Page:Cyclopaedia, Chambers - Volume 1.djvu/836

 FRA

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FRA

Again, twenty nine Sixtieths is wrote 44 ? where the Nu- merator 2 9 expreffes 29 Parts of an Integer divided into Sixty; and the Denominator (So gives the Denomination to thefe Parts, which are call'd Sixtieths.

The real Deiign of adding the Denominator, is to ffiew what aliquot Part the broken Number has in common with Unity* See Denominator, &c.

In all FraBions, as the Numerator is to the Denomina- tor ; fo is the FraBion it felf • to the Whole, whereof it is a FraBim?.

Thus, fuppofing $ of a Pound equal to 1 5 s. 'Tis evi- dent, that 3 : 4. : : 15 : 20. Whence it follows i" That there may be infinite FraBions of the fame Value, one with another • inafmuch as there may be infinite Numbers found, which fhall have the Ratio of 3 : 4. See Ri-
 * io.

Fractions are either Troper, or Improper.

A 'Proper Fraction is that where the Numerator is lets than the Denominator ; and confequently the FraBion lei's than the Whole, or Integer ; as |i.

An Improper Fr action is, where the Denominator is ei- ther equal to, or bigger than the Denominator ; and, of courfe, the FraBion, equal to, or greater than the Whole, or Integer, as Ji ; or 14 i or K-

FraBions, again, are either Simple, 01 Compound.

Simple Fractions are filch, as confift of only one Nu- merator, and one Denominator ; as y, or j\%, &c.

Compound Fractions, call'd alto FraBions of FraBions, are fuch as confift. of feveral Numerators, and Denomina- tors ■ as i of || of f of |4 Sec— —

Of FraBions thofe are equal to each other, whofe Nu- merators have the fameRatio to their Denominators. Thofe are greater, whofe Numerators have a greater Ratio ; and thofe left, which have lefs : Thus, | = f =r |=r|-| = l. But sf-J is greater than 4 ; and -J lefs than ||.

Hence, if 1 both the Numerator, and Denominator of a FraSion, as 4, be multiply'd, or divided by the fame Num- ber, 2 • the Facta in the former Cafe, 44, and the Quotients in the 'latter, -J, will conftitute FraBions, equal to the foil FraBion given. ■

The Arithmetic of FraBions confifis in the ReduBion, Addition, SubtraSion, and Multiplication thereof.

I. ReduBion of Fractions.

i° To reduce a given whole Number into a Fraction of any given 'Denominator: Multiply the given Integer, by the given Denominator: The Factum will be the Nume- rator.

Thus we fhall find 3=44; and 5— i| ; and 7=4* &c.

If no Denominator be given, the Number is redue'd to a FraBion, by writing 1 underneath it, as a Denominator. Thus f $ 44. ..

a ° To reduce a given Fraction to its loweft Terms ; 1. e. to find a FraBion, equivalent to a given FraBion, QfO but exprefs'd in lefs Numbers : Divide both the Numerator 20, and Denominator 48 by fome one Number, that will divide them both without any Remainder, as here by 4.. The Quotients 5 and 12 make a new FraBion, °4, equal to JJ.

And if the Divifion be perform 'd with the greater! Number that will divide them both ■ the FraBion is redue'd to its loweft Terms.

Now To find the greateft common Divifor of two Quanti- ties ■ Divide the greater by the lefs : Then divide the Divifor of the Divifion by theRemainder thereof: Again, divide the Divifor of the fecond Divifion by the Remainder of the fe- cond ; and fo on, till there remain nothing. The laft Di- vifor is the grcatcft common Mealure of the given Numbers.

If it happen that Unity is the only common Meafure of the Numerator and Denominator ; then is the FraBion in- capable of being redue'd any lower.

3 To reduce two, or more Fractions to the fame Deno- mination ; i. e. to find FraBions equal to the given ones, and with the fame Denominators : If only two FraBions be given, multiply the Numerator, and Denominator of each, by the Denominator of the other : The Products given are the new FraBions rcquir'd.

Thus 5) j anil 3) i niake || and fj. If more than two be given, multiply both the Numerator and Denominator of each- into the Product of the Denominators of the reft. Thus, 14) f 12)-}. iSH=fr- } ii- 44-, ,

4° To find the Value of a Fraflion in the known Tarts of its Integer : Suppofc, e. gr. It were requir'd to know <t what is -°4 of a Pound ; Multiply the Numerator by 20, the Number of known Parts in a Pound, and divide the Produfl by the Denominator id. The Quotient gives n s. Then multiply the Remainder 4 by 12, the Number of known Parts in the next inferior Denomination; and divid- ing the Produfl by itf, as before, the Quotient is 3 d. So that 4-} of a Pound — in. 3 d.

