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2. What is the product of 520°3 and 417 ?

Ans. 216'9651. 3. What is the product of 51.6 and 21? Ans. 1083'6. 4. What is the product of 217 and 0431 ?

Ans. 0093527.

5. What is the product of 051 and c09f?

Ans. 0004641.

NOTE. When decimals are to be multiplied by 1c, or 100, or 1000, &c. that is, by 1 with any number of cyphers, it is done by only moving the decimal point so many places further to the right hand, as there are cyphers in the said multiplier; subjoining cyphers if there be not so many figures.

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When the product would contain several more decimals than are neccessary for the purpose in hand, the work may be much contracted, and only the proper number of decimals retained.

RULE.

1. Set the unit figure of the multiplier under such decimal place of the multiplicand as you intend the last of

your

product

product shall be, writing the other figures of the multiplier in an inverted order.

2. Then, in multiplying, reject all the figures in the multiplicand, which are on the right of the figure you are multiplying by; setting down the products so that their right-hand figures may fall each in a straight line under the preceding; and carrying to such right-hand figures from the product of the two preceding figures in the multiplicand thus, viz. 1 from 5 to 14, 2 from 15 to 24, 3 from 25 to 34, &c. inclusively; and the sum of the lines will be the product to the number of decimals required, and will commonly be the nearest unit in the last figure.

EXAMPLES.

1. Multiply 27 14986 by 92°41035, so as to retain only four places of decimals in the product.

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2. Multiply 480 14936 by 2'72416, retaining four decimals in the product.

Ans. 1308.0036.

3. Multiply 73 8429753 by 4'628754, retaining five decimals in the product.

Ans. 34180097.

4. Multiply 8634875 by 8437527, retaining only the integers in the product.

Ans. 7285699.

L

DIVISION

DIVISION of DECIMALS.

RULE.*

Divide as in whole numbers; and to know how many decimals to point off in the quotient, observe the following rules:

1. There must be as many decimals in the dividend, as in both the divisor and quotient; therefore point off for decimals in the quotient so many figures, as the decimal places, in the dividend exceed those in the divisor.

2. If the figures in the quotient are not so many as the rule requires, supply the defect by prefixing cyphers.

3. If the decimal places in the divisor be more than those in the dividend, add cyphers as decimals to the dividend, till the number of decimals in the dividend be equal to those in the divisor, and the quotient will be integers till all these decimals are used. And, in case of a remainder, after all the figures of the dividend are used, and more figures are wanted in the quotient, annex cyphers to the remainder, to continue the division as far as necessary.

4. The first figure of the quotient will possess the same place of integers or decimals, as that figure of the dividend, which stands over the units place of the first product.

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*The reason of pointing off as many decimal places in the quotient, as those in the dividend exceed those in the divisor, will

2. Divide 3877875 by '675. 3. Divide 0081892 by *347. 4. Divide 7:13 by 18.

CONTRACTIONS.

Ans. 5745000. Ans. 0236. Ans. 39.

I. If the divisor be an integer with any number of cyphers at the end; cut them off, and remove the decimal point in the dividend so many places further to the left, as there were cyphers cut off, prefixing cyphers, if need be; then proceed as before.

EXAMPLES.

1. Divide 953 by 21000.

21.000)

3)'953

7).31766

*04538, &c.

Here I first divide by 3, and then by 7, because 3 times 7 is 21. 2. Divide 41020 by 32000.

Ans. 1281875.

NOTE. Hence, if the divisor be 1 with cyphers, the quotient will be the same figures with the dividend, having the deci mal point so many places further to the left, as there are cyphers in the divisor.

2173100 = 2.173.

EXAMPLES,

5.16 by 1000*00516,

419 by 10 = 41′9.
•21 by 1000 = '00021.

II. When the number of figures in the divisor is great, the operation may be contracted, and the necessary number of decimal places obtained.

RULE.

1. Having, by the 4th general rule, found what place of decimals or integers the first figure of the quotient will pos

sess;

easily appear; for since the number of decimal places in the dividend is equal to those in the divisor and quotient, taken together, by the nature of multiplication; it follows, that the quotient contains as many as the dividend exceeds the divisor.

sess; consider how many figures of the quotient will serve the present purpose; then take the same number of the left-hand figures of the divisor, and as many of the dividend figures as will contain them (less than ten times); by these find the first figure of the quotient.

2. And for each following figure, divide the last remainder by the divisor, wanting one figure to the right more than before, but observing what must be carried to the first product for such omitted figures, as in the contraction of Multiplication; and continue the operation till the divisor be exhausted.

3. When there are not so many figures in the divisor, as are required to be in the quotient, begin the division with all the figures as usual, and continue it till the number of figures in the divisor and those remaining to be found in the quotient be equal; after which use the contraction.

EXAMPLES.

1. Divide 2508 928065051 by 92'41035, so as to have four decimals in the quotient.In this case, the quotient will contain six figures. Hence

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