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Of Algebżalck Fractions, or Broken Duantitics.
Sect. 1. Notation of Fractional Quantities.
Raftional Quantities are expressed or fet down like Vulgar Fractions in common Arithmetick.
Th..sa 2 bc5b-4a Numerators.
• Dus I To ď: 4d +77 Denominators. How they come to be fo, fee Cafe 4, in the last Chapter of Divison. These Fractional Quantities are managed in all re: Spects like Vulgar Fractions in Common Arizhmetick. Y 2
Sect. 2. To alter or Change different fractions into
one Denomination, retaining the same Value,
RUL E. G U LTIPLY all the Denominators into each other for a
V new Denominator, and each Numerator inte all the Denominators but it's own for new Numerators.
E X A M P L ES.
First a xc, and d x b, will be the Numerators, and b xc will be the common Denominator, viz.com and are the two Fractions required : that is, te , and be
btc Again, let and be brought in one Denomination,
and they will be bat-bb-da-be
Sect. 3. To Bring whole Duantities into fractions
of a given Denomination.
VW nominator for a Numerator, under which subscribe the given Denominator, and you will have the Fraction required.
E Y A M P L E S. Let it be required to bring a tob into a Fraction, whose Denominator is d - A. First a +bxd--a is datbd-a b a:
Denominazominator for a Newhole Quantities
datbd--aaba is the Fraction required. Thendra
Again b ta will be 00:42. Andant - a will be a cada
aatbb Allo a tota
When whole Quantities are to be set down Fraction-wife, subscribe an Unit for the Denominator. Thus ab is -, And aa—bb, is a 4 B5, &c.
Sect. 4. To abbreviate, or Reduce Fractional Quantities into their lowest Denomination.
RUL E. Dlvide both the Numerator and Denominator by their greatest
common Divisor, viz. by such Quantities as are found in both; and their Quotients will be the Fraction in it's lowest Term.
аас, aa abbb. Bb . . bac Thusis do " abc "T
+7 =atd.. In such single Fractions as these, the common Divisors (if there be any) are easily discovered by Inspection only; but in compound Fractions it often proves very troublesome, and must be done either by dividing the Numerator by the Denominator, until nothing remains, when that can be done : or else finding their common Measure, by dividing the Denominator by the Numerator, and the Numerator by the Remainder, and so on, as in Vulgar Fractions (Sect. 4. Page 51.)
É X Å MP LE S. a acaad Suppose nd were to be reduced lower.
Hence it appears that - 2 a ab- 2 abb is the common Measure; by which a aa-abb being divided.
Viz. — 2 a ab- 2 abb) a aa-abb (-4 I
aaa taab 277
aab-abb -aabab b
Oo Then this the new Numerator ; and 67 - is the new Denominator. But + ==20 = 'the Numerator; and
- 2026 ==
4 ba = = the Denominator. Let both be multiplied with 2 ba,
zba and you will have a
-amb the Denominator.
waarab the Signs of all the Quantities, it will be
the new Frace:
aa — a b aaa — abь tion required. That is,
atba a + 2ab +66.
da bb Again, let it be required to reduce
a a tab the Numerator.
The common Measure of this Fraction will be the easiest found (as appears from Trials) by dividing the Denominator by the Numerator, &c. Thus,
Hence it appears that bdbb is the common Measure that will divide both the Numerator and the Denominator.