Vulgar Fractions.

ARITHMETIC

2.13

(29.) Definitions. A Fraction, or broken number, is an ex-pression of one or more parts of any number.

The number of parts into which the number is supposed to bedivided, is called the Denominator ; and the number of thoseparts expressed by the fraction, is called the Numerator. Thesetwo numbers are in general named the terms of the fraction.

[f the number of which the fraction is a part, or parts, be 1, itis called a Simple Fraction; and is denoted by the numera-tor written above the denominator, with a smali line betweenthem: Thus, -y denotes one third of 1 ; and J denotes seveneights of 1.

But if the number be different from 1, it is called a Compound,and is denoted by the word of, and the number subjoined to thenumerator and denomenator, expressed as before. So, -J of 6,denotes rue half .of (5; f of S, denotes two thirds of 8 ; and | of+ denotes three fourths of five eighths of 1.

Simple fractions, whose numerators are less than their denomi-nators, are called Proper Fractions. And those whose numera-tors are equal to or' greater than their denominators, are calledImproper Fractions .

The expression formed from an integer and a fraction joined to-gether, is called a Mixed Number. If both the numerator anddenominator of a fraction be multiplied or divided by the samenumber, the fraction will retain its original value. All computa-tions in fractions are founded on this principle.

A fraction denotes division, the numerator is the dividend,the denominator is the divisor, and the fraction itself expresses12

the quotient: thus — = 3 = the quotient of 12 divided by 4.

REDUCTION OF FuLGAR FRACTIONS.

(30.) Rule I. To abbreviate or reduce Fractions to less Terms.

Divide the terms of the given fraction by any number whichwill divide them without a remainder : the quotients will be theterms of a new fraction, equal in value to the former: this maybe abbreviated again, and the next again, and so on, till it appearthat there is no number greater than 1 that will divide them, inwhich case, the fraction is said to be in its least terms.

Example.

42

Let — be proposed to be abbreviated.

72

42 21 7

— = — = —, by dividing, first by 2 and then by 3.

72 36 12

If the numerator and denominator be expressed by the conti-nual product of several numbers, and any two of these numbersone in the numerator, and the other in the denominator can be di-vided exactly by any number, the quotients may be used insteadof them without changing the value of the fraction.

26x3x8 2x3x 1 6

Thus, -=-= —.

13X8X11 IXlXll 11

Sometimes the figures are cancelled by a dash, and the quoti-ents wrote above.

Thus,

/J3'X3X /$' 6

M3-x^xTT- U

I t

Tf the fraction must be brought to its least terms at one division,divide its terms by their greatest common measure, which com-mon measure is found by dividing the greater term by the less,and this divisor by the remainder, till 0 remain ; then is the lastdivisor the greatest common measure required.

Example. Reduce to its least terms at one division.

First, 246)372(1 ‘ 246-1-6 41

126)246( 1 Then-= —

120)126(1 372-1-6 62

The common measure 6)120(20

(31.) Rule II. To reduce an improper Fraction to its equivalentll'hole or mixta Number.

Divide the numerator by the denominator, and the quotientwill be the integer or mixed number required.

* Examples.

12 37 4 15 1

3 11 117 7

(32.) Rule 111. To reduce an Integer to an equivalent Fructionof a given Denominator .

Multiply the integer by the given denominator, and the pro-duct will be the numerator required.

Example.be 4.

Reduce 7 to a fraction whose denominator shall

7x4 28

7 =-= —.

4 4

(33.) Rule IV. To reduce a mixed Number to an equivalent iwproper Fraction.

Multiply the integer by the denominator of the fraction, to theproduct add the numerator; then the sum written above the de-nominator w ill form the fraction required.

Example. Reduce 2£ to a fraction.

3 2x7+3 14+3 17

7 7 7 7

(34.) Rule V. To reduce a compound Fraction to an equivalentsimple one.

Multiply all the numerators together for the numerator, and allthe denominators together for the denominator of the simple frac-tion required. If part of the compound fraction be an integer ora mixed number, reduce it to a fraction by one of the formercases.

Example. Reduce l of of -J of 5 to a simple fraction.

1 2 3 5 1 x 3X 3x 5 5

— of — of — of — =-= — by cancelling the com-

2 3 4 1 2x3X 4x 1 4

mon terms 2 and 3.

(35.) Rule VI. To reduce Fractions of different Denominatorsto equivalent Fractions of a common one.

If the fractions can be conveniently reduced to a common de-nominator, by multiplying or dividing their terms, proceed bythat method. But, if not, multiply each numerator continuallyinto all the. denominators, except its own, for each new numera-tor ; and multiply all the denominators together for the commondenominator. In this and several other operations, w hen any ofthe proposed quantities are integers, mixed numbers or compoundfractions, they must be reduced by their proper rules, to the formof simple fractions.

Example. Reduce ■§, J,and ^ to a common denominator.

12 3 12 16 18 68 9

Thus —, —, and — = —, —, and —, or == —, —, and —

2 3 4 24 24 24 12 12 12

(36.) Rule VII. To find the Value of proper Fractions in Num-

bers of inferior Denominations.

Multiply the integer by the numerator, and divide by the de-nominator.

1

Examples