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120. Definitions of Halves, Thirds, &c. (a.) Such parts as are obtained by dividing any thing or number into two equal parts, are called HALVES of that thing or number. One such part is called one half of it; two such parts are called two halves of it; three such, three halves of it; &c.

(6.) Such parts as are obtained by dividing any thing or number into three equal parts, are called THIRDS of that thing or number. One such part is called one third of it; two such, two thirds; three such, three thirds; four such, four thirds ; &c.

(c.) In like manner, such parts as are obtained by dividing any thing or number into four equal parts, are called FOURTHS of that thing or number ; such as are obtained by dividing it into five equal parts, are called FIFTHS ; into six, are called SIXTHS ; into seven, SEVENTHS; into eight, EIGHTHS ; into nine, NINTHS; into ten, TENTHS; &c.

(d.) 1. What are sevenths ?

Answer. - Sevenths of any thing or number are such parts as are obtained by dividing it into seven equal parts. 2. What are fifths ?

6. What are ninths ? 3. What are thirds ?

7. What are tenths ? 4. What are thirteenths ? 8. What are halves ? 5. What are fourths ? 9. What are twenty-firsts ?

(e.) From the method of obtaining halves, thirds, &c., it follows that

Halves are equal parts of such kind that two of them would equal a unit ; thirds are equal parts of such kind that three of them would equal a unit ; &c.

(f.) Let the pupil now answer all the preceding questions in this article according to the following model.

What are sevenths ?

Answer. - Sevenths of any thing or number are equal parts of srich kind that it will take seven of them to equal that thing or number.

Note. — The explanations of (d.) and (f.) are alike necessary to a thorough understanding of fractions, and the student should not rest satisfied till he has become so familiar with both, as to be able to use either.

121. Fractional Parts. (a.) Such parts as the above are called FRACTIONAL PARTS, and the arithmetical expressions for them are called FRACTIONS ; hence,

1. Fractional parts of any thing, quantity, or number are such parts as are obtained by dividing it into equal parts.

2. Fractional parts of any thing, quantity, or number are equal parts of such kind that a given number of them will equal a unit.

Note. — That from which the fractional parts are obtained is always, when considering the parts, regarded as a unit, and is called the unit of the fraction. It may be,

1. A single object, as an apple, an orange.
2. A unit of measure, as a foot, a yard, a bushel.
3. The abstract unit, one.

4. A number of single objects or units considered as a collection or whole, as 6 apples, 18 feet, 24 bushels.

5. An abstract number, as 5, 8, 12. 6. A fraction, as ,

When no particular unit is expressed, the abstract unit is the one referred to. (6.) It is obvious that the value of

any fractional part depends both on the nature of the unit and the number of such parts which it takes to equal it.

Illustrations. — 1. If a large apple and a small one be each divided into 2, 3, 4, or any other number of equal parts, the parts of the first will be larger than the corresponding parts of the second.

2. If several apples of the same size are divided, one into two equal parts, another into three, another into four, another into five, &c., the parts of the first apple will be larger than those of any other, the parts of the second will be smaller than those of the first, but larger than those of any other, &c.

(c.) If two equal units are divided, one into two equal parts, and the other into four, each part of the first will be equal to two parts of the second ; if one be divided into two equal parts, and the other into three times as many, each part of the first will be equal to three parts of the second ; and, generally, each part obtained by dividing a unit into any num

ber of equal parts, will be twice as large as each would be if the unit had been divided into twice as many equal parts ; three times as large as if divided into three times as many; four times as large as if divided into four times as many; &c.

(d.) In other words, while the unit of the fraction remains the same, each half will be twice as large as each fourth, 3 times as large as each sixth, 4 times as large as each eighth, &c.; each third will be twice as large as each sixth, 3 times as large as each ninth, 4 times as large as each twelfth, &c.; each fourth will be twice as large as each eighth, 3 times as large as each twelfth, &c.

