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Exam. 1. The compound Fraction of is reduced to the fimple Fraction, by Art. 3; fubtract the Numerator of this fimple Fraction from the Denominator, and under the Remainder place the Denominator, and we have the Fraction 17, to be annexed to the whole Number 29, lefs by the Unit that was borrowed: So is 287 the Remainder fought.

The other Example is done in the fame Manner.

Article 24. To fubtract a mixed Number from an Integer greater than the Integer of the mixed Number.

Rule. Subtract the Numerator of the fractional Part of the mixed Number from the Denominator, and place the Remainder over the Denominator, as in Art. 22; then carrying the I which was borrowed to the integral Part of the mixed Number, fubtract that from the Integer from which the mixed Number was to be taken, and annex the former remaining Fraction to this laft Remainder; this is the Answer sought. Ex. 1. Subtract 3 from 6. Ex. 2. Subtract 22 from 56.

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Exam. 1. The Numerator 2 fubtracted from the Denominator 5, there remains 3; that is, the Fraction subtracted from (which is borrowed) the Remainder is: Then say, the I that I borrowed and 3 is 4 from 6, there remains 2; to which join the; fo is 2 the Remainder required. The other Example is done by the fame Method.

Multiplication of Vulgar Fractions.

Article 25. To multiply fimple Fractions together.

Rule. Multiply all the Numerators together for a new Numerator; then multiply all the Denominators together for a new Denominator; fo is this new Fraction the Produc fought.

Exam. 1. Multiply by . Exam. 2. Multiply by .

IS

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32

Exam. 1. The Numerators 3 and 7 multiplied give 21; and the Denominators 4 and 8 multiplied are 32: Therefore the Fraction is the Product of by. The fame of the other Example. And, if there had been more than two Fractions given, the Numerators and Denominators must all have been multiplied together in the fame Manner.

Article 26. To multiply compound Fractions and fimple Fractions together, or compound with compound.

Rule. Multiply all the Numerators, both of the feveral Parts of which the compound Fraction confifts, and of the fimple Fraction, and the laft Product is a new Numerator; then multiplying all the Denominators in the fame Manner, the laft Product is a new Denominator. This new Fraction is the Product. The fame Rule is to be observed, if both the Fractions given to be multiplied are compound Fractions.

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These Examples, I think, do not require any Explication. But the Learner may probably obferve, that all Fractions decrease by Multiplication, the Product being less than either of the Fractions given to be multiplied; the Reafon of which is, that, all proper Fractions being only Parts of Unity, and therefore less than Unity, multiplying by a Fraction is only taking fo many Parts of the Thing multiplied, as are expreffed by the Fraction by which it is multiplied: For, as, in multiplying by a whole Number, the Product is fo many Times the Multiplicand as there are Units in the Multiplier, and therefore multiplying by 1 neither adds nor diminishes; fo, in multiplying by a Fraction, the Product is but fo many Parts of the Multiplicand as the Multiplier is Parts of an Unit: All Multiplication being in Geometrical Proportion, the firft

Term

Term of which is I; thus, as I is to the Multiplier, fo is the Multiplicand to the Product; therefore, if the Multiplier is less than 1 (as all proper Fractions are) the Product will be less than the Multiplicand.

Article 27. To multiply mixed Numbers and fimple or compound Fractions together.

Rule. Reduce the mixed Number to an improper Fraction, by Art. 2; then proceed to multiply the Numerators together for a new Numerator, and the Denominators for a new Denominator, as in the two laft Articles.

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Exam. 1. The mixed Number 12 is reduced to the improper, by Art. 2; then multiplying 51 by 5, the Product is 255; and the Denominators 4 and 6 multiplied give 24: Therefore 25 is the Product fought.

Exam. 2. The mixed Number 3 is reduced to the improper Fraction, by Art. 2; then multiplying the Numerators 25, 2, and 5 together gives 250 for a new Numerator; then the Denominators 7, 3, and 6 multiplied give 126 for the new Denominator: So is the Product fought.

126

Article 28. To multiply fimple Fractions and Integers together.

Rule.

Rule. Reduce the Integer to an improper Fraction, by Art. 1, that is, by placing an Unit under it for a Denominator; then proceed as in the foregoing Articles of Multiplication.

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Exam. 1. Placing 1 for a Denominator under the Integer 8 it makes; then the Numerators 8 and 5 multiplied is 40 for a new Numerator, and the Denominators 1 and 7 multiplied is 7 for a new Denominator: So is the Product fought. The other Example is done in the fame Manner. But it may be neceffary to obferve, that, if a compound Fraction was given to be multiplied by an Integer, it is to be done in the fame Manner as at Art. 26, as may be seen by the following Examples.

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Exam. 1. The Integer 26 is reduced to the improper Fraction 16, by Art. 1; then the Numerators 26, 2, and 1

26

multiplied

multiplied is 52 for the new Numerator; and the Denominators 1, 3, and a multiplied is 6 for a new Denominator: Thus is the Product fought.

The other Example is done in the fame Manner.

Article 29. To multiply mixed Numbers and Integers together.

Rule. Reduce the mixed Number to an improper Fraction, by Art. 2, and the Integer to an improper Fraction, by Art. 1; then proceed to multiply the Numerators together, and alfo the Denominators, according to the preceeding Articles.

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Thefe Examples being fo plain, their Explication is omitted.

Divifion of Vulgar Fractions.

Article 30. To divide fimple Fractions by fimple Fractions. Rule. Multiply the Denominator of the Divifor into the Numerator of the Dividend for the new Numerator; and the Numerator of the Divifor multiplied into the Denominator of the Dividend will be the Denominator of the Quotient.

Exam. 1.
Divide by
4)(Quotient.

Exam. 2.
Divide by 2.
)(Quotient.

Exam. 1. The Denominator of the Divifor 2 multiplied by by 3 the Numerator of the Dividend is 6, for the Numerator of the Quotient; and 1 the Numerator of the Divifor multiplied by 4 the Denominator of the Dividend is 4, for the

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