CHAP. XLVI. MULTIPLICATION of FRACTIONS, 599. B by a Fraction, is only meant the taking Y multiplying any Number, or Quantity, fuch Part, or Parts of it, as the Fraction expreffes. 600. The Rule to perform Multiplication of Fractions is If the Fractions are mixed, they must be first brought into improper Fractions; if compound, into fingle. Then multiply the Numerators together for a Numerator, and the Denominators for a Denominator. 601. Example 1. Multiply by. 6 the Numerator; and 4x 312 nator; the required Product; = 6 6 ΤΣ Here 3 x 2 = the Denomibut this may be reduced to lower Terms, viz. 4. See Cafe 8. 602. Example 2. Multiply 2 by, and this Product by 2, and this again by of. ༧ 2 5 g 1 2 T Solution. First 2 by Cafe 2; and 2 by Cafe 1; and of 3 = by Cafe 5. Hence, we are now 13 to multiply and and and together; ·.· by the Rule 5x1x2x5=50 the required Numerator, and 2 x 8 x 1 x 18 (the Product of the Denominators) 288 the Denominator. Hence the required Product = (by Cafe 8) 254 50 =288 603. Example 3. Multiply 10 Yards, 2 Feet, Inches, by 2 and of. This is the fame in Effect, as if it had been propofed to a Meafurer to find how many Yards are contained in a Piece of Pavement 10 Yards, 2 Feet, 3 Inches long, and 2 Yards, 1 Foot, 4 Inches broad; for the Rule, obferved by Measurers, is, to multiply the Length, taken as an applicate Number, by the Breadth confidered as an abftract Number. Solution. First, 1 Yard = 1 × 3 × 12 = 36 Inches, and 10 Yards, 2 Feet, 3 Inches=387 Inches; . the Length of a Yard; and, bringing the Breadth 8 into Inches, the Multiplier will be. Hence we are to multiply 397 of a Yard by 8, or, by Abbreviation, by Cafe 8, it is the fame to multiply 43 of a Yard by 22. 43 x 22 946 = the Numerator, and 4x9= 36 the Denominator; and ... the An260 Yards (by Cafe 3.) 946 fwer of a Yard, = 36 =26 Yards, by Cafe 8. 36 * 604. Scholium. In both whole Numbers, and Fractions, this Proportion holds good, viz. as one is to the Multiplicand, fo is the Multiplier to the Product. 605. Hence if the Multiplier be less than the Integer, that is a proper Fraction, the Product will be lefs than the Multiplicand. 3 606. The Reafon of the Rule for Multiplication of Fractions may be fhewn from the 1st Example. For, if it had been demanded to multiply by 2, it is evident at firft Sight, that X2 would be the Anfwer; but it was required to multiply by, that is, by the of 2; and.. the required Anfwer must be of the Product, that is, 1 = ÷ + · X 4 6 =ㄒ. CHAP. * Let m the Multiplicand, f = the Multiplier, p=the Product, then m fp; and, dividing by m, we have ƒ = t • ‡ as 1 : m :: ƒ: † 108. m 184. + Or the Truth of the Rule for Multiplication of Frac-.` N n tions may be shewn algebraically thus: Let and be the two Fractions, whofe Product is required; leta, b; hence multiplying the firft Equation by D, and the fecond by d, we fhall get N Da, and n=db by Art. 123; Nn * Ddab, and, dividing each Side of this Equation Nn by Dd, we have tab; but ab the Product of 607. CHAP. XLVII. DIVISION of FRACTIONS. A S Divifion of whole Numbers fhews howoften one Integer is contained in another Integer, fo Divifion of Fractions fhews how often one Fraction is contained in another. 608. If the Numerator of any Fraction be made a Denominator, and the Denominator a Numerator, the Fraction, fo made, is called the Reciprocal of the Former. Thus is the Reciprocal of 2. 2 3 * 609. In Divifion as the Divifor is to the Dividend, fo is an Unit to the Quotient (both in whole Numbers and Fractions.) 610. To divide one Fraction by another, the Rule is: Having made the fame Preparation as directed in Multiplication, multiply the Denominator of the Divifor by the Numerator of the Dividend, for a Nu-, merator; and the Denominator of the Dividend by the Numerator of the Divifor, for a Denominator. Or, which is in Effect the fame, change the Divifor into its Reciprocal, and then work by Multiplication of Fractions. 611. Example. Divide by 2. Solution. By the Rule, 5× 3 = 15 the Numerator; and 4 x 28 the Denominator; and fo = the required Quotient, I by Cafe the 3d. thus, the Reciprocal of the Divifor is; and x (by Multiplication of Fractions) before. 5 Or as 612. The Reason of the first Method of Operation, in the laft Article, may be easily fhewn: For * Let m = the Dividend, d= the Divisor, ent; then = 7; but this is the fame as 184, an Unit does not divide) therefore as d: m there q=the Quo = (because d m q I 3 thère it was required to divide 4 by 3. Now, if it had been required to divide by 2, it is manifeft the Quotient would be of, or ; but fince it was only required to divide by, that is, of 2, it is evident, of 2, or, must be contained 5 Times as often inas 2 is, that is, the Anfwer, which is ac cording to the Rule*. 3 613. As to the other Method of Operation, as it brings out as the first does, if the first be true, this laft Method must also. 614. When the Fractions to be divided have both the fame Denominator, it is fufficient to divide the Numerator of the Dividend by the Numerator of the Divifor, or, which is the fame in Effect, to set them like a Fraction. For by the Note to Art. 612. N n it appears, that the Quotient of by would be D 615. Divide by 4. The Quotient = 4. 616. By duly confidering the direct contrary Effects of Multiplication and Divifion, we have this ge * There are many other Methods of fhewing the Truth of the Rule, one of which is: Let be to be divided by N D these in one common Denominator are η ď 571. common Denominator, are in Proportion to each other as their Numerators, viz. as Nd: nD:: N Nd nD Nd Dd Dd nD † 108. 184. Nd nD Nd N D d Corollary. Hence, if was to be divided by D tient would be the Numerator of the Dividend, the Numerator of the Divifor, = N general Corollary: That it is the fame, in Effect, to multiply by any Number, whether integral or fractional, or to divide by its Reciprocal. For Inftance, 3×4 =3, or generally a x bat, each being (by ÷ their respective Rules of Multiplication or Divifion) b ➡ab; alfo a x =a÷ C that can be done by Multiplication, by taking the Reciprocal of the Multiplier, may be done by Divifion; and, on the contrary, any Thing that can be done by Divifion, may, by taking the Reciprocal of the Divifor, be done by Multiplication. 617. We fhall only add one Thing more under this Head, by Way of Corollary, and that is, that, if any Number, whether whole or fractional, be divided by a proper Fraction, the Quotient will be more than the Dividend; but, if the Divifor be an improper Fraction, the Quotient will be less than the Dividend. CHA P. XLVIII. The RULE of THREE DIRECT in FRACTIONS. 618. HE ERE, as in whole Numbers, the fecond and third Numbers must be multiplied together, and the Product divided by the first. But the Multiplication and Divifion must be performed by the Rules for Multiplication and Divifion of Fractions. Or the Anfwer, or fourth Number, may be found by this Rule, (which is the fame in Effect:) Multiply the Denominator of the firft by the Numerators of the second and third Numbers, for the Numerator; and the Numerator of the firft by the Denominators of the fecond and third, for a Denomi nator. |