other, units under units, tens under tens, and tenths under tenths, hundredths under hundredths, &c., and add as in whole numbers, and point off from the right for decimals as many figures as are equal to the greatest number of decimal places in any of the given numbers. The greatest number of decimals in the above example is six; six figures must therefore be pointed off for decimals. OBS. 1. The decimal point will always be exactly under the decimal points in the given numbers. OBS. 2. The learner must be very careful in placing the decimal points according to the rule. EXAMPLES. 25. Add 4.032, 64.5010, 96.081, 310.018, 1.0012. 26. Add 63.03036, 73.46030, .090345, 41.23101, 1.0109. 27. Add 340.14561, 84.960, .759112, .000012, 2.0345. 28. Add 9.03456, 1.23456, 12.34567, 123.4567, 1234.567. 29. Add 85.05376, 5.45405, 54.04345, 540.4345, 7.00034. 30. 46.13455+9.73456+.0009345+875.+34.5=? 31. 9671.03+5.05674+8.7561+750.12+87.34=? 32. 1.45610+67563.1+2.31267+91.234+.679=? 33. Add together 1 tenth, 1 hundredth, 1 thousandth, and 1 ten thousandth. 34. What is the sum of 5 hundredths, 5 ten thousandths, and 5 millionths? 35. What is the sum of 95 millionths, 1 hundred thousandth, and 1 tenth? 36. What is the sum of 784 thousandths, 347 millionths, 75 ten thousandths, and 99 hundredths? 37. What is the sum of 465, 78 hundred thousandths, 9 millionths, 99 ten thousandths, and 9 tenths? 38. What is the sum of 475, 9 ten thousandths, 83 hundred thousandths, and 9 tenths? MULTIPLICATION OF DECIMALS. 117. RULE. Multiply as in whole numbers, and point off from the right of the product as many figures as there are decimal places in both factors. PROOF. Multiplication of decimals may be proved in the same manner as multiplication of whole numbers, or by changing them to the form of common fractions. OBS. If there be not as many decimal places in the product as in both factors, make the number equal by prefixing ciphers. As there are eight decimal places in both factors, and only four in the product, four ciphers must therefore be prefixed to the product. The reason of this rule will be evident by changing the decimal multiplicand and multiplier to the form of a common fraction, and then multiplying. 118. When the multiplier is a whole number, the multiplicand is taken as many times as there are units in the multiplier; but when the multiplier is a fraction, only such parts of the multiplicand are taken as are indicated by the multiplier. When the multiplier is less than a unit, the product will be less than the multiplicand. What is the rule for the multiplication of decimals? When the multiplier is less than a unit, what is the product? G * EXAMPLES. Multiply the following decimals: 39. 45.601×7.456. 40. 31.735 x 9.735. 41. 41.244 × 1.642. 42. 784.67 x 9.641. 43. 46.043 x.0009. 44. 966.43 x.0061. 45. 56.300X.0312. 46. 67.456 × 1.245. 47. 96.314×2.103. 48. 814.21 x 36.142. 49. 204.101X 8.9614. 50. 56.421× 96.463. 51. 42.001x.00234. 52. 84.241x.00085. 53. Multiply 2 hundredths by 9 millionths. 54. Multiply 44 thousandths by 4 ten thousandths. 55. Multiply 1 ten millionth by 9 hundred thousandths. 56. Multiply 9999 by 9 hundred millionths. 57. Multiply 6 hundred millionths by 9 millionths. SUBTRACTION OF DECIMALS. 119. RULe. Write the less number under the greater, units under units, tenths under tenths, &c., so that the decimal points shall be exactly under each other. Subtract as in whole numbers, and point off from the right of the remainder as many places for decimals as are equal to the greatest number of places in either of the given numbers. PROOF. Subtraction of decimals may be proved in the same manner as subtraction of whole numbers. The principle of this rule is the same as that in subtraction of whole numbers. 58. From 961.345 take 2.456789. 961.345000 2.456789 958.888211 In this example three ciphers are annexed to the What is the rule for the subtraction of decimals? 1 minuend, to make the number of decimals equal to the number in the subtrahend, which does not change the value of the minuend, (ART. 108.) 68. From 1 take 1 millionth. 69. From 9999 take 9 hundred thousandths. 70. From 5 take 555 ten millionths. 71. From 222 take 22 hundred thousandths. 72. From 99 and 9 hundredths take 9 ten thousandths. 73. From 1 million take 1 hundred thousandth. DIVISION OF DECIMALS. 120. RULE. Divide as in whole numbers, and point off from the right of the quotient as many figures for decimals as the decimal places in the dividend exceed those in the divisor; and if there be not as many, make the number equal by prefixing ciphers to the quotient. PROOF. Division of decimals may be proved in the same manner as division of whole numbers, or by common fractions. OBS. 1. When the decimal places in the divisor and dividend are equal, the quotient will be a whole number. OBS. 2. When there are not as many decimal places in the dividend as in the divisor, ciphers may be annexed, and the division continued indefinitely. The ciphers thus annexed must be considered as decimal places. What is the rule for the division of decimals? OBS. 3. Unless great accuracy is required, it is not necessary to have more than five places of decimals in the quotient. 74. Divide .000288072 by 3.6. 3.6).000288072 (.00008002 288 072 Since the dividend has nine decimal places, and the divisor but one, the quotient must have eight decimal places; four ciphers must therefore be prefixed to the quotient. OBS. When there is a remainder after division, the sign + should be annexed to the quotient, to denote that the division may be continued farther. 91. Divide one hundred and twenty-five, and nine ten thousandths, by six, and fifty-four ten thousandths. 92. Divide four hundred and eleven, and seven hundred and six millionths, by fifty-five, and ninety-three ten thousandths. 93. Divide seven million and one hundred and ten thousand, and ninety-four millionths, by eight hundred and forty-five ten thousandths. 94. Divide two hundred and twenty-four, and nine ten millionths, by three hundred and twenty hundred millionths. |