Operation. 0.4673 0.15140 2271 265 11 Ans. 0.17687 4. Multiply 000524486 by 0.99993682. Operation. 0.00524486 0.99993682 0.004720374 472037 47204 4720 157 31 4 Ans. 0.005244527 5. Multiply 108-2808251671 by 19614591767. Operation. 108.2808251671 1.9614591767 108.2808251671 6 4968495100 433123301 54140412 9745274 108281 75796 6497 758 Ans. 212.3884181846 6. Multiply 0-009416517988 by 0.999936883996. Ans. 0.0094159236548. 7. Multiply 00000375229 by 0.0000275177. Ans. 0.000000001032543. 8. Multiply 0.999936883996 by 0.999955663612. Ans. 0.9998925504063. 9. Multiply 0.587401052 by 0018468950. Ans. 0.0108486807. 10. Multiply 91.6264232009 by 0·0172021234. Ans. 1.576169038601. 11. Multiply 212.3880258928 into itself. Ans. 45108 67354264. DIVISION OF DECIMALS. 40. IN multiplication, we have seen that there are as many decimal places in the product as there are in both the factors; and, since division is the reverse of multiplication, it follows that the number of decimal places in the quotient must equal the excess of those in the dividend, above those of the divisor. Hence, to divide one decimal expression by another, we have this RULE. Divide as in whole numbers, and point off as many places from the right of the quotient, for decimals, as the decimal places in the dividend exceed those of the divisor. If there are not as many figures in the quotient as this excess, supply the deficiency by prefixing ciphers. EXAMPLES. 1. Divide 3'475 by 4.789. Operation. 4.789)3-475000(0·725 3 3523 12270 9578 26920 23945 2975 In this example, the number of decimal places in the dividend, including the ciphers which were annexed, is 6, whilst the number of places in the divisor is 3; therefore, we make 3 places of decimals in the quotient. We might continue to annex ciphers to the remainder, and thus obtain additional decimal figures. 2. What is the quotient of 78.56453 divided by 4.78? Ans. 16'436. 3. What is the quotient of 1561.275 divided by 24.3? Ans. 64.25. 4. What is the quotient of 0.264 divided by 0·2? Ans. 1.32. 5. What is the quotient of 3.52275 divided by 3.355? Ans. 1.05. 6. What is the quotient of 901·125 divided by 2·25 ? Ans. 400.5. ABRIDGED DIVISION OF DECIMALS. 41. IF we divide 0-30679006 by 0.27610603, by the last rule, our work will be as follows: Operation. 0.27610603)0.30679006(1·1111313 27610603 3068403 0 2761060 3 307342 70 276106 03 31236|670 3626|0670 2761 0603 865 00670 828 31809 36 688610 27 610603 9 0780070 7948261 By simply inspecting the above work, it is obvious that all that part of the work which is on the right of the vertical line can in no way affect the accuracy of our quotient figures. By the following rule, we may perform the work of division so as to exclude all that part of the work on the right of the vertical line, thereby shortening the work, and still obtaining as accurate a result as by the last rule. To contract the work in the division of decimals, we have this RULE. Proceed as in the last rule, until we reach that point of the work where it would be necessary to annex ciphers to the remainder. Then, instead of annexing a cipher to the remainder, omit the right-hand figure of the divisor, and we shall obtain the next figure of the quotient ; and thus continue, at each successive figure of the quotient, to omit the right-hand figure of the divisor, until there is but one figure in the remainder. NOTE. If we regard the dividend as the numerator of a fraction whose denominator is the divisor, the quotient will be the value of such fraction. Annexing a cipher to the numerator of a fraction is equivalent to multiplying its value by 10, and omitting the righthand figure of the denominator is also equivalent to multiplying the value of the fraction by 10. Hence, in the operation of division of decimals, instead of annexing a cipher to the dividend, as in the ordinary rules, we may, instead thereof, omit the right-hand figure of the divisor, as in the foregoing rule. EXAMPLES. 1. What is the quotient of 365 424907 divided by 0.263803 ? Operation. 0.263803)365-424907(1385 21892 263 803 101 6219 79 1409 22 48100 21 10424 1 376767 57752 52761 4991 2638 2353 2110 243 237 6 5 1 |