FRACTIONS.

DEFINITIONS, NOTATION, AND NUMERATION.

7. If a unit be divided into 2 equal parts, one of the parts is called one half.

If a unit be divided into 3 equal parts, one of the parts is called one third, two of the parts two thirds.

If a unit be divided into 4 equal parts, one of the parts is called one fourth, two of the parts two fourths, three of the parts three fourths.

If a unit be divided into 5 equal parts, one of the parts is called one fifth, two of the parts two fifths, three of the parts three fifths, etc.

And since one half, one third, one fourth, and all other equal parts of an integer or whole thing, are each in themselves entire and complete, the parts of a unit thus used are called fractional units; and the numbers formed from them, fractional numbers. Hence,

78. A Fractional Unit is one of the equal parts of an integral unit.

79. A Fraction is a fractional unit, or a collection of fractional units.

80. Fractional units take their name, and their value, from the number of parts into which the integral unit is divided. Thus, if we divide an orange into 2 equal parts, the parts are called halves; if into 3 equal parts, thirds; if into 4 equal parts, fourths, etc.; and each third is less in value than each half, and each fourth less than each third; and the greater the number of parts, the less their value.

The parts of a fraction are expressed by figures; thus,

To write a fraction, therefore, two integers are required, one written above the other with a line between them.

81. The Denominator of a fraction is the number below the line. It shows into how many parts the integer or unit is divided, and determines the value of the fractional unit.

82. The Numerator is the number above the line. It numbers the fractional units, and shows how many are taken.

83. Thus, if one dollar be divided into 4 equal parts, the parts are called fourths, the fractional unit being one fourth, and three of these parts are called three fourths of a dollar, and may be written

3 the number of parts or fractional units taken.

4 the number of parts or fractional units into which the dollar is divided. 84. The Terms of a fraction are the numerator and denominator, taken together.

85. Fractions indicate division, the numerator answering to the dividend, and the denominator to the divisor. Hence,

86. The Value of a fraction is the quotient of the numerator divided by the denominator.

Thus, the quotient of 4 divided by 5 is, or expresses the quotient of which 4 is the dividend.

5 is the divisor.

1. What is 1 half of 8?

ANALYSIS. It is the quotient of 8 divided by 2, which is 4; or, it is a number which, taken 2 times, will make 8, which is 4. Therefore, 4 is 1 half of 8.

2. What is 2 thirds of 9?

ANALYSIS. Since 1 third of 9 is 3, 2 thirds of 9 is 2 times 3, which is 6. Therefore, 2 thirds of 9 is 6.

Hence, to obtain one half, one third, one fourth, or any fractional part of a number, divide that number by the denominator of the fraction expressing the parts; and to obtain any given number of such parts, multiply that part by the number of parts expressed by the numerator of the same fraction.

3. What is 1 fourth of 12? 3 fourths of 12?

4. What is 1 fifth of 20? 3 fifths? 4 fifths?

5. What is 1 eighth of 40? 3 eighths? 5 eighths?

6. What is 2 sevenths of 21 ? 5 sevenths of 35? 6 sevenths of 49 ?

7. What is 1 ninth of 63? 2 ninths of 27? 4 ninths of 36? 5 ninths of 45? 7 ninths of 81 ?

8. What is 1 twelfth of 48? 5 twelfths? 7 twelfths ? 9. If a pound of coffee cost 15 cents, what will 1 third of a pound cost? 2 thirds?

10. A farmer having 60 sheep, sold 1 fifth of them to one man, and 3 fifths to another? how many did he sell to both?

11. A boy having 48 cents, spent 3 eighths of them; how many had he left?

12. Paid 108 dollars for a horse, and 9 twelfths as much for a carriage; what did the carriage cost?

13. William had 120 pennies, and James had 7 tenths as many; how many had James?

87. It is often required to express by a fraction, what part one number is of another number.

1. What part of 5 is 3?

ANALYSIS. Since 1 is 1 fifth of 5, 3 is 3 times 1 fifth of 5, or 3 fifths of 5. Therefore, 3 is 3 fifths of 5.

The number preceded by the word of is generally made the denominator or divisor, and the other number called the part, the numerator or dividend.

2. What part of 6 is 3? 4? 5? 1?

3. What part of 9 is 2? 3? 5? 6? 1? 4?

4. What part of 10 is 7? 6? 3? 1? 9? 8? 4?

5. What part of 12 is 3? 5? 6? 8? 9? 7? 10? 11? 6. What part of 14 is 5 ?7? 9? 3? 6? 11? 8? 15? 7. What part of 15 bushels is 3 bushels? 7 bushels? 9 bushels? 11 bushels?

8. What part of 18 dollars is 7 dollars? 5 dollars? 9 dollars? 17 dollars?

9. If 6 oranges cost 30 cents, what part of 30 cents will 1 orange cost? 2 oranges? 3 oranges? 5 oranges?

EXAMPLES IN WRITING AND READING FRACTIONS.

Express the following fractions by figures :

1. 9 twelfths.

2. Eleven fifteenths.

3. Twenty-four forty-ninths.

4. Forty-four sixty-ninths.

Ans.

Ans. H.

Ans. 4.

Ans.

5. One hundred twenty four hundred fiftieths.

Read the following fractions :

84

6. 17, 18, 37, 146, 144, 15, 150, 119.

7. If the fractional unit is 28, express 9 fractional units; 16; 17; 22; 27.

8. If the fractional unit is 96, express 27 fractional units; 42; 75.

88. Fractions are distinguished as Proper and Improper. A Proper Fraction is one whose numerator is less than its denominator; its value is less than the unit, 1. Thus, ,,, are proper fractions.

90

An Improper Fraction is one whose numerator equals or exceeds its denominator; its value is never less than the unit, 1. Thus, 7, 8, 14, 35, 50, 182 are improper fractions. 89. A Mixed Number is a number expressed by an integer and a fraction; thus, 41, 1718, 9 are mixed numbers.

REDUCTION.

90. The Reduction of a fraction is the process of changing its form without altering its value.

CASE I.

91. To reduce fractions to their lowest terms.

A fraction is in its lowest terms when no number greater than 1 will exactly divide both numerator and denominator without a remainder.

1. Reduce to its lowest terms.

ANALYSIS. It is plain, that the numerator 2, and the denominator 4, are both divisible by 2, without remainders; hence

The terms thus obtained, viz., 1, the numerator, and 2, the denominator, are not both divisible by any number greater than 1, and therefore are the smallest terms by which the value of can be expressed.

2. Reduce to its lowest terms.