A Fraction, or Broken Number, is that which represents a Part or Parts of any thing proposed, (vide Page 3.) and is expreffed by two Numbers placed one above the other with a Line drawa betwixt them:

The Denominator, or Number placed underneath the Line, denotes how many equal Parts the thing is fuppofed to be divided into (being only the Divifor in Divifion). And the Numerator, or Number placed above the Line, fhews how many of those Parts are contained in the Fraction (it being the Remainder after Divifion). (See Page 29.) And these admit of three Diftinctions:

A proper, pure, or Simple Fraction, is that which is lefs than an Unit. That is, it reprefents the immediate Part or Parts of any thing less than the whole, and therefore it's Numerator is always less than the Denominator.

S is one Fourth Part As is one Third Part

And { is two Thirds, &c.

An Improper Fraction is that which is greater than an Unit. That is, it represents fome Number of Parts greater than the whole thing; and it's Numerator is always greater than the Donominator.

As or or 4 &c.

A Compound Fraction is a Part of a Part, confifting of feveral Numerators and Denominators connected together with the Word [of].

As of of, &c. and are thus read, The one Third of the three Fourths of the two Fifths of an Unit.

That is, when an Unit (or whole thing) is firft divided into any Number of equal Parts, and each of those Parts are

fubdivided

fubdivided into other Parts, and fo on: Then those laft Parts are called Compound Fractions, or Fractions of Fractions,

As for inftance, fuppofe a Pound Sterling (or 20 s.) be the Unit or Whole; then is 8s. the of it, and 6 s. the of those two Fifths, and 2 s. is the of those three Fourths; viz. 2s. = of of of one Pound Sterling.

All Compound Fractions are reduced into fingle ones, Thus,

RULE.

Multiply all the Numerators into one another for a Numerator, and all the Denominators into one another for the Denominator. Thus the of of will become . Or io.

For 1×3×2=6 the Numerator, and 3 × 4×560 the Denominator, but & or fo of a 1. Sterl. is 2 s. As above.

Sect. 2. To Alter or Change different Fractions into one Denomination retaining the fame Value.

IN

N order to gain a clear Underftanding of this Section, it will be convenient to premife this Propofition, viz. If a Number multiplying two Numbers produce other Numbers, the Numbers produced of them fhall be in the fame Proportion that the Numbers multiplied are, 17 Euclid 7.

That is to fay, If both the Numerator and Denominator of any Fraction be equally multiplied into any Number, their Products will retain the fame Value with that Fraction.

As in these,

2×2 4
3×2 6

Or

2x3:
3×3 9

6

That is, and. Or and. Or and Value, in refpect to the Whole or Unit.

are of the fame

From hence it will be eafy to conceive, how two or more Fractions that are of different Denominations, may be altered or changed into others that fhall have one common Denominator, and ftill retain the fame Value.

Example. Let it be required to change and into two other Fractions that fhall have one common Denominator, and yet re-. tain the fame Value.

According to the foregoing Propofition, if be equally multiplied with 7, it will become 1, viz. 27=14. Again, if † be

2×7

3×7 21

equally multiplied with 3, it will become , viz.

H

And

And by this means I have obtained two new Fractions, 1 and 2, that are of one Denomination, and of the fame Value with the two firft propofed, viz. 1 and 2=4.

And from hence doth arife the general Rule for bringing all Fractions into one Denomination.

RULE.

Multiply all the Denominators into each ether for a new (and common) Denominator. And each Numerator into all the Denominators but it's own, for new Numerators.

Example. Let the proposed Fractions be }, 3, 4, and 9.
Then, by the Rule,

A new Denominator

will be thus found..

And the new Numerators will

Hence 420 is the common Denominator; and 140. 168. 315. 360, are the new Numerators, which being placed Fraction-wife are 448.414.416.48 the New Fractions required. 42420.

"

140 I 168 2 315-3 360

That is,
420 3 420

420 4

5 420

and

6

420 7

Sect. 3. To bring mixed Numbers into Fractions, and the contrary.

