The lower number is called the denominator of the fraction, and it shows into how many equal parts the unit 1 is divided.

The upper number is called the numerator of the fraction, and it shows how many of the equal parts are expressed in the fraction.

In the fraction, 1 is the numerator and 2 the denominator.

In the fraction, 4 is the numerator and 3 the denominator.

In the fraction, 3 is the numerator and 4 the denominator.

Q. How is a fraction expressed? What does the lower number show? The upper? How is one-half expressed? Why is the lower number 2, and the upper number 1? How is the fraction two halves expressed? Three halves? Two thirds? What are the two numbers called which compose a fraction? Which is the numerator? What does the numerator show? Which is the denominator? What does it show? Which is the numerator and which the denominator in the fraction In ? Into how many equal parts is the unit divided in 2? In 3? How many parts are taken in ? How is the unit divided in the fraction, and how many parts expressed? In 16, 13, 4, 3, 10, 10? &c.

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61. A whole number may be expressed like a fraction, by writing 1 below it as its denominator. Thus 3 expressed fractionally, is . Which is read 3 ones. Since 3 ones make 3, the value of the number is not altered by writing 1 under it as a denominator.

Q. How may a whole number be expressed fractionally? How may 3 be expressed as a fraction? How do you read? Is the value of the number changed? Why not? Write 4, 5, 6, &c., in a fractional man

ner.

62. A fraction denotes division, and its value is equal to the quotient obtained by dividing the numerator by the denominator. Thus, is equal to 1; and since two halves make 1, the value of the fraction is represented by the quotient 1. In like manner, is equal to 2, 12 to 4, 16 to 4, 20 to 4, &c.

From this we conclude, that if the numerator be less than the denominator, the value of the fraction is less than 1. If the numerator be equal to the denominator, the value of the fraction is 1; and if greater than the denominator, the value of the fraction is greater than 1.

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Q. What does a fraction denote? What is its value equal to? What is the value of the fraction??0? ?? 16 ? When the numeraator is equal to the denominator, what is the value of the fraction? When greater? When less?

63. When the unit is divided into tenths, hundredths, thousandths, &c., the resulting fractions are called Decimal Fractions. All other fractions are called Vulgar or Common Fractions.

10, 120, 180, 180, 180, 1000, 1000, 100, &c., are decimal fractions.

f, 1, 4, f, f, &c. &c., are vulgar fractions.

The denominator of a decimal fraction is not usually expressed, and to distinguish the numerator, which is alone written, from a whole number, a point. is placed on its left, called the decimal point.

is written .2, is written .5, 25 is written .25. When the denominator of a decimal fraction is expressed, it is always 1, with as many cyphers annexed as there are figures in the numerator.

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Q. What are decimal fractions? What are other fractions called? What kind of a fraction is???? Are the denominators of decimal fractions usually expressed? How is the fraction written? When the denominator is expressed, what is it? What is .2? .5? .25? .05 .225? .2004? .100?

OF VULGAR FRACTIONS.

64. There are five kinds of Vulgar Fractions, viz: Proper, Improper, Simple, Compound and Mixed.

A Proper Fraction is one in which the numerator is less than the denominator; thus,,,,, &c., are proper fractions. A proper fraction is always less than 1.

An Improper Fraction is one in which the numerator is equal to or greater than the denominator; thus,,,,, are improper fractions.

A Simple Fraction is a single fraction, in which there is but one numerator and one denominator. A simple fraction may be either proper or improper. 1, 1, 3, 1, 3, &c., are simple fractions.

A Compound Fraction is a fraction of a fraction. Thus the following are compound fractions:

of ; of; 1 of 5 of 2.

A Mixed Number is composed of a whole number and a fraction. The following are mixed numbers:

21, 33, 48, 71, 91, &c.

which are read two and one-fourth, three and two-thirds, &c. Q. How many kinds of vulgar fractions? What are they? What is a proper fraction? Is its value greater or less than 1? Give an example of a proper fraction. What is an improper fraction? What is its value compared with 1? Give an example of an improper fraction. What is a simple fraction? What is a compound fraction? example of a simple fraction. Of a compound fraction. mixed number? Give an example of a mixed number? What kind of a fraction is?? of? of 2? } of ‡ of 4? 21? How is 21 read? 3? 50

Give an

What is a

65. If the numerator of a fraction be multiplied by a number, the denominator remaining the same, the value of the fraction will be increased as many times as there are units in the number.

