DEFINITIONS.

153. A Fraction is a number which represents one or more of the equal parts into which a unit is divided.

154. The Denominator of a fraction is the number which shows into how many equal parts the unit is divided.

155. The Numerator of a fraction is the number which shows how many equal parts are taken.

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

157. A Mixed Number is a number expressed by an integer and a fraction.

160.-1. If you should cut 1 apple into halves, how many halves would there be?

2. If you should cut 5 apples into halves, how many halves would there be? One half of 5 apples is how many halves of 1 apple?

3. How many apples are one half of 5 apples? 5 halves are what part of 5 apples?

4. Is greater or less than 1? Is greater or less than 1? What is the value of in ones? Of ? What is the value of

5. Is greater or less than 1?

in ones? Of ?

6. Considered as an expression of division, what is the value of? Of ?

7. Considered as an expression of division, what is the value of 2? Of 4?

DEFINITIONS.

161. A Proper Fraction is a fraction whose numerator is less than its denominator.

162. An Improper Fraction is a fraction whose numerator is not less than its denominator.

163. Reduction of Fractions is the process of changing their form or denomination without changing their value.

164. Principle.-The value of a fraction is the quotient arising from the division of the numerator by the denominator.

166. TEST QUESTIONS.-1. What is a fraction? What is the denominator of a fraction? The numerator of a fraction? What are the terms of a fraction? Take some fraction and show what are its terms.

2. What is a mixed number? Take some mixed number and show what expresses the integer and what the fraction.

3. What is a proper fraction? An improper fraction? What is reduction of fractions? What is the value of a fraction?

SECTION XIV.

REDUCTION OF FRACTIONS.

CASE I.

Integers or Mixed Numbers Reduced to Fractions.

167.-1. If 1 apple

be cut into halves, how many halves will there be?

2. In 2 apples, how many halves of 1 apple?

SOLUTION.-Since in 1 apple there are 2 halves, in 2 apples there must be 2 times 2 halves, which are 4 halves.

3. How many halves are there in 3? In 5?

4. In 1 orange, how many thirds of 1 orange? How many fourths of 1 orange?

5. In 1 dollar, how many fifths of 1 dollar? In 6 dollars? In 11 dollars?

6. In 31 apples, how many halves of 1 apple?

SOLUTION.--Since in 1 apple there are 2 halves, in 3 apples there must be 3 times 2 halves, which are 6 halves; 6 halves and 1 half are 7 halves. Hence, in 3 apples there are of 1 apple.

7. If you had 5 dollars to give some poor boys, to how many could you give of 1 dollar each?

8. In 9 dollars, how many halves? In 63, how many thirds? In 7, how many fourths?

9. In 4, how many fifths? In 10%, how many eighths? In 3, how many elevenths?

10. How may a number of ones be changed to halves? To ninths? To twelfths?

WRITTEN EXERCISES.

168.-1. Change 9 to sixths.

SOLUTION. Since in 1 there are 6 sixths, in 9 there must be 9 times 6 sixths, which are 54 sixths

=

- 54.

2. Change 19 to fourths. 25 to tenths.

3. Reduce 65 to sevenths.

4. Change 85 to ninths.

73 to eighths.

5. Reduce 13 to fifths. 19 to elevenths.
6. Reduce 27 to thirds. 57 to thirteenths.

7.

169. Rule for Reduction of Integers or Mixed Numbers to Improper Fractions.-Multiply the integer by the denominator, and, if there be a fractional part, add its numerator to the product. This result, written over the denominator, will be the required fraction.