(e.) From these considerations, it follows that in order that fractional parts may be of the same denomination, it is necessary,

1. That they should be parts of the same unit, or of equal

units;

2. That they should be obtained by dividing each unit into the same number of equal parts.

122. Definitions. Method of writing Fractions, &c.

(a.) A FRACTION expresses the value of such parts as are obtained by dividing a unit into equal parts ;

Or, by definition second,

A FRACTION expresses the value of such equal parts, that a certain number of them will equal a unit.

(6.) In writing fractions by figures, two numbers are necessary one to indicate the number of parts into which the unit is divided, or (which is the same thing) the number of such parts which it will take to equal the unit; the other to indicate how many of the parts are considered.

(c.) The first of these is called the DENOMINATOR, because it indicates the denomination of the parts. The second is called the NUMERATOR, because it numbers the parts.

(d.) The numerator is usually written above the denominator, and separated from it by a line.

Illustrations. Five sixths is written 6, 5 being the numerator, and 6 the denominator.

Seventh eighths is written 3, 7 being the numerator, and 8 the denominator.

34 twenty-firsts is written {1,-34 being the numerator, and 21 the denominator.

(e.) Write the following fractions by figures, and tell the numerator and denominator of each. 1. 3 fourths.

6. 1 third. 2. 9 twenty-seconds.

7. 18 thirteenths. 3. 7 halves.

8. 5 elevenths. 4. 9 seventeenths.

9. 15 thirty-firsts. 5. 22 ninths.

(F.) In DECIMAL FRACTIONS the numerator only is written, the denominator being determined by the position of the decimal point. (See 22, a.)

123. Exercises in explaining Fractions. 1. Explain the fraction ý.

Answer. — Five ninths, or five ninths of one, expresses the value of five such parts as would be obtained by dividing a unit into nine equal parts.

Another Form of Answer. - Five ninths, or five ninths of one, expresses the value of five equal parts, such that nine of them would equal

a unit.

2. Explain the fraction .07.

First Form of Answer. Seven hundredths, or seven hundredths of one, expresses the value of seven such parts as would be obtained by dividing a unit into 100 equal parts.

Second Form of Answer. - Seven hundredths, or seven hundredths of one, expresses the value of seven equal parts of such kind that 100 of them would equal a unit.

Explain the following fractions according to the first form of answer, and afterwards according to the second. 3. 4. 6. .09.

9. 7. 6.

10. .008. 8. .27.

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11. Po

* 25 forty-firsts, not 25 forty-oneths, nor 25 forty-ones.

12. 13.* 13. 11 14. 1. 15. 195.

l '16. 427.

17. .0628.
18. .000276.
19. 4.

I 20. 47

21. 25.
22. .0107.
23. .002006.

124. Various Kinds of Fractions. (a.) A simple fraction is one which has but one numerator and one denominator, each of which is a whole number, as g, 7, .

(6.) Simple fractions may be proper or improper.

(c.) A proper fraction is a fraction whose numerator is less than its denominator, as 11, , 13.

(d.) An improper fraction is one whose numerator is equal to, or greater than, its denominator, as g, 13, 143,

(e.) A mixed number is a whole number and a fraction, as 2}, which is read two and one fifth ; 5%, which is read five and three sevenths.

1. Which of the fractions in 123 are proper? 2. Which are improper ?

(f.) A proper fraction is less than 1, because it expresses less parts than it takes to equal a unit.

(g.) An improper fraction is equal to, or greater than, 1, because it expresses as many or more parts than it takes to equal a unit.

(h.) An improper fraction is so called because it expresses a value, a part or all of which may be expressed in whole numbers.

125. Illustrations of Operations on Fractions. Fractions may be added, subtracted, multiplied, and divided, as whole numbers are.

Thus, $ += %, just as 5 days + 4 days = 9 days. 8 + g = *°, just as 7 qts. + 3 qts. = 10 qts.

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