names and expresses how many fractional units are equal to a unit.

The numerator and denominator taken together are called terms of the fraction.

Fractions are of three kinds, common, decimal and duodecimal. One or more of the equal parts of a quantity, expressed by two numbers, one written above the other with a line between them, is a common fraction— , and 3.

Its denominator is other than ten, or some power of ten.

A fractional number, whose value is less than a unit, is a proper fraction, as 3, §. It is so termed because it expresses a value less than 1.

An improper fraction

is not properly a fraction of a unit, the value expressed being equal to or greater than 1.

A single fraction, either proper or improper, is a simple fraction, t, f.

A fraction of a fraction, or several fractions joined by of, is termed a compound fraction, as 2-4 of 7-8 of 3-12. A fraction in the numerator, or denominator, or both,

is termed a complex fraction, as

Unity divided by any number is termed a reciprocal ; thus the reciprocal of 4 is 4.

An integral number added to a fractional number is termed a mixed number, as 3+4, 7+%. The sign of addition is usually omitted.

GENERAL PRINCIPLES

1. Multiplying the numerator increases the value of the fraction.

Because it increases the number of fractional units while the value of the fractional unit remains the same.

2. Multiplying the denominator decreases the value of the fraction.

Because it diminishes the value of the fractional unit, while the number remains the same; it diminishes the value of the fractional unit because the unit of the fraction is divided into a greater number of fractional units, and each fractional unit is as many times less in value as there are units in the multiplier.

3. Multiplying both numerator and denominator by the same number does not alter the value of the fraction. Because it increases the number of fractional units, as many times as it decreases the value of the fractional unit; that is in the same ratio.

4. Dividing the numerator decreases the value of the fraction.

Because it diminishes the number of the fractional units, while the value of the fractional unit remains the same.

5. Dividing the denominator increases the value of the fraction.

Because it increases the value of the fractional unit, while the number remains the same; it increases the value of the fractional unit because the unit of the fraction is divided into a less number of fractional units, each fractional unit being as many times greater in value as there are units in the divisor.

6. Dividing both numerator and denominator by the same number does not alter the value of the fraction. Because it diminishes the number of fractional units as many times as it increases the value of the fractional unit.

7. If the numerator be multiplied by any number, the number of fractional units will be increased as many times as there are units in the multiplier.

8. If the numerator be divided by any number, the number of fractional units will be diminished as many times as there are units in the divisor.

9. If the denominator be multiplied by any number, the fractional units will be diminished as many times as there are units in the multiplier.

10. If the denominator be divided by any number, the value of the fractional units will be increased as many times as there are units in the divisor.

Naming the quantity or unit divided, the value of one of its fractional units, the number of fractional units, the denominator, numerator

Analysis of a fraction.

and the terms of the fraction, is to analyze a fraction. Thus:

Analyze the fraction .

is a fraction because it expresses 4 of the equal parts of a unit. 1 is the unit of the fraction, or the unit that is divided to form the fraction. is the fractional unit, or one of the equal • parts of the unit divided. 5 is the denominator; it names the parts; it shows that the unit is divided into 5 equal parts; it tells the size or value of each part. 4 is the numerator; it numbers the parts taken to form the fraction; it is written above the line. 4 and 5 are the terms of the fraction, and its value is 4÷5.

Dividing by =; as the numerator and denominator are prime to each other, the fraction is reduced to its lowest terms. This depends upon the following principle: Dividing both terms of the fraction by the same number does not alter the value of

the fraction, because the number of fractional units is decreased as many times as the value of the fractional unit is increased. (Deduce the rule.)

Reduction of

improper

Improper fractions are reduced to integer

fractions. or mixed numbers as follows:

Reduce 125 to an integral number.

125 ÷ = 25, or 5, 15 = 45 = 25.

In 1 there are 5 fifths; in 125 fifths, as many ones as 5 is contained times in 125, or 25. This depends upon the following principle: Dividing both terms of the fraction by the same number does not alter the value of the fraction; the same reason as when we reduce fractions to their lowest terms. (Deduce the rule.)

Integers reduced.

Integers or mixed numbers are reduced to improper fractions as follows:

Reduce 49% to fifths.

§ 49 = 245

245+2 = 247

In one there are 5 fifths; in 49 ones, 49 times 5 fifths, or 245 fifths; plus 2 fifths equals 247 fifths. This depends on the following principle: Multiplying both terms of the fraction by the same number does not alter the value of the fraction, because the number of fractional units is increased as many times as the value of the fractional unit is decreased. (Deduce the rule.)

Common

Fractions are reduced to a common dedenominator. nominator as follows:

The least common multiple of the denominators is 120; dividing the least common multiple by the denominator of the first fraction, we have the quotient 24; multiplying both terms of the fraction by 24, we have. This depends upon the following principle: multiplying both terms of the fraction by the same number, does not alter the value of the fraction, because it increases the number of fractional units as many times as it decreases the value of the fractional unit. (The same analysis for the re

As the fractions have the same fractional unit, we may add the numerators; +†

Add and.

=

§ = 1}.

As the fractions and have different fractional units, first reduce them to fractions having the same fractional unit. § is equal to ; equal to ; now as the fractions are of the same fractional unit value, we may add the numerators:

= 18. (Deduce the rule.)

=

One fraction is subtracted from another as

follows:

Subtract from 4.

=

20 = 30.

83

Subtraction.

The fractions and have different fractional units. First reduce the fractions to the same fractional unit value.equals 15; is equal to ; as the fractions are of the same fractional unit value, we may subtract one numerator from the other, giving us. (Deduce the rule.)

Fractions are multiplied by an integer as

Multiplication by an integer,