by 0.14403. 34. Multiply 0.00369 by 0.24683. 35. Multiply three hundred fifty-six thousandths by one hundred forty-five ten-thousandths. 36. Multiply thirty-four millionths by twenty-six ten-millionths. 37. Multiply eight hundred forty-two thousandths by five hundred thousand. From these examples we derive the following Rule. Divide as in whole numbers, and point off as many figures for decimals in the quotient as the number of decimal places in the dividend exceeds the number in the divisor. NOTE 1. If there are not as many decimal places in the dividend as in the divisor, make as many by annexing ciphers. NOTE 2. If the number of figures in the quotient is less than the excess of decimal places in the dividend over those of the divisor, supply the deficiency by prefixing ciphers to the quotient. NOTE 3. The pointing off is in reality the dividing of the denominator of the dividend by the denominator of the divisor, or it shows what that quotient is. NOTE 4. The rule for pointing in the quotient is also evident from the rule in multiplication if we notice that the dividend is a product whose factors are the divisor and quotient. 41. Divide 1.2575125 by 2.5. 48. Divide 58.794 by 12.3. 49. Divide 3647 by 0.125. 50. Divide 90321.6 by 3.642. 51. Divide 72 by 0.064. 52. Divide 0.13 by 8. 53. Divide 7.2 by 0.16. 54. Divide 8.7 by 0.25. 55. Divide 3.6 by 7.5. 56. Divide 0.34 by 0.24. 57. Divide 0.73 by 1.5. 58. Divide 4.63 by 2.9. 59. Divide three thousand eight hundred fifty-three hundredthousandths by thirty-two millionths. Ans. 1204.0625. 60. Divide eighty-four, and eighty-four hundredths by fortyeight thousandths. 61. Divide four hundred by four hundredths. 62. Divide sixteen thousandths by forty-five hundred. 63. Divide seventy-five thousand eight hundred one by two thousand two hundred ninety-seven ten-thousandths. 163. To reduce a common fraction to a decimal. 64. Reduce to a decimal. 8) 7.000 The value of a fraction is the quotient arising from dividing the numerator by the denominator (Art. 118 a). 778; but as 8 is not contained in 7 we reduce the 7 to tenths, viz. to 70 tenths; 70 tenths divided by 8 gives 8 tenths, the first quotient figure, and 6 tenths remainder; 6 tenths: = 60 hundredths; 60 hundredths divided by 8 gives 7 hundredths, and 4 hundredths remainder, and so on as in division of decimals. Hence, Rule. Annex one or more ciphers to the numerator and divide the result by the denominator, continuing the operation until there is no remainder, or as far as desirable. Point off as in division of decimals. 65. Reduce to a decimal. 66. Reduce to a decimal. 67. Reduce to a decimal. 68. Reduce to a decimal. 69. Reduce to a decimal. |