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Exam. 17. Here I place a Cypher to the left Hand of the 72, to have as many Fractions in the Product as there are in the Divifor; then I place a Dot, and a Cypher to the left of that, for the Place of Units; which would have been under the third Place of Integers, if the Dividend had been continued to the left Hand; and therefore I place the Dot in the Quotient accordingly.

The eighteenth Example is done in the fame Manner.

Now the Reader is a little familiar with thefe Operations, he may learn to fix the Quotient before the Divifion is ended; which, in fome Cafes, may fave Labour; for, as foon as the firft Subtraction is made, the Reader may diftinguish what will be the Denomination of the firft Figure in the Quotient; and fo know to how many Figures he fhould continue the Quotient to have two Places of it Decimals, which, in most Cafes, are fufficient. As fuppofe,

24.7) 5498.6734 (2

49.4

5

I obferve, that the q in the Place of Units ftands under the Place of Hundreds in the Dividend; which being in the third Place of Integers, I muft continue the Divifion to five Figures in the Quotient, to have two of them Decimals; and, if all the Dividend fhould not be taken down, the rest may be neglected, if two Places of Fractions are near enough; and,

if there are not Figures enough in the Dividend to continue the Quotient to fo many Places as are wanted, then that must be fupplied by taking down Cyphers, as in the former Examples.

There is another Way of determining the Number of Decimals in the Quotient; which is this, count the Number of Decimals in the Divifor, and then the Number of Decimals in the Dividend; and, if there are more Places of Decimals in the Dividend than there are in the Divifor, mark off so many Places for Decimals in the Quotient as the Number of Decimals in the Dividend exceed the Divifor. Thus, in Example 1, there are five decimal Places in the Dividend, and but three in the Divifor, therefore there must be two in the Quotient. In the fecond Example, there are four Places of Decimals in the Dividend, and but two in the Divifor, for which Reason there are two Places cut off in the Quotient. But, in this Cafe, it must be carefully observed, whether in the Operation any Cyphers have been added to the Dividend; for, if there have been Cyphers added, they muft he reckoned as fo many Places of Fractions,

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Exam. 19. There being no Units Place in the Divifor, if it was fupplied by putting a Cypher in the Units Place, to the left Hand of the firft Product 32, that Cypher would ftand one Place higher than the firft Figure, 4, in the Dividend, that is, in the Place of Thousands; therefore the 2, the first Figure in the Quotient, must be in the Place of Thousands; but, as there are but two Figures in the Quotient, and the Divifion being ended, the Places of Tens and Units must be fupplied with Cyphers: So is 2700. the Quotient fought.

Exam. 20. The Units Place in the firft Product stands under the 8 in the Dividend, which is the fourth Place of Integers, or Place of Thousands; therefore the firft Figure in the Quo

tient must be Thousands; but, as there are no more than three Places of Figures in the Quotient, and the Divifion being ended, a Cypher must be annexed to fupply the Units Place.

To give the Learner a clearer Idea of Divifion of Decimals, let 925 be propofed to be divided by 25; firft in whole Numbers, and afterwards by reducing firft the Divifor, and then the Dividend, gradually, to Decimal Fractions; by which the Reason of the Difference of the Quotients will appear.

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By thefe eight Examples it may be feen, that, tho' the Figures in the Quotient are ftill the fame, yet they vary in their Denomination or Power, as the Divifor or Dividend are

made

made Fractions; increasing, by making the Divifor a Decimal, from 37 to 37000.; and decreafing, by making the Dividend a Fraction, from 37 to .0037. The Truth of any of thefe Operations may be proved by multiplying the Divifor and Quotient, and marking the Decimals in the Products according to the Rule for Multiplication; which will then agree with the Dividend, in Integers and Fractions, if the Work is true.

When there are many Figures in the Divifor in Divifion of Decimals, and the Quotient is required only to be very near the Truth, the Work may be contracted, fo as to fave Trouble, without any confiderable Lofs.

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Article 45. To contract Divifion of Decimals.

Rule. Firft cut off from the Divifor all the Places beyond the fourth, at leaft, if not to the third; then begin dividing the Dividend, and find the Denomination of the first Figure in the Quotient, by Art. 44: By this is known how many Figures are required in the Quotient to bring out the Number of Decimals defigned: As for Example, if you defign to have three Places of Decimals, and the firft Figure in the Quotient is in the Units Place, then you must have four Figures in the Quotient; if it be Tens, there will be five Places; if it be Hundreds, there will be fix, &c. that is, you must have so many Figures in the Quotient, that, after the required Number of Decimals in the Quotient are fet off, the first Figure in the Quotient may be of that Denomination as is required by the Rule for Divifion of Decimals. Now, knowing how many Places of Figures are required in the Quotient, obferve whether the Divifor has the fame Number of Figures as are required in the Quotient; and, if it has, proceed to dividing, after having cut off the Figures in the Dividend (if there be any) beyond where the firft Product reached. The reft fhall be fhewn in the Examples.

Exam. I.

Divide 3946.87593

By 4.2753, fo as to have three Places of Decimals,

4.27513) 3946.875197 (9)
38.475

Exam. I,

Exam. 1. Having reduced the Divifor to three Places of Decimals, by cutting off the 3 in the fourth Place with a Dafh, I proceed to divide as in the usual Manner; and, by placing the Quotient Figure, 9, in the Quotient, and multiplying the Divifor by 9, I have 38475 for the firft Product; three Places of which being marked off as Decimals, because there are three decimal Places in the Divifor, I find the 8 in the Place of Units in the Product ftands under the 9 in the Place of Hundreds in the Dividend; therefore the firft Figure, 9, in the Quotient must be in the Place of Hundreds ; fo that the Quotient must have fix Figures, viz. three Places of Integers and three of Decimals; which being more Figures by two than there are in the Divifor, there must be two Figures left in the Dividend beyond where the firft Product reached, and but two cut off, viz. the 9 and the 7, at the End: But, if the Divifor had confifted of as many Places as are required in the Quotient, the laft four Figures might have been cut off: I fhall, therefore, fet it down exclufive of those two Figures, and perform the Remainder of the Work.

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When the two remaining Figures of the Dividend are taken down, as in common Divifion, I put a Dot under the 5, the laft Figure in the Divifor, to denote that it is not to

be

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