CHAPTER II

FRACTIONS.

14. A FRACTION is an expression representing a part of a unit.

VULGAR FRACTIONS.

15. A VULGAR FRACTION Consists of two numbers, the one placed above the other, as in division.

The number above the line is called the numerator; the number below the line is called the denominator. Thus, is a vulgar fraction, whose numerator is 5, and whose denominator is 8; it is read five-eighths.

The denominator shows how many parts the unit is divided into, and the numerator shows how many of these parts are used.

Thus, denotes that the unit is divided into 8 equal parts, and that 5 of these parts are used.

When the numerator is equal to the denominator, the fraction is equivalent to a unit. Thus, &,,, and 1, are each equivalent to 1.

3

When the numerator is less than the denominator, the value of the fraction is less than a unit; it is then called a proper fraction.

Thus,,, 4, and %, are each proper fractions. When the numerator is larger than the denominator, its value is then more than a unit; it is, therefore, called an improper fraction.

Thus,,,, and 4, are each improper fractions.

A fraction of a fraction, connected by the word of, is called a compound fraction.

2

Thus, of of 2 of 7, 1 of 3 of 9, and of of , are compound fractions.

A fraction is said to be inverted, when the numerator and the denominator change places.

Thus, the fractions,, and, when inverted, become 1, IT, 17 5 and.

Any integer may take the form of an improper fraction, by writing a unit for its denominator.

Thus, 6, 5, 3, and 11, are the same as the improper fractions,,, and .

A number consisting of an integer and a fraction, is called a mixed number.

Thus, 41, 57, 63, and 1314, are mixed numbers. They may also be written 4+1, 5+7, 6+, and 13+1.

REDUCTION OF FRACTIONS.

16. SINCE the value of a fraction is the quotient arising from dividing the numerator by its denominator, we may infer the following

Propositions.

I. That, multiplying the numerator of a fraction by any number, is the same as multiplying the value of the fraction by the same number.

II. That, multiplying the denominator of a fraction by any number, is the same as dividing the value of the fraction by the same number.

III. That, multiplying both numerator and denominator by the same number, does not alter the value of the fraction.

IV. That, dividing the numerator of a fraction by any number, is the same as dividing the value of the fraction by the same number.

V. That, dividing the denominator of a fraction by any number, is the same as multiplying the value of the fraction by the same number.

VI. That, dividing both numerator and denominator by the same number, does not alter the value of the fraction.

17. When the numerator and denominator of a fraction have no common measure, it is said to be in its lowest terms.

To reduce simple fractions to their lowest terms,* we have the following

*The following properties may frequently be of assistance in abbreviating vulgar fractions:

1. If any number terminate on the right with zero, or an even digit, the whole will be divisible by 2.

2. If any number terminate on the right with zero, or 5, the whole will be divisible by 5.

3. When the number expressed by the two right-hand figures are divisible by 4, the whole will be divisible by 4.

4. When the number expressed by the three right-hand figures are divisible by 8, the whole will be divisible by 8.

5. If the sum of the digits of any number be divisible by 3 or 9, then the whole number will be divisible by 3 or 9. This has already been shown under Art. 5.

6. When the difference between the sum of the digits occupying the odd places, counting from the right towards the left, and the sum of the digits occupying the even places is zero, or divisible by 11, then the number will be divisible by 11.

RULE.

Divide both numerator and denominator by their greatest common measure (found by one of the Rules

under Art. 8, or 9.)

This division will not alter the

value of the fraction.

(Prop. VI., Art. 16.)

EXAMPLES.

1. Reduce 1 to its lowest terms.

In this example, we find the greatest common measure of 375 and 425 to be 25.

Dividing both numerator and denominator by 25, we

Were we to resolve the numerator and denominator into their prime factors, we should then at once discover the factors common to the numerator and denominator, and could, therefore, strike them out, and then the frac tion would be in its lowest terms.

728

728

For example, let it be required to reduce the fraction 1. Resolving the numerator and denominator into their prime factors, the fraction will become = 23 × 7×13 22 × 7× 29

812

Here we discover that the factors 22 x 7

are common to both numerator and denominator. Striking them out, we obtain

It is obvious that this method may be used in all

cases.

Whenever we discover, by inspection, any number which will divide both numerator and denominator without a remainder, we may use it as a divisor, before resorting to either of the above methods.

18. To reduce an improper fraction to a mixed number, we have this

RULE.

Divide the numerator by the denominator; the quotient will be the integral part of the mixed number. The remainder, placed over the denominator of the improper fraction, will form the fractional part.

The correctness of the above rule is obvious from considering that the value of a fraction is the quotient arising from dividing the numerator by the denominator.

EXAMPLES.

1. Reduce to a mixed number.

Dividing 17 by 4, we obtain the quotient 4, with the remainder 1; .. the mixed number equivalent to, is 41, or 4+1.