7. If a cake is cut into six equal parts, what part of the cake is in each piece? In three of the parts?

8. How many sixths are in a cake? How many sixths are in a unit, or in any thing?

9. What is one of five equal parts of a melon called? What are two parts? How many fifths in a unit?

10. Into how many halves can a unit, or any thing, be divided? Into how many thirds? Fourths? Fifths? Sixths? 11. What is meant by one-half of a unit? By one-third? One-fourth? One-fifth? One-sixth?

12. What is meant by two-thirds of a unit? By threefourths? Two-fifths? Four-fifths? Three-sixths? Fivesixths? Four-sevenths? Six-sevenths?

13. What part of a unit is one of two equal parts of the unit? One of three equal parts? Two of four equal parts? Three of five equal parts?

14. Which is the greater, one-half or one-fourth of an apple? One-third, or one-sixth? One-fourth, or one-eighth ? One-fifth, or one-sixth?

15. Which are the smaller parts of any thing, halves or fourths? Thirds, or fourths? Fourths, or fifths? Fifths, or sixths? Sevenths, or eighths?

Definitions.

201. A Fraction is one or more of the equal parts of a unit.

Thus, 1 third, 3 fifths, 5 sixths are fractions.

Fractions are divided into two classes: Common Fractions and Decimal Fractions.

A Common Fraction is expressed in figures by two numbers, written one above the other, with a line between them.

202. The two numbers used to express a fraction are called the Denominator and the Numerator.

203. The Denominator of a fraction is the number written below the line.

The denominator shows into how many equal parts the unit is divided, and denominates, or gives name to, the parts. Thus, 10 is the denominator of the fraction ; it shows that the unit is divided into ten equal parts, which are named tenths.

9

204. The Numerator of a fraction is the number written above the line.

The numerator shows the number of the equal parts of the unit expressed by the fraction.

9

Thus, 9 is the numerator of the fraction; it shows that 9 of ten equal parts of a unit are expressed by the fraction.

205. The Terms of a fraction are its numerator and denominator.

9

Thus, 9 and 10 are the terms of the fraction 10

.

206. A Fractional Unit is one of the equal parts into which the unit is divided.

Thus, is the fractional unit of; of; $of $2.

Note. - The value of a fractional unit depends upon the number of equal parts into which the integral unit is divided.

Thus, if an apple is divided into halves, each part is larger than it would be if the apple were divided into fourths, or into eighths.

207. Principles.

I. The less the number of equal parts into which the unit is divided, the greater is the value of each of the parts.

-II. The greater the number of equal parts into which the unit is divided, the less is the value of each of the parts.

Notation and Numeration of Fractions.

208. A common fraction is written by placing the numerator above the denominator, with a line drawn between them.

6

7

Thus, six sevenths is written seven tenths, 10

209. A common fraction is read by stating the number of parts shown by the numerator, and then giving the name of the parts indicated by the denominator.

8

9

5 12'

123 456'

Thus, is read eight ninths; five twelfths; one hundred and twenty-three four hundred and fifty-sixths.

Exercises.

Express in figures, or write, the fraction six ninths.

210. Model. - The fraction six ninths is expressed in figures by writing 6 as the numerator, and 9 as the denominator, giving, the fraction required.

1. Express in figures the following fractions:

:

Two fifths; three eighths; four sevenths; five eighths; three ninths; six tenths; two elevenths; ten twelfths; nine fifteenths; sixteen twentieths.

Eleven thirtieths; fourteen thirty-fifths; eighteen forty-thirds; twenty fiftieths; twenty-five sixty-thirds; thirty-one seventieths; one hundred one hundred and sixths; two hundred two hundredths; one hundred and five two hundred and fifty-seconds; two hundred and fifty three hundred and sevenths; three hundred and forty-nine four hundred and seventy-firsts.

7

11

211. Model. The fraction is read by stating the number of equal parts shown by the numerator, seven, and then the name of the equal parts shown by the denominator, elevenths, giving seven elevenths, the expression required.

3. In the preceding exercises, what does the denominator of each fraction show and name, and what does the numerator show?

The Value of a Fraction.

1. If an apple is cut into halves, how many halves of an apple are there?

2. Is a half of an apple greater, or less than 1 apple? Why? 3. Is the value of $3 greater, or less than $1? Is / greater, or less than 1? ? ??

4. Are two halves of an apple greater, or less than 1 apple? Why? Three thirds? Why?

5. Which has the greater value, $1 or $1? of a yard, or 1 yard? of a gallon, or 1 gallon? Why?

6. One half of 2 apples is equal to how many halfapples? To how many whole apples?

7. If 3 apples are divided into halves, how many halfapples are there in all?

8. 3 half-apples are equal to how many apples? Why? 3 half-apples are what part of 3 whole apples?

9. What is the value of 5 half-apples? Of of a dollar? of a yard? 8 of a cake? Of 25? Of 13?

10. Compare the value of each of the following fractions with a unit, or 1: 3, 5, 7, 12, 12, 45.

24 15

11. Indicate in the form of a fraction the division of 1 by 2; 6 by 1; 3 by 8; 12 by 4; 15 by 7; 8 by 18; 24 by 12.

12. As an expression of division, what is the value of §?

Of ? Of ? Of 4? Of 15? Of ? Of 18?
12?
Of?

212. From the preceding examples, it appears that I. The value of a fraction is less than 1, when the numerator is less than the denominator.

Thus, the value of is less than 1, because not all the equal parts, or fourths, into which the unit, or 1, is divided, are taken.

II. The value of a fraction is 1, when the numerator and the denominator are alike.

4
4

Thus, the value of is 1, because all the equal parts, or fourths, into which the unit, or 1, is divided, are taken.

III. The value of a fraction is greater than 1, when the numerator is greater than the denominator.

Thus, the value of is greater than 1, because more equal parts, or fourths, are taken than are in a unit, or 1.

IV. The value of a fraction depends upon the number of equal parts into which the unit is divided, and the number of equal parts expressed by the fraction.

Definitions.

213. According to value, fractions are divided into two kinds: Proper Fractions and Improper Fractions.

214. A Proper Fraction is a fraction whose value is less than 1.

The numerator of a proper fraction is always less than its denominator.

215. An Improper Fraction is a fraction whose value is equal to, or greater than 1.