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And the Quotient is +* - 1, by which I Divide the aforeSaid Divilor thus,

*1) xx - 2x + (x - 1

X

And the Remainder is

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6c +42b by Divi

Wherefore x I is the greatest common Divisor sought, by which the given Fraction will be reduc'd to

* + c

by this Se&t.

If by proceeding in the foregoing Manner, you find the greate eft common Divisor to be i ; then the Fraction is in its leaft Terms already.

Note, Some Fraitional Quantities are reduc'd to their least Terms by Dividing the Numerator and Denominator severally, by the greatest common Measure of sheir several Members ; thus 49ed 4200 + 294bd

is reduc'd to 359bd + 28dc

596 + 40, ding the given Numerator and Denominator severally by 7d, the greatest common Divifor of their several Members.

SeEt. 4. To reduce a Compound fracion, to a Simple one of the same Value.

Hule. Multiply the Numerators by each other continually for a new Numerator, and the Denominators continually for a new Deno. minator; so this new Fraction is that required.

Examples : 1. Let it be required to reduce i of of

3

to a Simple

b Fraction of the same Value.

Fiift, 3 xixa -34, the new Numerator.-
Secondly bxcxd=bed, the new Denominator.

b Consequently

34

is the new Fraction required. bod

a + b 2. Let it be required to reduce of

十d 80 a Single Fraction of the same Value,

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atbXIXIXq=aq + bq, the new Numerator. fxct'd Xp Xr = fopr tfdpr, the new Denominator. Consequently

aq + bg

is the Simple Fraction required. fcpr + tdpr

CH A P. III. and IV.

Addition and Subftration of fracional Quans

tities. W?

Hat bath been done by the Rules in the next foregoing Chape

ter, is chiefly to fit and prepare Fractions of different Denominations for Addition, or Subftra&ion, as occasion requires ; viz.

1. If the Fractions given to be added, or Subftracted, be Compound ones, they must be reduc'd to Simple or Pure Fractions (by Sect. 4. of the next foregoing Chap.)

2. If they have not a Common Denominator, they must be reduc'd to Fraâions of the same Value, that will have a Common Denominator (by setti !. Chap. 2.) That being done, Addition and Substradion are thus perform'd.

Hule. Add or Substract their Numerators, as occafion requires; and un. der their Şum or Difference, subscribe the Common Denominaror.

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P

Suppose it was required to add but i +-2, andgt, inco

its one Sum. Firit the Fractional Parts are

P Secondly, d xDxp=dp, the common Denominacar.

Thirdly,

Li and to :P

d

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dpP

Thirdly, cxpxp=op.
Fourthly, -qxdxp=- qdp.

Fifthly, cx dxp=cp; Wherefore Epe gdp + cdp g-gdtod

(by Se&. 3. Chap. II.) is the Sum of the Fractional dp Parts Consequently the Sum required is 6++

1 ep - qdt od

dp Again, let it be required to add ; of

ba Firft, ixd=d. Secondly, b xp=bp; therefore of

T

is (by Se&. 4.

P Chap. II.) = ; Consequently

2d-9 is the Sum require

1

d-9.

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d

=

bp

Examples in Subftra&ion, 14+b+44 at 6

24 d

dto dta a + b

26 A

4+b-d d

dta
b

.6 td
23
d
dto

d+a

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dt.

ad

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aptd

Let it be required to take a + from

d a +
P

b + ď
d
First, at is (by Se&t. 2. Chap. 11.) =

P
Secondly,

+
and
apta

are ( by Sect. 1. Chap. II.)
btd P

pe + pc equal to and bap #bd + dapt do

respectively. pbt pd

pbt pa Consequently pat pc— bap bd-- dap-d*

is the Remain

pbt pd der required.

b of 3 of

b. Again, Let it be required to take

-- from

4 b

36 3 을 of

is (by Sect. 4. Chap. II) = 4

84

And

a

of

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b

86 And is

86-36 sb Therefore

is the Remainder soughc.

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1

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84

Note, The universal manner of Adding and Subftra&ting either Whole, or Fracted Quantities, is by + and refpe&tively.

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CHA P. V. uşultiplication of fracional Quantities. First prepare mix'd Quantities (if there be any

to be Multiplyed) by reducing them to Fractions of the same Denomination. (by Se&. I. Chap. II.) And whole Quantities by subscribing an Upit under each of them : then

Hule. Multiply the Numerators together for a new Numerator, and the Denominators together for a new Denominator ; chea this new Fraction is the Product required.

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ab

ad to d

44 F.2b f

id abd 12a

zab - 466 ct 2dd t dc

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2131 Suppose it was required to Multiply 34 +by 36+46.

These prepar'd for the Work as above directed, will stand thus,

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N.B. Any Fra&tion is Multiplyed by its Denominator, by casting off, or taking away tbe Denominator. b

b

be Thus xa=b; for

b.

6 4 + Again, Let it be required to Multiply

of

by f.

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X-
A

+c

First, 6x a F5X75baf rtbf, the Numerator
Sec. cxb toxi=cb + co, the Denomin.
Product required.

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CHA P. VI.

Division of fracional Quantities. THe Fractional Quantities being prepar'd as directed in the laft Chapter, then

Hule. Multiply the Numerator of the Dividend, by the Denominator of the Divisor, for a new Numerator; and Multiply the Denominator of the Dividend by the Numerator of the Divisor, for a new Denominator ; so this new Fraction, is the Quotient requi- . red.

Examples. abd Let be Divided by at the Work may stand

of

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