ART. 112.-The unit 1 is the measure of all numbers, hence, 1 is the unit of a fraction; one of the equal parts into which the unit is divided is a fractional unit; and more than one of the equal parts is a collection of fractional units.

ART. 113. To analyze a fraction is to name the unit of the fraction; the fractional unit; the number of fractional units taken; and the value of the fraction.

Thus, in analyzing the fraction we say: The unit of the fraction is 1; the fractional unit, 1; the number of fractional units taken, 3; the value of the fraction, of one.

In analyzing the fraction we say: The unit of the fraction is 1; the fractional unit, ; the number of fractional units taken, 4; the value of the fraction, f of one.

Analysis of the fraction : The unit of the fraction is 1; fractional unit, ; number of fractional units, 5; value of the fraction, of one.

ART. 114.-Analyze, according to the models given, the following fractions:

PRINCIPLES AND DEFINITIONS.

ART. 115.-Multiplying the numerator of a fraction by any number multiplies its value by that number, because it increases the number of fractional units. ↓ × 3 = 3; 2 × 4 = }; † × 5 = 45.

Thus:

ART. 116.-Dividing the denominator of a fraction by any number multiplies its value by that number, because it increases the size of the fractional unit. Thus, 2 + 2 = ; { + 2 = { ; } + 4 = £• 용

ART. 117.-Multiplying the denominator of a fraction by any number divides the value of the fraction by that number, because it decreases the size of the fractional unit. Thus, × 3 = 3; 4 × 2 = 4; 7 × 3 = 4. 4×

ART. 118.-Dividing the numerator of a fraction by any number divides the value of the fraction by that number, because it decreases the number of fractional units. Thus, + 2 = }; } + 3 = † ; } + 5 = ƒ•

3

ART. 119.-Multiplying both numerator and denominator of a fraction by the same number does not change its value, because the increase in the number of fractional units is equaled by the decrease in their size. Thus, 용

1 × 3 = 1; } × 4 = 1; } × } =

8 12

ART. 120.-Dividing both numerator and denominator of a fraction by the same number does not change its value, because the decrease in the number of fractional units is equaled by the increase in their size. Thus, 4 ÷ 1 = 1; } ÷ } = 1; 35 ÷ } = }·

ART. 121.-A Fraction is one or more of the equal parts of a unit. It is an expression of division.

ART. 122.-A fraction is expressed by two numbers, written one above a short horizontal line, and the other below it. It is a quotient.

ART. 123.-The Denominator of a fraction shows into how many equal parts the unit is divided, and is written below the line. It is the divisor.

ART. 124. The Numerator of a fraction shows how

many of the equal parts of the unit or number are taken, and is written above the line. It is the dividend.

ART. 125.-The Terms of a Fraction are the numerator and denominator.

ART. 126.-A Proper Fraction is one whose numerator is less than its denominator; as, 3, 4, 18.

ART. 127.-An Improper Fraction is one whose numerator is equal to or greater than its denominator; as, &, 13, 1o.

ART. 128.—A Simple Fraction is one having a single integral numerator and denominator; as, t, 30, 28.

ART. 129.-A Complex Fraction is one whose numerator, or denominator, or both, are fractional; as, 3 7 4 21 + 3

7' 7' 3' 6

ART. 130.-A Mixed Number is a whole number and fraction united; as, 31, 5, 19.

ART. 131.-The Reciprocal of a Number is 1 divided by that number. Thus, the reciprocal of 5 is 1 ÷ 5 or ; of 11, it is 111 or. The reciprocal of a fraction is the fraction inverted. Thus, the reciprocal of is ; of is 2.

ART. 132. To change Fractions to their Lowest Terms.

ART. 133.-A fraction is in its lowest terms or simplest form when no number greater than 1 will exactly divide its terms, as 4, 5, 7.

Solution. It is readily seen that 3 exactly divides both terms of the fraction. Dividing the terms by 3 we obtain 3. We cannot divide the terms of by any number greater than 1; hence, changed to its lowest terms is . Thus,

Change to their lowest terms :

÷ 3

= 용.

ART. 134.-Rule for changing Fractions to their Lowest Terms.-1. Find the greatest common divisor of the terms of the fraction, and 2. Divide both terms of the fraction by their greatest common divisor.

255

11. 2, 1888, 1188, 1835, to their lowest terms.

928 12849

ART. 135.—To change Mixed Numbers to Improper Fractions.

ORAL EXERCISES.

1. How many fourths are in 123 ?

Solution. Since there are 4 in 1 unit, in 12 units there are 12 times or 48, and added make 1. Proof: 51 ÷ 4 = 123.

Change the following mixed numbers to improper

ART. 136.-Rule for changing Mixed Numbers to Improper Fractions.-Multiply the integer by the denominator of the fraction, add the numerator to the product and write the sum over the given denominator.