For since the numerator of a fraction expresses the number of the parts taken in the fraction (Art. 60), if it be multiplied by a number, the number of parts taken will be increased as many times as there are units in the number, and the new fraction will be in like manner increased.

Thus, in the fraction, the unit is divided into halves, and 3 halves are taken; but if the numerator 3 be multiplied by 2, the fraction becomes §, which is 2 times greater than it was before, since 6 halves are taken instead of 3; if the numerator be multiplied by 3, the fraction becomes 2, which is 3 times greater than before, &c.

Q. What effect will there be in multiplying the numerator of a fraction by a number, the denominator remaining the same? How do you explain this? What does the numerator of a fraction express? What effect will be produced by multiplying it by a number? How is the unit divided in 2? How many halves are taken? If you multiply 3 by 2, how many halves will be taken? Is greater than? How many times greater? How many times is greater than? How was obtained from 3?

66. If the numerator of a fraction be divided by a number, the denominator remaining the same, the value of the fraction will be diminished as many times as there are units in the number.

For dividing the numerator is equivalent to diminishing the number of parts taken in the fraction, which will be as many times smaller as there are units in the number.

Thus, in the fraction, 9 halves are taken, but if the numerator 9 be divided by 3, the fraction becomes 3, in which but 3 halves are taken. is therefore 3 times smaller than 2.

Q. If the numerator of a fraction be divided by a number, the denominator remaining the same, will the value of the fraction be changed, increased or diminished? Why diminished? If the numerator of the fraction be divided by 3, what will the resulting fraction be? Is 3 greater or smaller than ? How many times smaller? Why?

67. If the denominator of a fraction be multiplied by a number, the numerator remaining the same, the value of the fraction will be diminished as many times as there are units in the number.

For the denominator expresses into what parts the unit is divided (Art. 60), and multiplying by a number will diminish the magnitude of these parts as many times as there are units in the number, and of course so many more of them will be required to make up the given unit.

Thus, in the fraction, the unit is divided into halves, and multiplying the denominator by 2, the fraction becomes 1, in which the unit is divided into fourths. But there are 2 halves in 1, and 4 fourths in 1; hence the fraction is 2 times greater than 1.

Q. If the denominator of a fraction be multiplied by a number, the numerator remaining the same, will the fraction be increased or diminished? Why diminished? What does the denominator express? If the denominator be diminished, will the unit contain more or less of the parts expressed? How is the unit divided in? In? How many halves in 1? How many fourths? Is greater or less than 7? How many times less? Why?

68. If the denominator of a fraction be divided by a number, the numerator remaining the same, the value of the fraction will be increased as many times as there are units in the number.

For by dividing the denominator by a number, we increase the magnitude of the parts into which the unit is divided, and the number of the new parts required to make up the unit will be less in proportion as the parts have been increased.

Thus, in the fraction the unit is divided in sixths, and 6 sixths are required to make 1; but if the denominator 6 be divided by 3, the fraction becomes, in which the unit is divided into halves; and since but 2 halves make 1, the fraction has been increased 3 times.

Q. If the denominator of a fraction be divided by a number, the numerator remaining the same, will the fraction be increased or diminished? Why increased? How is the unit divided in? In? How many sixths in 1? How many halves in 1? Is greater or less than ? Why less?

69. The value of a fraction is not changed by multiplying its numerator and denominator by the same number.

For when we multiply the numerator by this number, the value of the fraction is increased as many times as there are units in the number multiplied by (Art. 65), and when we multiply the denominator by the same number, the value of the fraction is in the same proportion diminished (Art. 67). Thus,,,,, &c., are equivalent fractions, since they are obtained by multiplying the terms of the fraction by 2, 3 and 4.

Q. Is the value of a fraction changed by multiplying its terms by the same number? Why not? Is greater or less than?

70. The value of a fraction is not changed by dividing its numerator and denominator by the same number.

For, when we divide the numerator by a number, the value of the fraction is diminished as many times as there are units in the number (Art. 66); and when we divide the denominator by the same number, the value of the frac tion is in like manner increased (Art. 68).

Thus 18,,, are equivalent fractions.

Q. Is the value of a fraction changed by dividing its terms by a number? Why not?