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Multiplication of Decimal Fractions.

Article 42. To multiply Decimal Fractions.

Rule. This is performed in the fame Manner as in Multiplication of whole Numbers; only, when the Multiplication is ended, there must be as many Places cut off in the Product, for Decimals, as there are decimal Places in the Multiplier and Multiplicand.

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Exam. 1. The Multiplication being performed as in whole Numbers, feven Places must be marked off for Decimals, because there are four decimal Places in the Multiplicand, and three in the Multiplier; therefore the Product is 1875.951763, the Cypher on the right Hand, or laft Place, being of no Signification.

Exam. 2. Multiply as in whole Numbers, and the Product is 1875, which must be all marked off for Decimals, because there are four decimal Places in the given Numbers, viz. two in the Multiplier and two in the Multiplicand..

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Exam. 3. There are three decimal Places in the Multiplicand, and two in the Multiplier, therefore there muft_be

five Places marked off in the Product for Decimals, which are all the Figures of the Product: Whence the Product is .25, rejecting the Cyphers as infignificant.

In the fourth Example, there are but two decimal Places in the Multiplicand, and none in the Multiplier, therefore there are but two Places to be marked off for Decimals in the Product.

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In each of these Examples, the Number of Decimal Places in the respective Multipliers and Multiplicands exceed the Number of Places in the Products; therefore Cyphers are annexed to the left Hand, to make the Number of Places of Decimals in the feveral Products equal to the decimal Places in their given Multipliers and Multiplicands.

The Learner will fee, from the next Examples, that, if Cyphers were not to be prefixed to the left Hand of the Product, to make the Number of Places of Decimal Fractions equal to thofe in the Numbers multiplied, the Product will be erroneous.

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Now, if a Cypher was not prefixed to the ninth Example, the Product of both Multiplications would be the fame, which is abfurd.

In multiplying whole Numbers with 3 or 4 decimal Places annexed both to the Multiplier and Multiplicand, the Work is somewhat tedious; and it often is not neceffary to have the

1 2

Product

Product nearer than to 3 or 4 Places of Decimals, a thousandth or a ten-thousandth Part of the Integer being near enough in moft Cafes; it will be proper, therefore, to inftruct the young Mathematician how to contract Multiplication, fo as to have but just fo many Places of Decimals in the Product as are thought neceffary.

Article 43. To contract Multiplication.

Rule. Place the Units of the Multiplier under the fame decimal Place of the Multiplicand, as is required to be the laft Place in the Product, and invert all the other Figures of the Multiplier; then begin multiplying each Figure, at that which stands immediately over it, adding what is to be carried from the Figure that ftands next to it on the right Hand, as will be fhewn in the following Examples..

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Exam. 1. The Figures of the Multiplier being inverted, and the Units fet under the fourth Place of Decimals, multiply the 7 into the 5 which is over it, and the Product is 35; but, as there is a 2 to the right Hand, which being multiplied into 7 is 14, from which I ought to be carried, and, as the Excefs above 10 is but 4, there must be only I carried, which makes 36, fet down the 6 and carry the 3, proceeding to multiply the 7 thro' all the other Figures in the Multiplicand, as in common Multiplication: Then begin multiplying by the 4, first into the 5, which ftands next to the

Figure that is over it, in order to know what must be carried, and the Product is 20; therefore 2 must be carried; then fay, 4 times 6 is 24, and 2 that I carried is 26; now set down this 6 under the former 6, carrying 2 to the next, and proceed as in common Multiplication. In the fame Manner every Figure must first be multiplied into that which stands next to the Figure that is over it, in order to know what is to be carried; and, if it makes 5 above any Number of Tens, one more must be carried for that 5, i, e. if it amounts to 15 you must carry 2, if to 25 you must carry 3, &c. because, in Cafe all the Figures were multiplied, the odd ones in caftingup increase to Tens, For which Reason, in multiplying by the 3, tho' that, being multiplied into the 8 which ftands next to it, makes but 24, yet, as there was a 4 rejected in the first Line, i. e. I carried but I for 14, therefore I here carry 3, tho' it is not more than 24. And, if this be carefully obferved, there will scarce be a ten-thoufandth Part of an Unit loft in this Method; as is the Cafe here, for all the Figures being multiplied in the common Way, the Product is 4328.70575804281.

Exam, 2. This being done in the fame Manner, the Product is 11004.1286; and, if done at full Length, is 11004.1287144916, little more than one hundred-thoufandth Part of an Unit more than comes out in the Contraction, which is nearer than is neceffary in most common Affairs; where it will be fufficiently near if contracted to two Places of Decimals. Inftead of giving other Examples, I fhall fet down the two former contracted to two Places, by which the Learner will fee that the Difference is fo little, as not to be worth the Trouble in common Cafes.

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Exam. 1. In multiplying by the 7 it must be observed, that, tho' 7 times 6 is but 42, yet, as there is a 5 in the next Place, which being multiplied by 7 is 35, and from which, according to this Rule, 4 fhould be carried, this will make it 46; fo that 5 must be carried to the 21, arifing from multiplying the 7 into the 3 that stands over it, which makes it 26; fet down the 6, and proceed in the Multiplication as before; in fuch fhort Contractions always taking Care not to omit carrying to the full Extent of the Rule. The Product here' falls fhort but .005; which, if it were in Foot-Measure, is not the of an Inch; but, in the Fraction of à Pound Sterling, is fomewhat more than one Penny.

Exam. 2. In multiplying by the 2 I carry 1, but in multiplying by the 8 I carry 8; because the 7 being multiplied by 8 makes 56; then multiplying the 9 by 8, and carrying 5 from the 56, it makes 77, which is more than is required for carrying 8 by the Rule. The fame Rule being obferved in the other Figures, the Product is 11004.13, which is but .0013 more than before; and which, in the Fraction of a Foot, is one hundredth Part of an Inch, or little more; but, in the Fraction of a Pound, is fomewhat more than one Farthing.

Art. 44. Divifion of Decimal Fractions.

Rule. Divide, in all Respects, as in whole Numbers; only there is fome Difficulty in afcertaining the Number of Decimal Places in the Quotient, which fhall be taught in the Examples.

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