133. To divide a quotient by an integer.

(1) Divide 1 seventh of 6 by 3.

RULE. Divide the dividend or multiply the divisor by

What will be the cost:

10. Of 1 hat, at the rate of 5 hats for $13? 11. Of 1 gun, at the rate of 3 guns for $78?

12. Of 1 knife, at the rate of 8 knives for $3*?

134. To divide an integer or quotient by a quotient.

135. To invert a quotient is to interchange the places of

RULE.-Invert the divisor and proceed as in multiplica

tion.

15. Divide 6 by ; 8 by ; 12 by .

5 contain ?

22. Does 1 contain?contain 5?

23. Does 1 third of 5 contain 1 fifth of 9?

24. If 4 knives are worth $8, what is one knife worth? 25. If I pay $4 for 4 books, what do I pay for 1 book?

26. If 5 tops are worth $3, what is one top worth? 27. If 7 melons cost $11, what will 1 melon cost?

How many:

28. Tops can I buy for $5, at $1 per top?
29. Books can I buy for $4, at $3 per book?
30. Slates can I buy for $18, at $3 per slate?

31. Pencils can I buy for $33, at $ per pencil?

32. How many hours will it take a boy to travel ≥ of 15 miles, at the rate of 5 miles in 3 hours?

33. A gentleman paid § of $6825 for a farm, at the rate of $25 for 2 acres; how many acres were in the farm?

QUESTIONS.

What is a fractional quotient? The value of a fractional quotient? The terms of a quotient? The principle of quotients? Reduction? How do you reduce a quotient to its lowest terms? Quotients to a common divisor? Add quotients? Subtract? Multiply? Divide?

COMMON FRACTIONS, OR EQUAL PARTS OF UNITS.

136. 1. Can any integral unit, as 1 apple, 1 bar of soap, or 1 dollar, be divided into equal parts ? What is the name of one of the parts when one is divided into two equal parts? Into three equal parts? Into four equal parts?

2. How many halves in one? How many thirds? Fourths? Fifths? Sevenths? Tenths? Twentieths?

3. How is one half expressed by figures? One third? One fifth? One twelfth? One fifteenth ?

4. Does

5. Does

express 1 of 2 equal parts of one?

express 3 of 5 equal parts of one?

6. What does express? ? ? ? 5? 3 ?

137. A fractional unit is one of the equal parts of an integral unit.

(1) of a yard, of a dollar, , are fractional units.

138. A fractional number, or fraction, is a fractional unit, or a collection of fractional units.

(1) of a mile, of a pound, }, are fractions.

139. A common fraction is expressed by two numbers, one above and the other below a horizontal line.

140. The terms of a fraction are the two numbers used to express it.

141. The denominator is the term below the line. It shows the number of parts into which the integral unit is divided, and thus names the parts expressed by the fraction.

142. The numerator is the term above the line. It shows how many parts are taken, and thus numbers the parts expressed by the fraction.

(1) The terms of the fraction are 7 and 10; the denominator is 10, which shows the size of the parts tenths, or the fractional unit 1 tenth; the numerator is 7, which shows the number of parts taken.

143. There are two ways of considering any fraction, as 16.

1o. It may imply 1 tenth of seven. In this sense it is a quotient, 7 being the dividend and 10 the divisor. Fractions were so regarded and treated under the heading of Fractional Quotients.

2o. It may imply 7 tenths of one. It is in this sense that fractions are now to be regarded and treated.

Read each of the following fractions, and name the terms, the numerator, the denominator, the fractional unit, the number of fractional units, and the signification:

144. The value of a fraction is its relation to one.

145. When classified with respect to value, fractions are proper, improper, and mixed.

146. A proper fraction is one whose numerator is less than its denominator; an improper fraction is one whose numerator is equal to, or greater than its denominator; and a mixed number is a number composed of an integer and a fraction.

(3) 1, 43, 94, 133, are mixed numbers.

147. When classified with respect to form, fractions are simple, complex, and compound.

148. A simple fraction is one whose terms are integers; a complex fraction is one which has one or both of its terms fractional; and a compound fraction is a fraction of a fraction.

(1) 2, 4, 7, 15, are simple fractions.

149. An integer may be expressed as a fraction by writing 1 under it as a denominator.

(1) 8 = f, which is read eight ones.

150. The reciprocal of a fraction is the result of inverting it. See Art. 135.

PRINCIPLES OF FRACTIONS

151. 1°. Multiplying the numerator of a fraction by any integer multiplies the value of the fraction by that integer.

Explanation.--Multiplying the numerator by 2, doubles the number of parts taken.

Illustration.

g is 3 fifths of one.

2 x 3 is 6 fifths of one.

5

2°. Dividing the denominator of a fraction by any integer multiplies the value of the fraction by that integer.

3°. Dividing the numerator of a fraction by any integer divides the value of the fraction by that integer.

Explanation. Dividing the numerator by 2, halves the number of parts taken.

Illustration.

gis 6 fifths of one.

6 ÷ 2

is 3 fifths of one.

5

4°. Multiplying the denominator of a fraction by any integer divides the value of the fraction by that integer.

Explanation.—Multiplying the denominator by 2, halves the size of the parts taken.

5

Illustration.

§ is 5 thirds of one.

2 ᄉ3

is 5 sixths of one.