E",RA


( 89 )


wenty nine Sixtieths is wrote j4; where the Nu-
exprefWes P9 arts of an Integer divided into Sixty;
,enominator 6o gives the Denomination to thefe
.h are call'd Sixtieths.
a  Defign of adding the Denominator, is to {hew
ot Part the broken Number has in common with
:e DENOMINATOR, &C.
raffions, as the Numerator is to the Denomina-
the Fragtion it felf;  to the Whole, whereof it
on.
iuppofing I of a Pound equal to I 5 s. 'Tis evi-
3 : 4 ::  5: 2o.   Whence it follows I' That
be infinite FraPtions of the fame Value, one
er; inafmuch as there may be infinite Numbers
:~~~~ L.                      n    A A11tAL *  _


found, Wmcn mnall nave the ratio or 3: 4.   Q
VIO.
Fraffions are either Proper, or Improper.
A Proper FR kcTioN is that where the Numerator is lefs
than the Denominator; and confequently the FraCtion lefs
than the Whole, or Integer; as,450
An ImproperFFRACTION is, where the Denominator is ei-
ther equal to, or bigger than the Denominator ; and, of
courfe, the Fraftionj equal to, or greater than the Whole,
or Integer, as " ; or 2  or 9'
Fradtions, again, are either Simple, or Compound.
Simple FRACTIONS are fuch, as confifi of only one Nu-
merator, and one Denominator; as ,, or T-'4, &c.
Compound FRACTIONS, call'd alfo0 raftions of FraCtions,
are fuch as confifI of feveral Numerators, and Denomina-
tors; as EF of ,1 of 9t of ! &c.
Of Fra'tions thofe are equal to each other, whofe Nu-
merators have the fame Ratio to their Denominators. Thofe
are greater, whofe Numerators have a greater Ratio; and
thofe lefs, which have lefs: Thus,  2=      - = . But
W  is greater thank; and  lefs than f .
Hence, if both the Numerator, and Denominator of a
Fraftion, as #, be multiply'd, or divided by the fame Num-
ber, a; the Fafla in the former Cafe, f-, and the Quotients
in the latter, 2, will conflitute Fraftions, equal to the firit
.Fraffion given.-
The Arithnmetic of Fraftions confifns in the ReduCtion,
Addition, Subtraion, and Multiplication thereof.
I. Reduftion of FRACTIONS.
71 Toreduce a given whole Number into a Fra&ion of
try given Denominator: Multiply the given Integer, by
the given Denominator: The Faurum will be the Nume-
rator.
Thus we Mhall find 3=;4; and S=,; and 7-_S &c.
If no Denominator be given, the Number is reduc'd to
U: lo      , T    .tb it as a .. nnm.ngtnr


'duce a given Fraaion to its loweft Terms; i. e.
rat ion, equivalent to a given Fra lion, (2) but
i lefs Numbers: Divide both the Numerator
nominator 48 by fome one Number, that will
both without any Remainder, as here by 4. The
and In make a new Fraftion, f15 equal to 1j
the Divifion be perform'd with the greateft
at will divide them both; the Fradtion is reduc'd
Terms.
find the grearet  common f~ivifor of two 5)uanti-
ethe greater by the lefa: Then divide the bivifor
Lion by the Remainder thereof: Again, divide the
the fecond D~iviflon by the Renlainder of the fe-
fo on, till there remain nothing. The lafil Di-
greatef* common Meafure of the given Numbers.
pen that Unity is the only common Meafure of
ator and Denominator; then is the FraCtion in-
being reduc'd any lower. ,
'duce two, or more Fraffions to the fame Deno-
i. e. to find Fradtions equal to the given ones;
ie fame Denominators: If only two Fradions be
:iply the Numerator, and Denominator of each,
nominator of the other: The Produ&s given are
xalions requir'd.
- and 3) 4 make 12 and 5!. If mote than two
kulti ly both the Numlerator and Denominator of
the Produ&t of the Denominators of the tell.
2* 157   8)-    - a   -
ad the Value of a Fraffion in the known Parts
rer: Suppofe, e. gr. It were requir'd to know
of a Pound; Multiply the Numerator 9 by 2o,
or of known Parts in a Pound, and divide the
the -Denominator .6. The Qfotient gives II S.
eiply the Remainder 4 by IX, the Number of
ti n the next inferior Denomination ; and divid-
idu bit x6, as before, the Quotient is 3d. So
a Pounded It $.- 3 A-
du a mix'd Number, as 4   into an impf~ ei
'the fame Value': Multiply the Integer, 4, by si


