Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/473

 M E T I C K, 391 g : 9 : : 4 : 12, then and it has been proved, that 3 X 12 = 9 X4. Therefore if, of four proportional numbers, any three be given, the fourth may eafily be found, viz. when one of the extremes is fought, divide the produft of the means by the given extreme; and when one of the means is fought, divide the produdt of the extremes by the given mean. 14—4 J 5. In multiplying fradtions, equal fadtors above and' below may be dalhed ordropt. Thus, ^ of yX^ of -£=; Multiplication of V.ulgar Frailicns. vX | X ^X ^; and dropping the fadtors 2, 3, 4, both In multiplication of fraftions there is no occafion to re- above and below, the produdt is -f. In like manduce the given fractions to a common denominator, as ner, to facilitate an operation, a fadtor above and anoin addition and fubtra&ipn: onlyj if a mixt number be ther below may be divided by the fame number : Thus, given, reduce it to an improper fraftion; if an integer — Or we may exchange onebe given, reduce it to an improper fra&ion, by putting 1 v r = * v? 7X2 an unit for its denominator; if a compound fra&ion be numerator for another: Thus, ^Xx-i—tXx given, you may either reduce k to a Ample one, or, inftead of the particle of, infert the lign of multiplications, 6. To take any part of a given number, is to multhen work by the following tiply the faid number by thefradtion. Thus, ^ of 32a Rule. Multiply the numerators for the numerator thus, -^X—°=iXJ-~:, = i-X‘V0==-LfJ0=200. of the product, and multiply the denominators for its. de- isIn found like manner, -fof 4J-|, is ■ f X454 = fX -f-i = |L nominator. X ~-=% X -r = 4—= 3°i. Hence, to reduce & Examp-, i. tV 7 =: := 3 ::= compound fradtion to a Ample one, is to multiply the 2. |-X V 'T4 4T 5 4iinto one another. Note i. If any number be multiplied by a proper frac- parts7. ofIf ita multiplicand or fhore denominations be tion, the product will be lefs than the multiplicand; for given to be multiplied byof atwofradtion, reduce the higher multiplication is the taking of the multiplicand as often part or parts of the multiplicand to the fpecies, as the multiplier contains unity; and confequeatly, if and then multiply. Thus, to multiply 8 1.lowed: io^s. by-^ <S J the multiplier be greater than unity, the produ<51 will be fay, 8l.=8X20Si=i6os. and i6o+io|=i7o|s. greater than the multiplicand; if the multiplier be unity, and yX 4* — —= 11 s.=L. 5:13:10. = -|Or the produift will. be. equal to the multiplicand; and if the without Reducing, you may multiply the given multipli multiplier be lefs than unity, the product will, in the cand by the numerator of the fradtion, and divide tho fame proportion, be lefs than the multiplicand. Thus, by the denominator. fuppofing the multiplier to be -|- or -f, the product, in produdt ^ by A'. Prod. 4vthis cafe, will be equal to one half or to one third of the Examp. 2.i. Multiply Multiply 74'by-f. Prod. multiplicand. 3. Multiply 8-4 by 94;. Prod. 84-1-.', 2. Mixt numbers may be multiplied without reducing them to improper fractions, by work-, The reafonancof the rule may be (hewn thus: 4X-£=T8T : ing as in the margin; where firft multiply the for 4=4-f-> * ? of tt tt 4 and confequently 4 of ■ integral parts, viz. 54 by 24; then multiply, xt i® it* the integral parts crofs-ways into their altern The truth of the rule may alfb be proved thus : Af1 fractions, viz. -54 by !■,. and the product 27 let fume two fradtions equal to two integers, fuch as, |.> and down; in like manner multiply 24 by f, and 4, equal to 2 and 3, and the produdt of the fradtions the produdt 6 likewife fet down ; then add ; and will 8 be equal to the produdt of the integers; for "1329s’ tofractions. the fum annex. the product of the two = =6, and.2X3—6. 3. In'multiplying a fraction, by > an integer, you have ITtvifion of Vulgar Frallions. only to multiply the numerator by the integer, the put- In divifion of fradtions, if a mixt number be given,. ting one for the denominator being only matter of form. reduce it to an improper fradtion; if an integer, be given, And to multiply a fraction by its denominator is to take away the denominator, the produft being an integer, put an unit for its denominator ; if a compound fradtion reduce it to a Ample one, and then work by the fame with,r or equal to the numerator. Thus, bethegiven, 5 following .JX8=7. F<> IX-^ ^?4. If the-numerators and denominators of two equal Rule. Multiply crofs-ways,-p/z. the numerator of the divifor into the denominator of the dividend, for the fractions be multiplied.crqfs-ways,. the produ&s will be denominator of the quot; and the denominator of thfequal. Thus, if ^.=T., then will 3 X 12 = 9X4; for divifor into the numerator of the dividend, for the numultiplying both by 9, we have 3; and multi- merator of the quot. (44= iTy= Itplying thefe by 12, we have 3 X 12 “9.X 4. Hence, if Examp. 2i. four numbers be proportional, the prodaft of the ex• x) 4x V (x-l = 5 A — STtremes will-be .equal to the product of- the means: for if, S’ Note,. _