2

taken in and, and each part in the former is three times as great as each part in the latter, by (72).

Hence, to multiply and divide a fraction by a whole number, we have only to multiply the numerator and denominator by it, respectively.

76. The same kind of reasoning will enable us to represent what is called a Compound Fraction in the form of a simple one.

A Compound Fraction is made up of two or more simple fractions connected together by the word of, as

which is a simple fraction of the ordinary form:

that is,

and from this, we infer that a compound fraction is equivalent to the simple fraction formed by multiplying together respectively the numerators and the denominators of its constituent simple fractions.

TRANSFORMATION OF FRACTIONS.

77. If the numerator and denominator of a fraction be both multiplied or divided by the same number, the value of the fraction will not be altered.

15

15

For, if the fraction be multiplied by 5, the product is and again if this be divided by 5, the quotient is by the last article but one: but since these two operations are the reverse of, and therefore neutralize, each other, it follows that

35,

and also, that

15 3 15÷5

35 7 35÷5

By means of this article, a whole number may be expressed in the form of a fraction with any denominator we please: thus,

transformation

of Fractions

7

Also, a fraction may be transformed into another with a given denominator, provided it be a multiple of the denominator of the proposed fraction: thus, may be transformed so as to have 96 for its denominator, because 7 x 12 84

78. Since

7

8

96*

5 × 4

=

2 × 4 2

for the Multiplication of a fraction by an integer, it appears
to be immaterial whether the numerator be multiplied,
or the denominator be divided, by it: and inasmuch as

for the Division of a fraction by a whole number, it
amounts to the same thing whether we divide the nu-
merator, or multiply the denominator, by it.

79. A quantity made up of two others, one of which is
an integer and the other a fraction, may be represented in
the form of a fraction alone.

Let us take 3, which is called a mixed quantity, and is intended to express the integer 3 and the fraction taken together, and must be read three and four-fifths: then, since

3

=

3 15
1 5'

=

the mixed quantity 3 is equivalent to 15 and

19

taken

together, or, to by the second definition: and this
operation put in the form,

RULE. Multiply the integer by the denominator of the fraction: to the product add the numerator, and the result will be the required numerator, which placed over the denominator will form the improper fraction required.

Examples for Practice.

(1) Express 25, 5, 12, and 54 in the forms of improper fractions.

602

and

11

(2) Reduce to fractional forms, the mixed quantities, 41, 123, 275 and 374,5

80. A compound fraction formed of mixed quantities, may therefore by the last article be exhibited in the form of a simple fraction: thus,

Exhibit the compound fractions, 13 of 7; of of 12, and 15 of 8 of 13%, as improper fractions. 2684 25 64672 and

Answers:

9
27 6

35

81. By means of the preceding articles, what is called a Complex Fraction may be reduced to a simple one; thus,

a simple fraction, obtained by multiplying both the compound numerator and denominator by the product of the denominators of their fractional parts.

82. A quantity in the form of an improper fraction may always be expressed by a mixed quantity.

35
8

We see immediately that is equivalent to

32

32

32+ 3

8

or, to and taken together: but is equal to the

integer 4, and therefore the required mixed quantity will be equal to the integer 4 and the proper fraction taken together, which is sometimes expressed by

3

4+ but more generally in the form 43.

8'

This process is evidently the same thing as dividing both the numerator and denominator by the denominator, and noticing the remainder of the former: and stated in the form,

8) 35
4

it suggests the following rule.

RULE. Divide the numerator of the fraction by the denominator, and the quotient will be the integral part; and the fractional part will be formed by making the remainder the numerator of a fraction having the same denominator as the one proposed. If there be no remainder, the fraction is equivalent to the integer thus found.

Answers: 33, 127, 130% and 823.

(3) Represent the following fractional quantities,

4

5

8357 18793 1
of of 6%, and of
278' 359 3 7

in the forms of mixed quantities.

Answers: 30 521, 1 and 27.

215

83. A fraction may be reduced to its lowest terms, by dividing both its numerator and denominator, by their greatest common measure.

For, since the value of a fraction is not altered by dividing its numerator and denominator by any factor

common to them both, it will necessarily be expressed in its lowest or simplest terms, when that factor is the greatest common measure, determined by the Rule of Article (53).

If the greatest common measure be 1, the numerator and denominator are prime to each other, and the fraction is already in its lowest terms.

Ex. Reduce the fraction

825 960

to its lowest terms.

By Article (53) above mentioned, we have

8 2 5 9 6 0 (1
825

135) 8 2 5 (6

810

1 5) 1 3 5 (9

135

and 15 is therefore the greatest common measure: and dividing each of the terms of the fraction by it, as

55

75

we have for the equivalent fraction expressed in the least terms possible.

The terms of the original fraction are equal multiples, or equimultiples, of those of the equivalent reduced one.

84. In many instances it is unnecessary to find the greatest common measure at first, the fractions being reducible to lower terms by successive divisions of the numerators and denominators by common factors discovered by inspection.

Thus,

4968 2484 1242 621 207 69

=

5904 2952 1476 738 246 82'

from three successive divisions of the numerator and denominator by 2, and then from two successive divisions