ERi  A
Fk a


the Denominator of the Fragion: and to the Produce 41
add the Numerator: The Sum 59 fet over the former De-
nominator, TV conilitutes the Fraffion requir'd.
6f Lo reduce an improper Fra'tion into its equivalent
mixt Number: Suppofe the given Fraffion -  divide the
Numeratorb theDenominator; the Quotient 4- is the
Number fought.
70 To reduce a Compound Fracion 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 .2. of 4 of ; reduc'd, will be 45.
II. Addition of Vulgar FRACTIONS'
IO If the given Fradtions have different Denominators, re-
duce them to the fame. Then, add the Numerators to-
gether, and under the 4um write the common Denomina-
tor. Thus, e. gr. .-1-     _ 0 7 - I 27. And 2 -1- -- 4
= 7tt 7Z l7t-h  4 =7 I 72 =I17
2f If CompoundFra~fionsare given to be added; they muffo
firfi be reduc'd to fimple ones: And if the Fra~tions be of
different Denominations, as X of a Pound, and s-~ of a Shil-
ling, they mull firfi be reduc'd to Fra~tions of the fame De-
nomination of Pounds.
30 Pro add mixt Nunzbers: The Integers are firil to be
added; then the fraaional Parts: And if their Sum be a
proper Fradion, only annex it tb the Sum of the Integers.
if it be an improper Fradion, reduce it to a mix'd Num-
ber; adding the integral Part thereof to the Sum of the In-
tegers, and the fracional Part after it. Thus, 5 A- 49
-IO2.
III. Subftraflion of FRACTIONS.
1f If they have the fame common Denominator, fub{1raff
the leffer Numerator from the greater, and fet the Re-
mainder over the common Denominator.
Thus from y9 take T  and there remains 4.
e If they have not a common Denominator, they muff
be reduc'd to Fra~tions of the fame Value, having a com-
mon Denominator, and then as in the firft Rule.
'Thus A - __ 1z -  4   .6
3 To JubtraAt a 'whole Number from a mix'd Number;
or one mix'd Number from another: Reduce the whole, or
mix'd Numbers to improper Fraffions, and then proceed as
in the firfc and fecond Rule.
IV. Multiplication of FRACTIONS.
I0 If the Fralqions propos'd be both fingle, multipli the
Numerators one by another for a new Numerator, and the
Denominators for a new Denominator.
Thus -4 into W produces 2.
z0 If one of them be a mix'd, or whole Numberi it muft
be reduc'd to an improper Fradtion; and then proceed as
in the laft Rule.
Thus T into 532, gives i-4; and A into 1 =
In Multiplication of Frations offerve that the Produa 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 Product: Or, as
Unity, is to either Factor; fo is the other Faator, to
the Produ&. But Unity is bigger than either Fador, if the
Fraftions be proper; and therefore either of them mufl be
greater than the Produff.
Thus in whole Numbers, if 5 be multiply'd by 8, it wilt
be, as i : 5 :: 8 : 40; or I: 8:: 5 : 40. Wherefore
in Fraftions alfo, as I          or as I:: r :  :
But I is greater than either W or us Wherefore either of
them mufl be bigger than 3.-2
V. 7Divivjon of FRACTIONS.
I' If the Fraftions propos'd be both fimple, multiply the!
Denominator of the Divifor, by the Numerator of the Di-
vidend; the Produ&i is the Numerator of the Quotient.
Then multiply the Numerator of the Divifor, by the De-
nominator of the Dividend, the Produ& is the Denomina-
tor of the Quotient.
Thus 35) 4 (.
;f If either Dividend, Divifor, or both, be whole or
mix'd Numbers, reduce them to improper Fradtions : Andi
if they be compound Fradtions, reduce them to fimple ones;
and proceed as in the firm Rule.
In Divifioti of Fraifions, obferve that the Quotient is al-
ways greater than the Dividend; becaufe in all Divifofi, as
the Divifor, is to Unity; fo is the Dividend, to the
Quotient; as if 3 divide iz, itwill be, as 3:  z :  . II.: 4.
Now 3 is greater than I; wherefore rz muat be eater
than 4: B~ut in Fradtions as -A : :: 9 : .9y; whr,~ 518
elfo be leis 4hanC
Fr 9AC-


a


H
g
g
I
9