MIX'D Numbers are brought into improper Fractions by the following Rule.

RULE.

Multiply the Integers, or whole Numbers, with the Denominator of the given Fraction, and to their Product add the Numerator, the Sum will be the Numerator of the Fraction required.

Example. 9 by the Rule will become 42. For 9 x 5. And, the improper Fraction required. Again, 13 will become. For 13 x 15. And+2. And fo for any other as occafion requires. To find the true Value of any improper Fraction given, is only the Converfe of this Rule. For if 29, as before is evident:

Then

Then it follows that if 49 be divided by 5, the Quotient will, give 9. And if 206 be divided by 15, it will give 131, &c. confequently it follows, that

If the Numerator of any improper Fraction be divided by it's Denominator, the Quotient will discover the true Value of that Fraction.

EXAMPLES.

5. And=4. And = 60. Or 1=34, &c. When whole Numbers are to be expreffed Fraction-wife, it is but giving them an Unit for a Denominator. Thus 45 is $1 92, and 25 is 22, &c.

Sect. 4.

To Abbreviate or Reduce Fractions into their Loweft or Leaft Denomination.

THIS is done, not out of any neceffity, but for the more convenient managing of fuch Fractions as are either proposed in large terms; or fwell into fuch, either by Addition or otherwise: befides it is most like an Artist to exprefs or fet down all Fractions in the lowest Terms poffible; and to perform that, it will be neceffary to confider thefe following Propofitions.

Numbers are either Prime or Compofed.

1. A Prime Number is that which can only be measured by an Unit. Euclid 7. Defin. 11.

That is, 3, 5, 7, 11, 13, 17, &c. are faid to be Prime Numbers, because it is not poffible to divide them into equal Parts by any other Number but Unity or 1.

2. Numbers Prime the one to the other, are fuch as only an Unit doth measure, being their common Measure. Euclid 7. Defin. 12.

For instance, 7 and 13 are Prime Numbers to each other, because they cannot be divided by any Number but an Unit. And 9 and 14 are alfo Prime Numbers to each other, for altho' 3 will measure or divide 9 without leaving a Remainder; yet 3 will not meafure 14 without leaving a Remainder; Again, altho' 2 will measure 14 without any Remainder, yet 2 will not measure 9 without leaving a Remainder, &c.

3. A compofed Number is that which fome certain Number meafureth. Euclid 7. Defin. 13.

For inftance, 15 is a compofed Number of 3 and 5, for 5 × 3 =15, confequently 3 or 5 will juftly measure 15. Allo 20

is compofed of 5 and 4, viz. 5 x 420, therefore 5 and 4 will each juftly measure 20.

4. Numbers compofed the one to the other, are they which fome Number being a common Measure to them both doth meafure. Euclid 7. Defin. 14.

That is, if two or more Numbers can be divided by one and the fame Divifor; then are thofe Numbers faid to be compofed one to another.

For Inftance, 14 and 21 are Numbers compofed the one to the other, because they can both be measured or divided by 7. For 7 x 214, and 7 x 321; therefore 7 is a common Meafure to 14 and 21. So that if it were propofed to be abbreviated, it will become 3.

2

And how thofe greatest common Meafures may be found, comes from Euclid 7. Prob. 1, 2, 3, and is thus:

RULE.

Divide the greater Number by the leffer, and that Divifor by the Remainder (if there be any) and fo on continually until there be no Remainder left: Then will that laft Divifor be the greatest common Meafure. (and if it happen to be 1, then are thofe Numbers Prime Numbers, and are already in their lowest Terms; but if otherwife) Divide the Numbers by that laft Divifor, and their Quotients will be their leaft Terms required.

EXAMPLE.

Let it be required to find the greatest common Measure of 72 and 108, viz. of.

72) 108 (1

72

Here because there is no Remainder ;

36) 72 (236 is the greateft common Measure.

72

(0)

Therefore, {

77

36) 722 Hence is abbreviated 36)108=3

I to the lowest Terms.

Again, to find the greatest common Measure of 744 and 899.

Thus,