CHAP. VIII.

Of the Addition, Subftraction, Multiplication, and Divifion, of vul gar or common Fractions.

Fractions,

or common Fractions are for. call'd, because they were vulgarly Common or commonly made Ule of, before that why so cal decimal Fractions were invented and led, made known.

so

2.

Difference

and com

mon Fraions,

The great Difference between common and decimal Fractions lies in this, that The great the denominative Values, or in one between Word, the Denominators of decimal Fra. Decimal tions (like as of Integers) are known by the Places of the decimal Figures, and proceed regularly in a like Proportion ; whereas the Denominators of common Fractions are not to be known by the Places of their Figures, forafmuch as they proceed not in any regular uniform Proportion.

wherein it

es

3.

Hence, in the Notation of common Fractions it is requifite, to write down The Dend not only their numerative Values or Nu- of common merators, but alfo their denominative Fractions

F

minators

Values are to be

wric

down;

Values or Denominators. For Inftance, 2 tenth Parts are by a common Fraction denoted ufually thus, whereas the 12 3 fame is exprefs'd by a decimal Fraction thus, 2, mely, (as has been above obferv'd,) the Place of the Decimal 2 being known to be the firft Place of Decimals, thereby it is known alfo, that the faid 2 denotes 2 Tenths. In like manner, 2 hundredth Parts are exprefs'd by a common Fraction thus, and 2 thousandth Parts thus,, whereas the former is exprefs'd as a decimal Fraction thus, 92, and the latter thus, .002: Namely, the decimal Figure 2 standing in the former Fraction in the fecond Place, is known thereby to denote 2 hundredth Parts ; and in the latter Fra&tion ftanding in the third Place, is known thereby to denote 2 thoufandth Parts. Hence, what is exprefs'd by the common Fractions, and, and, is exprefs'd much shorter by the Decimal 222. And this fhorter Way of Notation by Decimals, than by common Fractions, is one Reason which has made the Ufe. of decimal Fractions be fo generally preferr'd to the Ufe of common Fracti

ons.

But

decimal

mon,

lies.

But that, wherein decimal Fractions 4. chiefly excell common Fractions, is this, The Excel that decimal Fractions (by Reason of lency of their Denominators proceeding regularly Fractions in the like Proportion, and fo being de- above Com noted by their Places, as are the Deno- wherein it minators of Integers) are work'd after the chiefly fame Manner as Integers whereas common Fractions can't be fo work'd in many Cafes, (namely, as often as they have different Denominators, which proceed not regularly in a like Proportion); but must be first reduc'd to one and the fame common Denominator.

Fractions

may be

Method

As often as common Fractions have 5. the fame Denominators, they may be Common work'd as whole Numbers, and fo con- of the fame tain no Manner of Difficulty, as shall be DenomiThewn in the following Examples. Only nation, it may be of Ufe first to obferve here, work'd as that, whereas the common Way of wri- Integers, especially, ting common Fractions, is to place the if writ af Denominator under the Numerator with ter the a Line between, which occafions fome feeming Difference between the Operati⚫ons of Integers, and common Fractions of the fame Denomination; in order to take away even this feeming Difference,) it is very advisable to write common Fractions, as we write Numbers of feveral external Denominations, that is, to F 2 place

here shewn,

place the Denominators at the Head or Top of the Numerators of their refpeЯtive common Fractions. Namely, as we denote two Shillings and fix Pence

thus, 2 and 6, fo it feems better to denote the fame by common Fractions thus, 2**, and 6', than thus (as ufually), and. For the former Way being the fame, whereby Integers of feveral external Denominations are denoted, hence common Fractions being express'd after the fame Way, the Working of these will more appear to be agreeable to the Working of those, than if the common Fractions were exprefs'd the common Way. However, to comply with Cuftom, the common Way of Expreffing and Working common Fractions fhall not be altogether here omitted, but plac'd by the Side of the other Way.

Addition

Addition of common Fractions, of 6.

the fame Denomination.

Addition of common Fractions of the fame

tion.

EXAMPLE II. Denomina

8+3=112 2+5+6=13

EXAMPLE I.

8'

3 or

(*) 12 12 12 6 20 20 20 20

13

Substraction of common Fractions, of the fame Denomination.

7.

Subftra&ti

on of the Jume.

(*) In this Way of Writing and Working, Learners are apt to enquire, why only the Numerators are added and fubftracted, and not alfo the Denominators. Whereas the Reafon thereof is made evident by the other Way, which fhews the Manner of Working to be the fame as in Integers of feveral external Denominations.