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Parts; and the Numerator 32 fhews that 32 of those Parts are fignified by the Fraction, or to be taken for its Value.

There are two Sorts of Fractions common in Practice; namely, Vulgar, of which we are now treating; and Decimal, of which hereafter.

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Vulgar Fractions are of feveral Sorts; that is, either proper or improper, fimple or compound. A proper Fraction is that whofe Numerator is lefs than the Denominator, as,,,, or the like. A Fraction is called improper, when the Numerator is equal to, or greater than, the Denominator, as, 2, , or the like. Thefe are called improper Fractions, because they contain the Whole of an Integer when the Numerator is equal to the Denominator, and more than an Integer when the Numerator is greater than the Denominator.

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A fimple or fingle Fraction is that which has but one Numerator and one Denominator, as, or any of the Fractions above fet down. A compound Fraction is that which hath more than one Numerator and Denominator, and hath the Word of interpofed betwixt the feveral Parts of which it confifts: Thus of is a compound Fraction, expreffing one Half of three Fourths of an Integer: Soof of is a compound Fraction, fignifying three Fourths of five Sevenths of two Ninths of an Integer, or whole Thing. Which Kind of Fractions are ufually called Fractions of Fractions: Thus a Farthing is the one Fourth of a Penny, which is the one Twelfth of a Shilling, which is the one Twentieth of a Pound; and by Fractions is wrote thus, of of of a Pound Sterling.

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Proper fimple Fractions often arife in Divifion of whole. Numbers; for, when the Divifor will not equally divide the Dividend, but there is a Remainder, this Remainder is the Numerator of a Fraction, whofe Denominator is the Divifor : This Fraction, annexed to the integral Part of the Quotient, gives the true Quotient of the Work. Thus, if 26 Pounds were to be divided among 3 Men, the Quotient will be 8; and there will be a Remainder of 2 Pounds; which, being placed over the Divifor 3, makes the Fraction of a Pound, to be annexed to the 8; fo that the true Anfwer is 8 of a Pound, that is, 8 Pounds, and of a Pound, for each Man. The third Part of a Pound being 6s. 8d. two Thirds are 135. 4 d. fo that the true Answer is 81. 13. 4d.

When a Fraction is annexed to a whole Number, this is called a mixed Number. Thus, in the Example above, 8 is a mixed Number, 8 being the integral Part, and the fractional

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fractional Part; that is, it is 8 Pounds, and two Thirds of a Pound more. Alfo, if you would exprefs 187. 15 s. it is to be wrote thus, 18, or, which is the fame, 18, that is, 18 Pounds, and three Fourths of a Pound, which is 15s. for, theof a Pound being 5s. three Fourths will be 15s. This, therefore, is a mixed Number, 18 being the integral Part, and, or, the fractional Part.

Having fhewn the Method of the Notation of Fractions, and the feveral Kinds, I proceed to

The Reduction of Vulgar Fractions.

Article 1. To reduce a whole Number to an improper Fraction.

Rule. Place the given whole Number as a Numerator, and under it 1, or Unity, for a Denominator.

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Exam. 1. Under the 8, given to be reduced to an improper Fraction, place 1, with a Line drawn betwixt, which makes

8

and is to be read, 8 Units, or 8 Wholes. The other Examples are done in the fame Manner, and do not require Explanation. But it will be proper to obferve, if a whole Number be given to be reduced to an improper Fraction, whofe Denominator is given, that, in fuch Cafe, the Rule is, multiply the given Integer by fuch affigned Denominator, and place the Product over the given Denominator; fo will that be an improper Fraction, equal to the given whole Number,

Exam. 4.

Exam. 5.

Reduce 6 to an Reduce 8 to an improper Fraction, improper Fraction, whofe Denomina- whofe Denomina

tor is 2.

8

2

Exam. 6, Reduce 12 to an improper Fraction, whofe Denominator is 15.

15

12

tor is 4.

6

4

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Exam. 4. The Integer 6 being given to be reduced to an improper Fraction, whofe Denominator fhall be 4; multiply 6 by 4, the Product is 24; which placed over 4 gives, equal to 6; as appears hereafter by the Rule for finding the Value of a Vulgar Fraction.

Exam. 5. The Integer 8 being given to be reduced to an improper Fraction, whofe Denominator is 2; multiply 8 by 2, placing the Product 16 over the given Denominator 2; fo is 16 the improper Fraction fought.

Exam. 6. Multiply the given Integer 12 by the given Denominator 15 (or 15 by 12, which is the fame); the Product 180 placed over 15 is 180, the improper Fraction fought.

Article 2. To reduce a mixed Number to an improper Fraction.

Rule. Multiply the integral Part of the mixed Number by the Denominator of the fractional Part, and then add to the Product the Numerator of the Fraction; this Sum is the new Numerator; under which place the Denominator in the given mixed Number, and this will be the Fraction fought,

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Exam. 1. Multiply 10 by 5 the Denominator, and to the Product 50 add 4 the Numerator; the Sum 54 is the new Numerator, to be placed over 5 the Denominator of the given Fraction; fo is the Fraction fought.

Exam. 2. Multiply 18 by 4, and to the Product 72 add 3; the Sum 75 placed over 4 gives 25, the Fraction fought. The third Example is done in the fame Manner.

Exam. 4.

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Art. 3. To reduce compound Fractions to fimple Fractions. Rule. Multiply all the Numerators together for a new Numerator, and all the Denominators for a new Denominator; the Product of the Numerators, being placed over the Product of the Denominators, will be the fimple Fraction fought.

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Exam. 1. The two Numbers 1 and 3 multiplied give 3 for the new Numerator; then multiply the two Denominators 2 and 4, which makes 8, for the new Denominator; fo is the fimple Fraction fought.

Exam. 2. The Numerators, being all Units, in multiplying produce I for the new Numerator. The Denominators 4, 12, and 20, multiplied together, give 960 for a new Denominator: Thus is the Fraction fought.

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The following Examples are done in the fame Manner.

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Article 4. To reduce fimple Fractions having different Denominators to Fractions of the fame Denominator.

Rule. Multiply each Numerator into all the Denominators except its own; thefe Products are the new Numerators for each Fraction whofe Numerator was fo multiplied: Then multiply all the Denominators together; this Product is the common Denominator to all the Fractions.

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