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be used in the future Part of the Work: Then asking how many times 427, the other three Figures of the Divifor, may be had in the Remainder 1050, which is now the Dividend, or how many times 4 can be had in 10, according to the ufual Method, I find I can have it twice; therefore, putting the 2 in the Quotient, I multiply the Divifor 4.27 by 2, (having first multiplied the 2 into the 5 that is rejected, in order to know what Increase is to be carried, which is 1, for twice 5 is 10, from whence I must be carried) faying, twice is 14, and the I that I carried is 15; therefore I set down the 5, and carry the I; then say, twice 2 is 4, and the I that I carried is 5; laftly, twice 4 is 8, which I fet down; now fubtract this Product 855 from 1050, and the Remainder is the next Dividend: Then putting a Dot under the 7 in the Divifor, to denote that it is to be rejected, I ask how many times 42 I can have in 195, or how many times 4 can be had in 19, according to the ufual Method, which I find is 4 times; then fetting down the 4 in the Quotient, and mul-. tiplying the 7 which was laft marked off, to know what Increase must be carried, I fay, 4 times 7 is 28; which being more than 25, I carry 3, as was taught in contracted Multiplication; then fay, 4 times 2 is 8, and the 3 that I carried is 11, from which I must be carried; laftly, 4 times 4 is 16, and the 1 that I carried is 17, which I fet down; now fubtract this Product from 195, and the Remainder is 24, which is now the Dividend: Then, putting a Dot under the 2 in the Divifor, I ask how many times 4, the only remaining Figure of the Divifor, I can have in 24, and the Answer is 6 times: So that the Quotient to three Places of Decimals, as was required, is 923.246. But, in regard that 6 times 4 is 24, which is all that is in the Dividend, it appears from hence, that this Quotient rather exceeds than falls fhort of the true Quotient; because, according to the Rule of carrying the Increase arifing from the rejected Figures, it would make 25, if we regard only the multiplying the 6 into the 2; tho', the Figure beyond that being a 7, and, as 6 times 7 is 42, if that 4 is carried to the 6 times 2, it will make 16; from which carrying 2, according to the Strictness of the Rule, it will make 26, which is too much. By working this Example at Length, I find the Quotient is 923.24584, &c. which is but little more than one ten-thousandth Part of an Unit lefs than the other. This I thought proper to remark, to fhew the Learner how near the contracted Mehod comes to the Truth;

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and, at the fame Time, the Work itfelf fhews whether it be too little or too much, tho' only by fuch a Trifle.

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Exam. 2. The firft Figure in the Quotient appearing, by Art. 44, to be in the Place of Tens, there must be five Figures in the Quotient, in order to have three Places of Decimals, as required; and, there being five Places of Figures in the Divifor, I cut off the other four Figures in the Dividend: Then, proceeding to mark off one Figure in the Divifor each Time, I first inquire how many times 25.29 can be had in the Remainder 16493, which is the prefent Dividend, or how many times 2 can be had in 16, as ufual; and I find I can have it 6 times; therefore, I fay 6 times 3, the Figure that is marked off, is 18, for which I carry 2, it being near 20; then fay 6 times 9 is 54, and the 2 that I carried is 56; now put down the 6 and carry the 5, as in common Multiplication; and, the Product being obtained, fubtract it from the Dividend, and the Remainder 1317 is the next Dividend: Then marking off the 9, the fecond Figure in the Divifor, I afk how often 25.2 can be had in this Dividend, and I find I can have it 5 times;

faying 5 times 9 is 45, for which I carry 5; then fay 5 times 2 is 10, and the 5 that I carried is 15; now fet down the 5, and proceed to the reft; and the Product is 1265, which, being fubtracted from the Dividend, leaves 52 for the next Dividend: Then marking off the 2, the third Figure in the Divifor, I afk how many times 25 may be had in 52, which is twice; then multiply and fubtract as before, and there remains 2: Laftly, mark off the 5, and then the Divifor is reduced to the 2, which may be had once in 2, and the Work is ended. The other Example is done the fame Way; always obferving to have Regard to what is to be carried on account of the Figures marked off in the Divifor.

Thus having gone thro' the Addition, Subtraction, Multiplication, and Divifion of Decimal Fractions, it will now be proper to inftruct the Learner in the Articles of Reduction; which, not being neceffary before, could not have been fo well understood, as the Method of doing them requires the Knowledge of fome of thofe Rules.

Reduction of Decimal Fractions.

There are three Rules of Reduction of Decimals, viz. To reduce Vulgar Fractions to Decimal Fractions; To find the Value of Decimal Fractions in the accuftomary Divifions of Coin, Weights, Measures, &c. And to reduce the customary Divifions of Coin, Weights, Measures, &c. into Decimals of the refpective Integers,

Article 46, To reduce Vulgar Fractions to Decimals of the fame Value.

Rule. Annex Cyphers, at Pleasure, to the Numerator, marking them as Decimals; then divide by the Denominator, and fet off the Quotient according to the Rule for Divifion of Decimals, confidering the Divifor as Integers; and this will be the Decimal fought in lieu of the Vulgar Fraction: And, if the Divifion fhould not be ended, the Quotient may be continued to any Degree of Exactnefs by taking down Cyphers, as has been taught in the Divifion of Decimal Fractions.

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Exam. 1. Annex a Cypher to the Numerator 2, and mark the Cypher for a Fraction; then divide by the Denominator 5, which must be confidered as an Integer, and the Quotient is 4; which must be in the firft Place of Decimals, as the o, in the Units Place in the Product which is fubtracted from the Dividend, ftands under the firft Place of Fractions in the Dividend.

Exam. 2. Here three Cyphers are annexed, but a greater or lefs Number might have been annexed; the Divifion, being ended, the Quotient is .125, according to the Rule for Divifion of Decimals.

So, in the third Example, the Quotient is .3125, a Decimal Fraction in lieu of the given Vulgar Fraction.

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Exam. 4. The Divifion is complete, by the fecond Subtraction; therefore the laft Cypher needs not be taken down.

Exam. 5. The first Product 96 being placed under the Dividend, and the 6, which is in the Place of Units, ftanding under the second Place of Decimals in the Dividend, therefore the first Figure in the Quotient must be in the second Place of Decimals, by Art. 44; for which Reason a Cypher muft be prefixed, and then .0625 is the Decimal Fraction equal to 18.

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Exam. 6. The Remainder being the fame at every Subtraction, if Cyphers were continued to be annexed, the Quo tient Figures would be 3, or the fame, continually, however far continued: In all fuch Cafes, it is ufual to fet down as many Places of Fractions in the Quotient at once as may be thought neceffary.

Exam. 7. After two Figures are obtained in the Quotient, (which are the fecond and third Places of Decimals, by Art. 44) the Remainder is but 8; fo that the next would be a Cypher; therefore the Divifion is ftopped, as it may not be neceffary to continue the Divifion by annexing more Cyphers.

Article 47. To find the Value of Decimal Fractions in the accuftomary Divifions of Coin, Weights, Measures, &c.

Rule. Multiply the Decimal Fraction by the next lower Denomination, which is to be confidered in the Multiplication as Integers, and mark off the fame Number of Places for Decimals, by a Dot, as taught in Multiplication of Decimals, at Art. 42; and the whole Numbers, if any, are of the fame

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