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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 3, 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:

First Operation. Second Operation. Third Operation.

0.333 0.3333 0.33333 0° 166 0-1666 O' 16666 0-0333 0-03333 0-033333 200 2000 20000

20 200 2000 0.0553 20 200 0-05553 20

* 0-055553

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.

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6. Multiply 0-009416517988 by 0.999936883996. Ans. 0-0094159236548. 7. Multiply 0.0000375229 by 0.0000275177. Ans. 0.000000001032543. 8. Multiply 0.999936883996 by 0.999955663612. Ans. 0.9998925504063. 9. Multiply 0.58740.1052 by 0.018468950. - Ans. 0-0108486807. 10. Multiply 91-6264232009 by 0.0172021234. Ans. 1:57.6169038601. 11. Multiply 212:3880258928 into itself. Ans. 45108.67354264.

DIWISION 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

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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.

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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 030679006 by 0.27610603, by the last rule, our work will be as follows:

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|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 R U L E. Proceed as in the last rule, until we reach that point of the work where it would be necessary to annea ciphers to the remainder. Then, instead of annering a cipher to the remainder, omit the right-hand figure of the divisor, and we shall obtain the next figure of the quotient;

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