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June 3. Jonathan Pollard, 6 bags Pernambucca cotton, No. 10, I cwt. I qr. 13 lb.; N°. 11, 1 cwt. 2 qr. 7 lb.; N°. 12, 1 cwt. I qr.; No. 13, 1 cwt. 2 qrs. 19 lb.; No. 14, 1 cwt. 17 lb.; No. 15, 2 cwt. I qr. 21 lb.; draught and tare as above, at 2s. 74d.

10. Forbes & Bell, 4 bags ditto, No. 16, 1 cwt. 2 qr. 14 lb.; No. 17, I cwt. 2 qr.; No. 18, I cwt. 2 qr. 8 lb.; No. 19, 1 cwt. 2 qr. 20 lb.; No. 20, 1 cwt. 2 qr.; draught and tare as above, damage on No. 17, 3 lb; at 2s. 6d. Charges, viz. freight from Greenock, L.3: 14:5; towns duty, 5s. 9d.; mending bags and canvas, 7s. 3d.; cartage to weigh-houfe, 3s. 6d.; weighing and delivering, 2s. 8d.; cellar rent, 7s. 6d.; poltages, 2s. 9d.; commission, I per cent.

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CHAP. VIII.

VULGAR FRACTIONS.

INTRODUCTION.

As the idea of numbers is acquired by obferving several objects collected, fo that of fractions by obferving an object divided into feveral parts; to understand, therefore, the nature of fractions, we muft fuppofe a unit, or any thing confidered as a whole, divided into feveral equal parts, one or more of these parts is called a fraction, and is expreffed by two numbers; one placed above a line, the other below it: for example, three fourth parts is a fraction and written thus. The number [4] under the line is called the DENOMINATOR, and fhews into how many parts the unit is divided; the number [3] above the line is called the NUMERATOR, and thews how many of these parts are expreffed.

And because the whole is always equal to the fum of its parts, the value of a fraction becomes greater, when the numerator is increased, and lefs, when the denominator is increased. Hence it follows, that if both terms be multiplied by the fame number, or divided by any number which measures both, the fraction obtained is ftill of equal value. Thus, both terms multiplied by 2 produces, and divided by 4, gives, which is of the fame value as. On this principle most of the following problems depend.

A fraction having one numerator and denominator as, is called a SIMPLE FRACTION. A fraction of a fraction, as of 2, is called a COMPOUND FRACTION. An integer with a fraction annexed, as 3 is called a MIXT NUMBER *. A fraction whose numerator is greater than its denominator, as is called an IMPROPER FRACTION.

* See page 1, Def. 3d.

Integers may be written or understood to be written in the form of vulgar fractions whofe numerator is 1; for example, 5 may be expreffed by. As fractions can, in moft cafes, be neither added nor fubtracted, till they be reduced, it is proper to begin with reduction.

Sect. I. REDUCTION OF VULGAR FRACTIONS.

PROBLEM I. To reduce mixt numbers to improper fractions, multiply the integer by the denominator of the fraction, and to the product add the numerator, under which fum place the given denominator.

Ex. Red. 3 to an improper 1. Red. 57

fraction?

3 × 4=12

add the numerator,

6. Red. 21,

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Thefe examples are reversed by those in next problem.

As every integer is equal to twice as many halves, or four times as many quarters, and fo on; therefore, every integer may be written in the form of an improper fraction, having any given denominator: the numerator is found by multiplying the integer into the denominator: thus in the above example, 3 reduced to an improper fraction whofe denominator is 4, makes and added 14 to this, as above..

PROBLEM II. To reduce improper fractions to whole or mixt numbers; divide the numerator by the denominator.

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This problem being the reverse of the former, the reason of the one may be inferred from that of the other.

PROBLEM III. To reduce fractions to their lowest terms: divide both numerator and denominator by their greatest common measure, which is found thus: divide the greater term by the lefs, and the divifor by the remainder continually, till nothing remain. The laft divifor is the greatest common meafure.

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That by dividing both the numerator and denominator by the fame number, we obtain another of equal value is evident. And if thefe divifions be performed as often as can be done, the lat divifor will be the greateft common measure, which may be fhewn from the above example. For any number which measures 617, and 540, will also measure 72 their difference, and 504 which is a multiple of 72 (being 72 × 7) and any number which measures 540, and 504, will alfo measure their difference 36, confequently, no number greater than 36, can measure 612, and 540. Again, 36 will measure them both, for it measures 72, and therefore meafures 504, and 540 (or 504 + 36) and because it measures 540 and 72 it will also measure 612 their fum; and the fame will ap ply to any other numbers.

As the above method of reducing fractions to their lowest terms is rather inconvenient in practice, we may therefore in general divide by fuch numbers as are found to anfwer on infpection: To facilitate which we obferve,

ift, Any number whofe unit's place is 2, 4, 6, 8 or o, is meafured by 2.

2d, Any number whose unit's place is 5, or o, is measured by 53d, Any number whofe figures when added, amount to an even number of 3's and 9's, is measured by 3 or y respectively.

PROBLEM IV. To reduce fractions to others of equal value, that have the fame denominator, multiply each numerator into all the denomina tors except its own, for a numerator, and multiply all the denominators together for a common denominator.

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By placing the numbers properly under one another, it will be feen that both terms of each fraction are multiplied by the fame number, and therefore their value is not altered.

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Thus in the

7 × 3 × 6
8×3+6

The lower that fractions are reduced the better is their value apprehended, and the more fimple and easy will any operation be, wherefore, after the fractions are reduced to a common denominator, if the feveral numerators and denominators be measured by the fame number, divide by that number, and write the quotients inftead of the original terms. In the above example all the terms are measured by 6.

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PROBLEM V. To reduce lower denominations to fractions of higher, place the given number for the numerator, and the value of the higher for the denominator.

If the given quantity be in more denominations than one, reduce it to the lowest for the denominator. See Ex. 2d.

If a fraction be given, multiply its denominator by the value of the higher denomination. See Ex. 3d.

If a mixt number be given reduce it to an improper fraction. See Ex. 4th.

Ex. 1ft, Red. 7d. to the fraction of a fhilling. 78. 'Anf.

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Ex. 2d, Red. 16s. 5d. to the fraction of a pound. 16s. 5d. × 12

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127 Anf.

Ex. 4th, Red. 4}d. to the 285° fraction of a pound.

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In example 2d, we reduce 16s. 5d. to 197 pence for the numerator. In example 3d. we multiply the denominator 6 by 12 and 20, or 240 the value of a pound, and the product 1440 is the denomination required, or when reduced to its lowest terms 288. In example 4th we reduce 44d. to an improper fraction by problem I. and proceed as before. The following examples are the reverfe of those in next problem: we have expreffed the answer in its lowest terms.

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PROBLEM VI. To find the value of fractions; reduce the numerator to the next lower denomination, and divide by the denominator; if there be a remainder, reduce it to the next denomination, and divide as before, and so on as far as necessary.

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In example ift, we multiply the numerator by 20, and divide the product 380 by 40, the denominator of the fraction, and obtain a quotient of 9's.; then we multiply the remainder 20, by 12, which produces 240, divided by 40, quotes 6d. without a remainder. In example 2d, we proceed in a like manner; but as there is a remainder, the quotient is completed by a fraction.

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