EXAMPLES. Let it be required to divide 192,1 by 7,684, and 441 by,7875. Cafe 4. If after Divifion is finished, there are not fo many Figures in the Quotient, as there ought to be Places of Parts by the general Rule; fupply their defect by prefixing Cyphers to it. EXAMPLES. Let it be required to divide 7,25406 by 957. 957) 7,25406 (,00758 the true Quotient required. Note, When Decimal Numbers are to be divided by 10. 100. 1000. 10000. &c. that is, when the Divifor is an Unit with Cyphers; Divifion is performed by removing or placing the feparating Point in the Dividend, fo many Places towards the Left-hand, as there are Cyphers in the Divifor. EXAMPLE. 10) 5784 (578,4 1000) 5784 (5,784 100) 578,4 (57,84 10000) 578,4 (,05784 Note, Thefe Operations are the direct Converse to thofe in page 62. I prefume it needlefs to give more Examples at large; only I fhall infert a few Dividends, and Divifors, with their Quotients, wherein are contained all the Varieties that can happen in Divifion of Decimals. 574) 493,066 (859 574) 493,066 (,859 574) 49,3066 (,0859 5,74) 4930,66 (859 5,74) 49,3066 (8,59 5,74) 493066,00 (85900 0574) 493,0665 (8590 0574) 493066 (8,59 K 2 There There is alfo a compendious Way of contracting Divifion, like that of Multiplication, page 64, by which much Labour may be faved; efpecially when the Divifor hath many Places of Decima I Parts in it: And it is thus performed. Having determined how many Places of whole Numbers there will be in the Quotient, if any at all; or if none, of what Value or Place the firft Figure in the Quotient will be: Then omit, or prick off one Figure of the Divifor at each Operation; viz. for every Figure you place in the Quotient, prick off one in the Divifor; having a due Regard to the Increase which would arife from the Figure fo omitted. EXAMPLE. Let it be required to divide 70,23 by 7,9863. The Work contracted I prefume is fo obvious (if compared with the fame at large) that it is needless to give any farther Explanation of it. Sect. 5. To Reduce Uulgar Fractions into Decimals, and the contrary. ANY Vulgar Fraction being given, it may be reduced, or rather changed, into Decimal Parts equivalent to it. Thus, Rule Annex Cyphers to the Numerator, and then divide it by the Denominator, the Quotient will be the Decimal Parts equivalent to the given Fraction; or at least fo near it as may be thought necessary to approach.. EXAMPLE. EXAMPLE. 4 It is required to change or reduce into Decimals. Again,5; thus 2)1,0 (,5. And 4,25; 4) 1,00 (,25 7) 4,0000000000 (5714285714 &c. =$. Nate, When the laft Figure of the Divifor, (that is, the Denominator of the propofed Fraction) happens to be one of these Figures; viz. 1.3.7. or 9. (as in the Example) then the Decimal Parts can never be precisely equal to the given Fraction; yet by continuing the Divifion on, you may bring them to be very near the Truth. As in this Example; Suppose it was required to change into Decimal Parts. 13) 1,0000 ),07692307692307 &c. ad infinitum. These being understood, it will be easy to find the Decimal Parts equivalent to any known Part or Parts of Coin, Weights, Measures, Time, &c. If you firft reduce the given Parts of Coin, &c. into a Vulgar Fraction, whofe Denominator is the Number of thofe known Parts contained in the Integer, and the given Parts it's Numerator. Examples in Coin, &c. 1. Let it be required to find the Decimals of 16 s. 6d. Firft 16 s. of one Pound, and 6 d. 4 of 1 l. But 18+4=48. Then 40) 33,000 (,825 the Decimal Parts required: That is, 825 = 16 s. 6 d. Again, Suppose it were required to find the Decimals equal to 31.135.44. Here Here 3. is 3 Integers, and 13 s. = 28 of 1 l. and 4 d. 140. But 2018. Then 240) 160,000 (0,666666 . Hence 3. 13 s. 4 d. 3,666666 &c. As was required. 2. What are the Decimals equal to 74 Inches, one Foot being made the Integer. Firft, 7 Inches are 2 of 1 Foot, and of Inch are 48. But +4. Then 48) 31,000 (,64583 &c.—7 Inches. 3. Let it be required to change 8 Oz. 19 Pwt. 8 Grains into Decimals; one Pound Troy being the Integer. These being reduced into the leaft Terms, and added together, will become of 1 Pound, Then 5760) 4304,000 (74722 &c. The Decimals required. And thus may any propofed Parts of Coin, Weights, Measures, &c. be reduced or changed into Decimal Parts; which perhaps may at first seem somewhat tedious in Practice, but being a little acquainted with them it will be found very eaf; and the ingenious Practitioner will (with a little Confideratio) foon find how to reduce them almoft mentally; or with the help of a very few Figures, without the Ufe of fuch large Tables as are ufually inferted in Books of Decimal Arithmetick; or at moft they may be contracted into fuch as thefe following, which if duly applied to those Tables in Chap. 3. will be found very useful. In English Coin. 0,05.....=IS. 10,0046667 = 1 d. Decimal Tables. 0,00104167 = 1 Farthing. 11. being the Integer. Troy Weight. 10,05... = Put. 0,00208333 = 1 Grain. Oz. being the Integer. Apothecaries Weight. 10,125..... = 1 Dram. 0,04166667 = 1 Scruple. 10,00208333 = 1 Grain. Oz. being the Integer, 0,04166667 Hour." The Ufe of thefe Tables will be evident by the following EXAMPLE EXAMPLE. Let it be required to find the Decimal Parts equivalent to 17 s. 9 d. 2 Farthings. Firft 0,05=IS. Therefore 17x,05=,85 . . . .=17 s. And,004166=1d. Therefore ,004166x9=,037494-9 d. Confequently their Sum, viz. 0,889577=175.9%d. Now to find the Value of Decimals in known Parts of Coin or Weights, &c. is only the Converfe of the former Work, and is thus performed. Multiply the given Decimals with the Denominator of the Vulgar Fraction required: That is, multiply the Decimals with fuch a Number of Units, as are contained in the next lower Denomination of that Kind or Species' which your Decimal is of; and the Product will be the Number required. EXAMPLE. 1. What is the Value of 0,825 Decimals of 1 Pound Sterling. That is, how many Shillings, Pence, &c.,825. First, the next lower Denomination is 20, because 20 s. make one Pound. Therefore 0,825 20 Shillings 16,500 and Parts of 1 Shilling, 12 6,00 Answer 0,825 = 16 s. 6 d. Again, What are the known Parts of English Coin equal to 3,666666 Decimals. Here the 3 Integers are 3 Pounds. Then,666666 20 3,9998404 near. What is the Value of 0,74722 Parts of 1 lb Troy. Firft, 74722 Then,,96664 Again,,33280 12 20 24 |