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CH A P IV.

Of Wulgar Fractions.

Se&. 1. Of Notation. A Fraction, or Broken Number, is that which represents a Part

or Parts of any thing proposed, (vide Page 3.) and is exprefsed by two Numbers placed one above the other with a Line drawn betwixt them :

3

4 . The Denominator, or Number placed underneath the Line, denotes how many equal Parts the thing is supposed to be divided into (being only the Divisor in Division). And the Numerator, or Number placed above the Line, lhews how many of those Parts are contained in the Fraction (it being the Remainder after Division). (Sec Page 29.) And these admit of three Distinctions:

Proper or Simple
Viz. 3 Improper

Fractions.

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

Sis one Fourth Part As

one š is one Third Part

i is two Tbirds, &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 Do nominator.

As or ļor 4'} &c. A Compound Fraction is a Part of a Part, consisting of several 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

subdivided

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subdivided into other Parts, and so on: Then those last Parts are called Compound Fractions, or Fractions of Fractions,

As for instance, fuppose a Pound Sterling (or 20 s.) be the Unit or Whole ; chen is 8s. the of it, and 6 s. the of those two Fifths, and 2 s. is the of those three Fourths; viz. 2 s. = of of į 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 į of of will become 2. Or i. For 1x3x2=6 the Numerator, and 3*4*5=60 the Denominator, but % or is of a I. Steri. is 2 so As above.

Sect. 2. To Alter or Change different Fractions into one

Denomination retaining the fame Value.

2x2

3x 2

IN N order to gain a clear Understanding of this Section, it will

be convenient to premise this Proposition, viz. If a Number multiplying two Numbers produce other Numbers, the Numbers produced of them shall be in the fame Proportion that the Numbers multiplied are, 17 Euclid 7.

That is to say, 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 there,

"Or
2*3. 6

Or?*5_50, &c.

3x39 3x5 15 That is, and. Or and. Or and 1} are of the same Value, in respect to the Whole or Unit.

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

Example. Let it be required to change and į into two other Fractions that shall have one common Denominator, and yet re« , tain the fame Value. According to the foregoing Proposition, if į be equally multipli

2x7 14 ed with 7, it will become **, viz.

Again, if I be

3 x 7 equally multiplied with 3, it will become A, viz. 343–2

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21

7*3 219

And

And by this means I have obtained two new Fractions, it and , that are of one Denomination, and of the fame Value with the two first proposed, viz. ** = and i=}.

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

RUL E. Multiply all the Denominators into each cther 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 }, }, }, and %.

Then, by the Rule, A new Denominator And the new Numerators will will be thus found.

be thus found. 3

2. 3.

6
-5

5
3
3

3
15

5
6
9

18
4

4
4 5

5
60

24 45 90 !...7

7 ! 7 7

4 420

140. 168

315 360 · Hence 420 is the common Denominator; and 140 . 168. 315. 360, "are the new Numerators, which being placed Fraction-wise are 118.499.214.498 the New Fractions required. 140

168 That is,

315 3

6 420 3 420 5.420 4 420 7

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20

1

2

360

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Sect. 3. To bring mixed Idumbers into Fractions, and

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

..visas RU L E. 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 . For 9x55.
And, 41+= the improper Fraction required.
Again, 13it will become 4 For 13 x 15=14
"And +1}=. And so for any other as occafion requires.

To find the true Value of any improper Fraction given, is only the Converse of this Rôle. For if = 9, 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 13 is, &c. consequently 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.

E X A M P L E S. 1=5. And *=41. And =66. Or !=3, &c.

When whole Numbers are to be expressed Fraction-wise, it is but giving them an Unit for a Denominacor. Thus 45 is 41 9}, and 25 is ?, &c.

Sect. 4.

To abbreviate or Reduce Fractions into

their Lowest or Leaft Denomination. THIS is done, not out of any necessity, but for the more con

venient managing of such Fractions as are either proposed in large terms; or swell into such, either by Addition or otherwise: besides it is most like an Artist to express or set down all Fractions in the lowest Terms poffible; and to perform that, it will be necessary to consider these following Propositions.

Numbers are either Prime or Composed. 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 said to be Prime Numbers, because it is not possible to divide them into equal Parts by any other Number buc Unity or 1.

2. Numbers Prime the one to the other, are such 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 also Prime Numbers to each other, for altho' 3 will measure or divide 9 without leaving a Remainder ; yet 3 will not measure 14 without leaving a Remainder; Again, altho' 2 will measure 14 without any Remainder, yet 2 will noc measure 9 without leaving a Remainder, &c. .

3. A composed Number is that which fome certain Number measureth. Euclid 7. Defin. 13.

For instance, 15 is a composed Number of 3 and 5, for 5*3 3,15, consequently 3 or 5 will justly measure 15. Allo 20

H 2

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is composed of 5 and 4, viz. 5 * 4 = 20, therefore 5 and 4 will cach juftly measure 20.

4. Numbers composed the one to the other, are they which some Number being a common Measure to them boch doth measure. Euclid 7. Defin. 14.

That is, if cwo or more Numbers can be divided by one and the same Divisor; then are those Numbers said to be composed one to another.

For Instance, 14 and 21 are Numbers composed the one to the other, because they can both be measured or divided by 7. For 7 *2= 14, and 7*3=21; therefore 7 is a common Meafure to 14 and 21. So that if it were proposed to be abbreviated, it will buome

7) 14 = 2 Thus

)21=3 And how those greatest common Measures 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 so on continually until there be no Remainder left: Then will that last Divisór be the greatest common Measure. (and if it happen to be r, then are those Numbers Prime Numbers, and are already in their lowest Terms ; but if otherwise) Divide the Numbers by that last Divisor, and their Quotients will be their leaft Terms required.

EXAMPLE

Let it be required to find the greatest common Measure of 22 and 108, viz. of it.

72) 108 (1.

72
36) 72 (2 . Here because there is no Remainder ;

72 36 is the greatest common Measure,
(0)

36) 72 = 2 Hence is abbreviated Therefore, {

36)108 = 3 7 to the lowest Terms. Again, to find the greatest common Measure of 744 and 899.

Thus,

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