5° To reduce a mix'd Number, as 444 into an improper Fraflion of the fame Value : Multiply tha Integer, 4, by 12,'

the Denominator of the FraBion : arid to the Product 4$ add the Numerator : The Sum 59 fet over the former De- nominator, 44, conftitutes the FraBion requir'd.

6° To reduce an improper Fraction into its equivalent mixt Number : Suppofe the given FraBion 11 ; divide the Numerator by the Denominator; the Quotient 444 is the Number fought.

7° To reduce a Compound Fraction into a Simple one: Multiply all the Numerators into each other for a new Nu- merator ; and all the Denominators for a new Denominator.' Thus | of % of 4 redue'd, will be JJ.

II. Addition of Vulgar Fractions.

i° If the ghenFraBions have different Denominators, re- duce them to the fame. Then, add the Numerators to- gether, and under the Sum write the common Denomina- tor. Thus, e.gr. | -I- 4 := ff + tt = «t-. • And. £-1-4 -1- -J

45. _X 1* _1_ s 4 __ xi4 — - [4£ — - i°2..

2° Il l CompoundFrac?;w«are given to be added; they mutt firft be redue'd to fimple ones : And if the FraBions be of different Denominations, as £ of a Pound, and £ ot a Shil- ling, they muft firft be redue'd to FraBions of the fame De- nomination of Pounds.

3 To add mixt Numbers : The Integers are firft to be added ; then the fraflional Parts : And if their Sum be a proper FraBion, only annex it to the Sum of the Integers. If it be an improper FraBion, reduce it to a mix'd Num- ber ; adding the integral Part thereof to the Sum of the In, tegers, and the fraflional Part after it. Thus, 5J+4J

III. SubftraBion of Fractions.

i° If they have the fame common Denominator, fubflrafl the leffer Numerator from the greater, and fet the Re- mainder over the common Denominator.

Thus from T J take ^, and there remains T |.

2 If they have not a common Denominator, they muft be redue'd to FraBions of the fame Value, having a com- mon Denominator, and then as in the firft Rule.

rpi 6 ^ jo. ,1* i± t

-° To fiibtraB^a whole Number from a mix'd Number ; or one mix'd Number from another : Reduce the whoie, or mix'd Numbers to improper FraBions, and then proceed as in the firft and fecond Rule.

IV. Multiplication of Fractions.

i° If the FraBions propos'd be both fingle, multiply the Numerators one by another for a new Numerator, and the Denominators for a new Denominator.

Thus I into 4 produces ||.

2 If one of them be a mix'd, or whole Number, it muft be redue'd to an improper FraBion ; and then proceed as in the laft Rule.

Thus i into jf, gives i-J ; and J into ' °- — s T ...

In Multiplication of FraBions obferve that the 1 rodua is lefs in Value, than either the Multiplicand, or Multiplicator ; becaufe in all Multiplications, as Unity, is to the Multipli- cator ; fo is the Multiplicand, to the Prcdufl : Or, as Unity, is to either Factor ; fo is the other Factor, to' the Produfl. But Unity is bigger than either Faflor, if the FraBions be proper ; and therefore cither of them muft be greater than the Produfl.

Thus in whole Numbers, if 5 be multiply'd by 8, it will be, as 1 : 5 : : 8 : 40; or 1 : 8 : : 5 : 40. Wherefore in FraBions alfo, as 1 : 4 : : 4 : s '4j or as 1 : | :. 4; : \i._ But 1 is greater than either - 4 or f: Wherefore citner ot them muft be bigger than \i.

V. Divifion of Fractions.

i° If the FraBions propos'd be both fimple, multiply the' Denominator of the Divilor, by the Numerator of the Di- vidend ; the Produfl is the Numerator of the Quotient. Then multiply the Numerator of the Divifor, by the De- nominator of the Dividend, the Produfl is the Denomina- tor of the Quotient.

Thus 4) 4 (4?. , ,

2 If either Dividend, Divifor, or both, be whole or mix'd Numbers, reduce them to improper FraBions : And if they be compound FraBions, reduce them to fimple ones j and proceed as in the firft Rule. .

In Divifion of FraBions, obferve that the Quotient is al- ways greater than the Dividend - becaule in all Divifion, as the Divifor, is to Unity; fo is the Dividend, to the Quotient; as if ; divide 12, it will be, as 3 : 1 :: 12 : 4. Now 3 is greater than 1 ; wherefore 12 muft be great- * than 4: But in FraBions as 4 : 1 : : 3 ■ », 5 ™ ne -' c > 1 ls lefs than 1 ; wherefore 4 muft alfo be lcls than T7.


 * -z, Frac-