DEFINITIONS, NOTATION, AND NUMERATION.

87

114. To write a fraction, two integers are required, one to express the number of parts into which the whole number is divided, and the other to express the number of these parts taken. Thus, if one dollar be divided into 4 equal parts, the parts are called fourths, and three of these parts are called three fourths of a dollar. This three fourths may be written

3 the number of parts taken.

4 the number of parts into which the dollar is divided.

115. The Denominator is the number below the line. It denominates or names the parts; and

It shows how many parts are equal to a unit.

116. The Numerator is the number above the line. It numerates or numbers the parts; and

It shows how many parts are taken or expressed by the fraction.

117. The Terms of a fraction are the numerator and denominator, taken together.

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

Hence, 119. The Value of a fraction is the quotient of the numerator divided by the denominator.

120. To analyze a fraction is to designate and describe its numerator and denominator. Thus, is analyzed as follows:

4 is the denominator, and shows that the unit is divided into 4 equal parts; it is the divisor.

3 is the numerator, and shows that 3 parts are taken; it is the dividend, or integer divided.

3 and 4 are the terms, considered as dividend and divisor. The value of the fraction is the quotient of 34, or 2.

How many numbers are required to write a fraction? Why? Define the denominator. The numerator. What are the terms of a fraction? The value? What is the analysis of a fraction?

EXAMPLES FOR PRACTICE.

Express the following fractions by figures:

1. Seven eighths.

2. Three twenty-fifths.

3. Nine one hundredths.

4. Sixteen thirtieths.

5. Thirty-one one hundred eighteenths.

6. Seventy-five ninety-sixths.

7. Two hundred fifty-four four hundred forty-thirds.

8. Eight nine hundred twenty-firsts.

9. One thousand two hundred thirty-two seventy-five thou sand six hundredths.

10. Nine hundred six two hundred forty-three thousand eighty-seconds.

Read and analyze the following fractions:

11. ; fz; 25; 13; 75; Tiz; 25; 11. 12. ; ; 12; 10; 2880; 1275.

121. 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, 12, 16, fo,are proper fractions.

An Improper Fraction is one whose numerator equals or exceeds its denominator; its value is never less than the unit, 1. Thus, 7, 3, 42, 35, 48, 180 are improper fractions.

122. A Mixed Number is a number expressed by an integer and a fraction; thus, 44, 178, 9 are mixed numbers. 123. Since fractions indicate division, all changes in the terms of a fraction will affect the value of that fraction according to the laws of division; and we have only to modify the language of the General Principles of Division (87) by substituting the words numerator, denominator, and fraction, or value

What is a proper fraction? An improper fraction? A mixed number? What do fractions indicate?

of the fraction, for the words dividend, divisor, and quotient, respectively, and we shall have the following

GENERAL PRINCIPLES OF FRACTIONS.

124. PRIN. I. Multiplying the numerator multiplies the fraction, and dividing the numerator divides the fraction.

PRIN. II. Multiplying the denominator divides the fraction, and dividing the denominator multiplies the fraction.

PRIN. III. Multiplying or dividing both terms of the fraction by the same number does not alter the value of the fraction. These three principles may be embraced in one

GENERAL LAW.

125. A change in the NUMERATOR produces a LIKE change in the value of the fraction; but a change in the DENOMINATOR produces an OPPOSITE change in the value of the fraction.

REDUCTION.

CASE I.

126. To reduce fractions to their lowest terms.

A fraction is in its lowest terms when its numerator and denominator are prime to each other; that is, when both terms have no common divisor.

1. Reduce the fraction 8 to its lowest terms.

FIRST OPERATION.

48=31=13=t, Ans.

ANALYSIS.

Dividing both terms of a fraction by the same number does not alter the value

of the fraction or quotient, (124, III;) hence, we divide both terms of 48, by 2, both terms of the result, 4, by 2, and both terms of this result by 3. As the terms of are prime to each other, the lowest terms of 8 are . We have, in effect, canceled all the factors common to the numerator and denominator.

What is

First general principle? Second? Third? General law? What is meant by reduction of fractions? Case I is what? meant by lowest terms? Give analysis.

SECOND OPERATION.

12) 18, Ans.

In this operation we have divided both terms of the fraction by their greatest common divisor, (97,) and

thus performed the reduction at a single division. Hence the

RULE. Cancel or reject all factors common to both numerator and denominator. Or,

Divide both terms by their greatest common divisor.

12. Express in its simplest form the quotient of 441 divided by 462.

Ans.

13. Express in its simplest form the quotient of 189 divided by 273.

Ans.

14. Express in its simplest form the quotient of 1344 divided by 1536.

CASE II.

Ans.

127. To reduce an improper fraction to a whole or mixed number.

1. Reduce 324 to a whole or mixed number.

OPERATION.

32432415=21=213, Ans.

ANALYSIS. Since 15 fifteenths equal 1, 324 fifteenths are

equal to as many times 1 as 15 is contained times in 324, which is 21 times. Or, since the numerator is a dividend and the denom

Rule. Case II is what? Give explanation.

inator a divisor, (118,) we reduce the fraction to an equivalent whole or mixed number, by dividing the numerator, 324, by the denominator, 15. Hence the

RULE. Divide the numerator by the denominator.

NOTES. 1. When the denominator is an exact divisor of the numerator, the result will be a whole number.

2. In all answers containing fractions, reduce the fractions to their lowest terms.

EXAMPLES FOR PRACTICE.

2. In

of a week, how many weeks?

Ans. 19.

3. In 147 of a bushel, how many bushels ?

Ans. 23.

4. In 491 of a dollar, how many dollars?

Ans. 541.

5. In 872 of a pound, how many pounds?
6. Reduce 138 to a mixed number.
7. Reduce 13 to a whole number.
8. Change 112 to a mixed number.
9. Change 1321 to a mixed number.

10. Change 237040 to a mixed number.

225

11. Change 2531820 to a whole number.

CASE III.

Ans. 183.

Ans. 1053.

Ans. 7032.

128. To reduce a whole number to a fraction having a given denominator.

1. Reduce 46 yards to fourths.

OPERATION.

46
4

184, Ans.

ANALYSIS. Since in 1 yard there are 4 fourths, in 46 yards there are 46 times 4 fourths, which are 184 fourths 184. In practice we multiply 46, the number of yards, by 4, the given denominator, and taking the product, 184, for the numerator of a fraction, and the given denominator, 4, for the denominator, we have 184. Hence we have the

RULE. Multiply the whole number by the given denominator; take the product for a numerator, under which write the given denominator.

Rule. Case III is what? Give explanation. Rule.