Secondly. We multiply 0-3789, which is the multiplicand deprived of its right-hand figure, by 7, the second figure of the multiplier, observing to carry 3, since the figure cut off, multiplied by 7, gives 28, which is nearer 30 than 20; we thus obtain 26526 for the second partial product. Thirdly. Multiplying 0:378 by 4, observing to carry 4, we obtain 1516 for the third partial product. Fourthly. Multiplying 0·37 by 5, observing to carry 4, we obtain 189 for the fourth partial product. Fifthly. Multiplying 0-3 by 2, observing to carry 1, we get 7 for the fifth partial product. As a second example, we will find the product of by, using in the first operation, 3 decimal places in each factor; in the second operation, we will use 4, and in the third, we will use 5, as follows: From the above work, it will be seen that the results of these three operations have the same degree of accu racy as when performed by the usual rule. 3. Multiply 0-3785 by 0-4673. 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 1.9614591767. Operation. 108 2808251671 1.9614591767 108 2808251671 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. 0000000001032543. 8. Multiply 0.999936883996 by 0.999955663612. Ans. 0.9998925504063. 9. Multiply 0-587401052 by 0018468950. Ans. 00108486807. 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 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 27610 603 3626 0670 2761 0603 865 00670 828 31809 36 688610 27 610603 9 0780070 8 2831809 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; |