AN IMPROPER FRACTION is one in which the numerator is equal to, or exceeds the denominator. Such fractions are called improper fractions because they are equal to, or exceed unity. When the numerator is equal to the denominator the value of the fraction is 1; in every other case the value of an improper fraction is greater than 1.

The following are improper fractions:

A SIMPLE FRACTION is a single expression. A simple fraction may be either proper or improper.

The following are simple fractions:

A COMPOUND FRACTION is a fraction of a fraction, or several fractions connected together with the word of between them.

The following are compound fractions:

ofofof }, ↓ of 3, ‡ of of 4.

A MIXED NUMBER is made up of a whole number and a fraction. The whole numbers are sometimes called integers. The following are mixed numbers:

31, 4, 62, 53, 6, 31,

Q. How many kinds of Vulgar Fractions are there? What are they? What is a proper fraction? Is its value greater or less than 1 ? What is an improper fraction? Why is it called improper? When is its value equal to 1? What is a simple fraction? What is a compound fraction? What is a mixed number? Give an example of a proper fraction? Of an improper fraction? Of a simple fraction? Of a compound fraction? Of a mixed fraction? Is four-ninths a proper or improper fraction? What kind of a fraction is six-thirds? What is its value? What kind of a fraction is nine-eighths? What is its value? What kind of a fraction is one-half of a third? What kind of a fraction is two and one-sixth? Four and a seventh? Eight and a tenth ?

§ 78. The numerator and denominator of a fraction, taken together, are called the terms of the fraction. Hence, every fraction has two terms.

Q. What are the terms of a fraction? What are the terms of the fraction three-fourths? Of five-eighths? Of six-sevenths?

§ 79. A whole number may be expressed fractionally by writing 1 below it for a denominator. Thus,

3 may be written and is read, 3 ones.

But 3 ones are equal to 3, 5 ones to 5, 6 ones to 6, and 8 ones to 8. Hence, the value of a number is not changed by placing 1 under it for a denominator.

Q. How may a whole number be expressed fractionally? Does this alter its value? Give an example?

§ 80. If an apple be divided into 6 equal parts,

will express one of the parts,

and generally, the denominator shows into how many equal parts the unit is divided, and the numerator how many of the parts are taken.

Hence, also, we may conclude, that,

PROPOSITION I. If the numerator of a fraction be multiplied by any number, the denominator remaining unchanged, the value of the fraction will be increased as many times as there are units in the multiplier. Hence, to multiply a fraction by a whole number, we simply multiply the numerator by the number.

Q. If an apple be divided in six equal parts how do you express one of those parts? Two of them? Three of them? Four of them? Five of them? Repeat the proposition? How do you multiply a fraction by a whole number?

§ 81. If three apples be each divided into 6 equal parts, there will be 18 parts in all, and these parts will be expressed by the fraction 18. If it were required to express but one-third of the parts, we should take in the numerator but one-third of 18: that is, the fraction would express one-third of 18. If it were required to express one-sixth of the parts, we should take one-sixth of 18, and would be the required fraction.

In each case the fraction 18 has been diminished as many times as there were units in the divisor. Hence,

PROPOSITION II. If the numerator of a fraction be divided by any number, the denominator remaining unchanged, the value of the fraction will be diminished as many times as there are units in the divisor. Hence, a fraction may be divided by a whole number by dividing its numerator.

Q. If 3 apples be each divided into 6 equal parts, how many parts in all? If 4 apples be so divided, how many parts in all? If 5 apples be so divided, how many parts? How many parts in 6 apples? În7? In 8? In 9? In 10? What expresses all the parts of the three apples? What expresses one-half of them? One-third of them? One-sixth of them? One-ninth of them? One-eighteenth of them? What expresses all the parts of four apples? One-half of them? One-third of them? One-fourth of them? One-sixth of them? One-eighth of them? One-twelfth of them? One-twenty-fourth of them? Put similar questions for 5 apples, 6 apples, &c. Repeat the proposition. How may a fraction be divided?

EXAMPLES.

1. Divide 2 by 2, by 7, by 14.

Ans. 14, 15, 15.

4

2. Divide 112 by 56, by 28, by 14, by 7. 3. Divide 100 by 25, by 8, by 16, by 4.

Ans.

Ans.

§ 82. Let us again suppose the apple to be divided into 6 equal parts. If now each part be divided into 2 equal parts, there will be 12 parts of the apple, and consequently each part will be but half as large as before.

3

Three parts in the first case will be expressed by 3, and in the second by 12. But since the parts in the second are only half the parts in the first fraction, it follows that, one half of 3.

If we suppose the apple to be divided into 18 equal parts, three of the parts will be expressed by, and since the parts are but one-third as large as in the first case, we have

= one third of 3:

and since the same may be said of all fractions, we have

PROPOSITION III. If the denominator of a fraction be multiplied by any number, the numerator remaining unchanged, the value of the fraction will be diminished as many times as there are units in the multiplier. Hence, a fraction may be divided by any number, by multiplying the denominator by that number.

Q. If an unit be divided in 6 equal parts and then into 12 equal parts, how does one of the last parts compare with one of the first? Îf the second division be into 18 parts, how do they compare? If into 24 ? What part of 24 is 6? If the second division be into 30 parts, how do they compare? If into 36 parts? Repeat the proposition. How may a fraction be divided by a whole number?

$83. If we suppose the apple to be divided into 3 parts instead of 6, each part will be twice as large as before, and three of the parts will be expressed by 3 instead of. But this is the same as dividing the denominator 6 by 2; and since the same is true of all fractions, we have

PROPOSITION IV. If the denominator of a fraction be divided by any number, the numerator remaining unchanged, the value of the fraction will be increased as many times as. there are units in the divisor. Hence, a fraction may be multiplied by a whole number, by dividing the denominator by that number.

Q. If we divide 1 apple into three parts and another into 6, how much greater will the parts of the first be than those of the second? Are the parts larger as you decrease the denominator? If you divide the denominator by 2, how do you affect the parts? If you divide it by 3? By 4? By 5? By 6? By 7? By 8? Repeat the proposition. How may a fraction be multiplied by a whole number?

EXAMPLES.

1. Multiply by 2, by 4.

2. Multiply 18 by 2, 4, 8, 16, 32.

Ans.,

Ans. 18, 18, 18, V, Y.

16 16 16 1 4929

3. Multiply by 2, 4, 6, 8, 12, 16, 24, 48.

19 Ans. 12, 11, 12, &c., &c. Ans. 10 151, 151, 15.

5. Multiply 15 by 5, 10, 20. $84. It appears from Prop. I. that if the numerator of a fraction be multiplied by any number, the value of the fraction will be increased as many times as there are units in the multiplier. It also appears from Prop. III., that if the denominator of a fraction be multiplied by any number, the value of the fraction will be diminished as many times as there are units in the multiplier.

Therefore, when the numerator and denominator of a fraction are both multiplied by the same number, the increase from multiplying the numerator will be just equal to the decrease from multiplying the denominator; hence we have,

PROPOSITION V. If the numerator and denominator of a fraction be multiplied by the same number, the value of the fraction will remain unchanged.

Q. If the numerator of a fraction be multiplied by a number, how many times is the fraction increased? If the denominator be multiplied by the same number, how many times is the fraction diminished? If then the numerator and denominator be both multiplied at the same time, is the value changed? Why not? Repeat the proposition.