In this example, the multiplicand has 3 decimal places, and the multiplier has 2; therefore, by the rule, the product must have 5 places, and since the product consists of but 4 figures, we prefix one cipher before making the decimal poini. 2. Multiply 0-561 by 0.786. Ans. 0-440946. 3. Multiply 3.012 by 4.027. Ans. 12:129324. 4. Multiply 47.051 by 37·039. Ans. 1742:721989. 5. Multiply 33.33 by 66-66. . Ans. 22217778. 6. Multiply 125.125 by 5.5. . Ans. 688.1875. 53. A decimal number may be multiplied by 10, 100, 1000, &c., by removing the decimal point as many places to the right as there are ciphers in the multiplier; and if there are not so many figures, make up the deficiency by annexing ciphers. 110 121.2. 100 1212. 1000 12120. Thus, 12.12 multiplied by 7 10000 = 121200. 100000 1212000. | 1000000 | 12120000. How may a decimal number be multiplied by 10, 100, 1000, &c. ? When there are not as many decimal figures in the multiplicand as there are ciphers in the multiplier, how do you proceed? DIVISION OF DECIMAL FRACTIONS. 54. In multiplication of decimals, we know that the number of decimal places in the product is equal to the sum of those in both the factors. Now, since the product divided by one of the factors must produce the other fac tor or quotient, it follows, that in division the decimal places of the dividend must be equal to the number of places in the divisor and quotient taken together. Hence, the number of decimal places in the quotient must equal the excess of those in the dividend above those in the divisor. Divide 5.81224 by 5.432. Dividing 581224 by 5432 we find 107 for the quotient Since 5 figures of the dividend are decimals, and only 3 figures of the divisor are decimals, it follows that two figures of the quotient 107 must be decimals, so that 1.07 is the quotient sought. Hence tire following RULE. - - Divide as in whole numbers; give as many decimal places in the quotient as those in the dividend exceed those in the divisor ; if there are not as many, supply the deficiency by prefixing ciphers. How do you divide one decimal by another ? How many decimal places must the quotient have? If the whole number of figures in the quotient is not as great as the number of decimals required, how do you proceed ? EXAMPLES. OPERATION. 1. Divide 0-123428 by 11.8 11.8)0:123428(0-01046. 118 708 - In this example, the dividend contains 6 decimal places, and the divisor but 1; therefore, by the rule, the quotient ought to contain 5; but as there are but 4 figures in the quotient, we make up the deficiency by prefixing a cipher before making the decimal point. 2. Divide 3.810688 by 1.12. »Ans. 3.4C24. 3. Divide 0.109896 by 0.241. Ans. 0:456. 4. Divide 1.12264556 by 1:0012. Ans. 1·1213. 5. Divide 0·01764144 by 0:0018. Ans. 9.8008. 55. When there are not as many decimal places in. the dividend as in the divisor, we may, (by Art. 49,) annex as many ciphers to the dividend as we please, if we do not change the place of the decimal point. When the number of decimal places is the same in both dividend and divisor, the quotient will be a whole number. As for example, is divided by to=3, which is a whole number; that is, 0:6 divided by 0·2=3, a whole number. • When there are not as many decimal places in the dividend as in the divisor, how do you proceed? When the number of decimal places in the dividend is the same as in the divisor, what will the quotient be ? 6. Divide 244:431 by 1.2345. In this example, before OPERATION. performing the division, | 1.2345)244.4310(198 whole we annex a cipher to the 12345 number. dividend, so that it may 120981 have as many decimal 111105 places as the divisor has ; 98760 we then perform this 98760 7 Divide 122:418 by 3-4005 8. Divide 0.7 by 0.07. 9. Divide 0-25 by 0.0005. 10. Divide 0.125 by 0.000005. Ans. 36. 56. When there is still a remainder, and we wish a more accurate quotient, we may continue to annex ciphers and to divide as far as we please, observing the rule foi placing the decimal point. 11. Divide 20 by 0.003. OPERATION, By Short Division. 0.003)20.000 6666-6666, &c., to any extent When, in the quotient, we write the sign + it is to indicate that the quotient is still larger than is written. It frequently happens, as in this example, that the work will never terminate. When there is still a remainder, how may we proceed to obtain a still more accurate value for the quotient? What does the sign + at the right of a quotient indicate ? 57. We may, obviously, divide any decimal by 10, 100, 1000, &c., by removing the decimal point as many •places to the left as there are ciphers in the divisor; when there are not so many figures at the left of the decimal point, we may prefix ciphers. [ 10 ] [ 1.212. 1 0.1212. 0.01212. Thus, 12:12 divided by 3 3 10000 0:001212. 1 100000 0.0001212. | 1000000 0.00001212. How may we divide a decimal by 10, 100, 1000, &c. ? When in the decimal number there are not as many figures on the left of the decimal point as there are ciphers in the divisor, how do you proceed ? FEDERAL MONEY. 58. This is the currency of the United States. Its denominations, or names, are Eagles, Dollars, Dimes, Cents, and Mills. Eagles, * Seo noto at end of the subject of Federal Monoy. |