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RULE. Multiply the two factors after the same manner as in whole numbers, then point off from the right of the product as many figures for decimals as there are decimal places in both the factors. If there are not so many places of figures, supply the deficiency by prefixing ciphers.

Examples. 1. Multiply 3.753 by 1.656.

OPERATION,

3.753 1.656 22518 18765 22518

3753

Ans. 6.214968 2. What is the product of 0.005 into 0.017?

Ans. 0.000085. 3. What is the product of 0.376 into 0.0076894 ?

Ans. 0.0028912144. 4. What is the product of 0.576 into 0.3854 ?

Ans. 0.2219904. 5. What is the product of 0.43 into 0.65?

Ans. 0.2795. 6. What is the product of 3.9765 into 4.378 ?

Ans. 17.409117. 7. What is the product of 415.314 into 7.3004?

Ans. 3031.9583256. 8. What is the product of 7.42 into 11.1415 ?

Ans. 82.66993.

ABRIDGED MULTIPLICATION OF DECIMALS.

39. If we multiply 0.37894 by 0.67452, by the last rule, our work will be as follows:

OPERATION.

0.37894
0.67452

715768
189 470
151576
265258

227364

Ans. 0.2556026088 Here it is plain that the figures on the right of the vertical line can not be depended upon as accurate, since they are beyond the sixth place of decimals, whilst the factors are supposed to be true to only five decimals: so that if we wish to be accurate we must reject all the decimals on the right of the vertical line. By the rule below, we may perform the multiplication so as to exclude all that part of the work on the right of our vertical line, thereby shortening the work, and still obtaining the product with the same degree of accuracy as by the usual rule.

Our rule for contracting the work of multiplying decimals is as follows:

RULE. I. Multiply the multiplicand by the left-hand figure of the multiplier.

II. Multiply the multiplicand, deprived of its right-hand figure, by the second figure of the multiplier, counting from the left.

III. Multiply the multiplicand, deprived of its two righthand figures, by the third figure of the multiplie, counting from the left.

Continue this process until all the figures of the multiplier have been used. Observe to place the successive products so that their right-hand figures shall be directly under each other.

Note.-In omitting successively the different figures on the right of the multiplicand, we must so far use them as to determine what there would be to carry into the next column.

Examples. 1. Multiply 0.37894 by 0.67452.

OPERATION.

0.37894
0.67452
227364
26526
1516
189

Ans. 0.255602

EXPLANATION. First. We multiply the multiplicand 0.37894 by 6, the left-hand figure of the multiplier, which gives the first partial product 227364.

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. 2. Multiply 0.3785 by 0.4673.

OPERATION.

0.3785
0.4673
15140

2271

265

11

Ans. 0.17687 3. What is the product of 0.00524486 by 0.99993682?

OPERATION,
0.00524486
0.99993682
4720374
472037
47204
4720
157
31

Ans. 0.005244527 4. Multiply 108.2808251671 by 1.9614591767.

Ans. 212.3884181846. 5. Multiply 0.009416517988 by 0.999936883996,

· Ans. 0.0094159236548. 6. Multiply 0.0000375229 by 0.0000275177. 7. Multiply 0.999936883996 by 0.999955663612. 8. Multiply 0.587401052 by 0.018468950. 9. Multiply 91.6264232009 by 0.0172021234. 10. Multiply 212.3880258928 into itself.

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 so many figures in the quotient as this excess, supply the de. ficiency by prefixing ciphers.

Examples. 1. Divide 3.475 by 4.789.

OPERATION.
4,789)3.475000(0.725

33523
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.

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