If an orange be cut into 8 equal parts, what fraction will express 3 of the parts?

If 3 apples be divided equally among 4 boys, what part of an apple will be given to each boy?

What part of 1 apple, is a third part of 2 apples?
What expresses the quotient of 5, divided by 8?

ART. 127. A whole number may be expressed in the ト

form of a fraction, by writing 1 below it for a denominator.

Thus, 2 may be written
3 may be written

4 may be written

and is read two-ones.

and is read three-ones.

and is read four-ones.

But, 2 ones are 2; 3 ones, 3, &c.; hence, the value of the number is not changed.

ART. 128. If 2 apples be divided, each into four equal parts, there will be 8 parts in all. Three of the parts (fourths) are expressed by ; 4 parts by ; 5 parts by

4'

When the number of parts taken is less than the number into which the unit is divided, the value expressed is less than one, or the whole thing;

When the number of parts taken is equal to the number into which the unit is divided, the value, as, is equal to 1; When greater than the number into which the unit is divided, the value,, is greater than 1. Hence,

REVIEW. 126. Of the fractions to be read, which expresses parts of the largest size? Which the smallest? Which the least number? Which the greatest? Which the same? Which parts of the same size?

127. Ilow may a whole number be expressed in fractional form? Does this change its value? Why not? 128. When is the value of a fraction less than 1? When equal? When greater? ilustrato by examples.

1st. When the numerator is less than the denominator, the value of the fraction is less than 1.

2d. When the numerator is equal to the denominator, the value of the fraction is equal to 1.

3d. When the numerator is greater than the denominator, the value of the fraction is greater than 1.

DEFINITIONS.

ART. 129. 1. A Fraction is an expression of one or more of the equal parts of a unit, or one thing.

2. A Proper Fraction has a numerator less than the denominator; as, 1, 3, and §.

3. An Improper Fraction has a numerator equal to, or greater than the denominator; as, 3, and §.

REM.-A proper fraction is so termed, because it expresses a value less than one. An improper fraction is not properly a fraction of a unit, the value expressed being equal to, or greater than one.

4. A Simple Fraction is a single fraction, either proper or improper; as,,, and .

5. A Compound Fraction is a fraction of a fraction, or several fractions joined by of; as, of 1,

of of

6. A Mixed Number is a fraction joined to a whole number; as, 21, 31, and 57.

7. A Complex Fraction has a fraction in one or in both of

two equal parts, one of the parts, as A C, is termed onehalf (1): that is, one-half of 1, or the whole thing.

REVIEW.-129. What is a fraction? A proper fraction? An improper fraction? REM. Why is a proper fraction so termed? An improper fraction? What is a simple fraction? A compound fraction? A mixed number? A complex fraction? Give examples of each. 180. What is the half of one-half? Why? The third of one-half? Why?

COMMON FRACTIONS.

139

If a half be divided into 3 equal parts, as in the figure, one of the parts is one-third of one-half, expressed thus, of 1; and this is one-sixth of the whole line: that is, of is

In the same manner, it may be shown, that ofis ; that the ofis; that ofis ; that ofis ; &c.

If one apple be divided into three equal parts, and each of these parts into 3 other equal parts, into how many parts will the whole be divided?

What part of the whole will one piece be? What is of }?

If one orange be divided into 3 equal parts, and each part into 4 other equal parts, into how many equal parts will the whole be divided?

What part of the whole will I piece be? What is of ? What is of; of 1? of? of? of? Young Pupils may omit the Illustrations until they review the book.

GENERAL PRINCIPLES.

ART. 131. If the numerator of the fraction be multiplied by 3, without changing the denominator, the result will be .

ILLUSTRATION. Each of the fractions and having the same denominator, expresses parts of the same size: but, as the numerator of the second fraction (§), is 3 times that of the first, (), it expresses 3 times as many of those equal parts as the first, and is 3 times as large. The same may be shown of any other fraction. Hence,

PROPOSITION 1-If the numerator be multiplied without changing the denominator, the value of the fraction will be multiplied as many times as there are units in the multiplier.

Hence, a fraction is multiplied, by multiplying its numerator.

REVIEW.-131. What is the effect of multiplying the numerator of a fraction, without changing the denominator, Prop. I?

ART. 132. If the numerator of the fraction be divided by 2, without changing the denominator, the result will be.

denominator, expresses parts of the same size; but the numerator of the second fraction (3), is only one-half the numerator of the first (); therefore, expresses one-half as many of those equal parts as the first (), and is one-half the value. Hence,

PROP. II.—If the numerator be divided without changing the denominator, the value of the fraction will be divided as many times as there are units in the divisor.

Hence, a fraction is divided by dividing its numerator.

ART. 133. If the denominator of the fraction be multiplied by 2, without changing the numerator, the result will be .

3

Thus, 2-3

ILLUSTRATION. Each of the fractions

same numerator, denotes the same number of parts; but, in the second (3) the parts are one-half the size of those in the first (3): but, of (Art. 180); consequently, the whole value of the second fraction is one-half that of the first. Hence,

PROP. III. If the denominator be multiplied without changing the numerator, the value of the fraction will be divided as many times as there are units in the multiplier.

Hence, a fraction is divided, by multiplying its denominator.

ART. 134. If the denominator of the fraction be divided by 3, without changing the numerator, the result will be.

ILLUSTRATION. Each of the fractions and having the same numerator, denotes the same number of parts; but, in the

second fraction (3), the parts are 3 times the size of those in the first (3): but, }=} of { (Art. 130); consequently, the value of the second fraction is 3 times that of the first. Hence,

PROP. IV. If the denominator be divided, without changing the numerator, the value of the fraction will be multiplied as many times as there are units in the divisor.

Hence, a fraction is multiplied, by dividing its denominator.

ART. 135. If the numerator of a fraction be multiplied by any number, its value (PROP. I,) will be multiplied by that number; if the denominator be multiplied, the value (PROP. III,) will be divided by that number.

Hence, if both terms are multiplied by the same number, the increase from multiplying the numerator, equals the decrease from multiplying the denominator: and the value is not changed. 3X2 6 3X3 9

Thus,

=

and

=

5X2 10' 5X3 15'

ILLUSTRATION.—Multiplying both terms of the fraction by 2, gives in which the parts are twice as many, but only one-half the size. Multiplying both terms of by 3, gives; three times as many parts, each part one-third the size. Hence,

PROP. V. Multiplying both terms by the same number, changes its form, but does not alter its value.

ART. 136. If the numerator of a fraction be divided

by any number, its value (PROP. II,) will be divided by that number; if the denominator be divided, the value (PROP. IV,) will be multiplied by that number.

Hence, if both terms are divided by the same number, the decrease from dividing the numerator, equals the increase from dividing the denominator: and the value is not changed

REVIEW.-132. What is the effect of dividing the numerator of a fraction, without changing the denominator? 183. What of multiplying the denominator without changing the numerator?

134. What of dividing the denominator, without changing the numerator? 135, What of multiplying both terms by the same number ?