Divide anything, as the line A B, into two equal parts;
each part is one half of the line A B.

Divide the line A. B. into three equal

parts; each part is one third of the line A B.

231. Equal parts of anything are fractions of that thing. As halves of a dollar, twelfths of a foot, etc.

232. One half is expressed in figures thus:

or 1⁄2; one third

thus, or 3; two thirds, thus, or 23.

233. The figure written under the short line, as in or 3⁄4,
shows into how many equal parts the thing, or unit,

has been divided. A figure thus written is the denom-
inator of the fraction.

234. The figure written above the line, as in or 2/3, shows how many of the fractional parts, such as halves, thirds, etc., compose the fraction. A figure thus writ ten is the numerator of the fraction. Thus, means two of the three equal parts of the same unit.

REDUCTION OF FRACTIONS.

CASE I.

235. To reduce an integer or a mixed number to an improper

16

G to 10 To to to to to to to to to to to to to to H

1

1. If the line A B be divided into two equal parts, what is each part called? If a line is divided into four equal parts, what is each part called? If into eight? If into sixteen ?

2. If a unit be divided into three equal parts, what is
each part called? What are two of the parts called?
How many
fourths in the line CD? How many eighths in the
line EF? How many sixteenths in the line GH?
4. How many halves in any unit? How many thirds?
How many 5ths? How many 6ths ?
How many

3. How many halves in the line A B?

7ths ?

5. How many halves in one apple? How many halves in two apples? In three? In four? In five ?

6. How many thirds in two units? How many fourths, in two units? How many eighths in two units? In three units ?

Such fractions as,,, etc., are called improper fractions.

7. How may you find the number of eighths in any number? This process is called reducing an integer to an improper fraction.

8. State a general rule for reducing an integer to an improper fraction.

9. Reduce 4 to 4ths; to 5ths; to 6ths; to 7ths.

10. Change 8 to 3rds; 9 to 8ths; 12 to 10ths.

11. How many 5ths in 2? In 2? In 23? In 2? Numbers composed of an integer and a fraction are called mixed numbers.

12. State a general rule for changing a mixed number to an improper fraction.

13. Reduce 83 to 3rds; 73 to 4ths; 5% to 9ths.

14. Reduce 6 to 12ths; 9 to 10ths; 12 to 7ths.

15. Change 3 to 5ths; 5% to 6ths; 4 to 8ths.

Reduce to improper fractions:

236. To reduce an improper fraction to an integer or to a

mixed number.

1. How many 5ths in 1 ?
2. How many units in 10?
3. How many units in ?

How many units in ?

In 15? In 20? In 25.
In ? In? In ? ?

4. State a general rule for changing an improper fraction to an integer or to a mixed number.

5. When is the result an integer? When is the result a

237. To change a fraction to higher or lower terms.

1. How many 4ths in 1 ?

2. How many 8ths in 1 ?

In ?

In ? In ?

3. How many 16ths in 1 ? In? In? In ?

4. Which has the greater value, or ?

5. Is there any difference in the value of and? Is there a difference in their form?

6. How may be changed to ? How may be changed to?

is said to be in lower terms than, because the denom-
inator is a smaller number. fis in higher terms
than . Why?

A fraction may be considered as an indicated division.
Thus, means 25 divided by 5, and may be so read.
So, may be read, 1 divided by 5.

7. In the fraction 25, which term is the dividend? Which
term is the divisor? Divide both dividend and divisor

by 5; what is the quotient? Does 25 or 5? What general principle is here illustrated?

10. Change to lower terms. Reduce to lower terms. 11. State a general rule for reducing a fraction to lower terms.

8

13. Reduce to 5ths; 4 to 8ths; to 3rds.

14. How do you know what divisor to use in order to change 35ths to 5ths ?

15. State a general rule for examples such as example 13. 16. Reduce to 12ths; to 9ths;

72

25

to 15ths. to 9ths; to 24ths; to 5ths. 18. Change to 10ths; to 18ths; to 24ths; to 50ths. This

17. Change

is reducing a fraction to higher terms.
general principles does it depend?

Upon what

19. State a general rule for reducing fractions to higher

terms.

20. Reduce to 15ths; to 20ths; to 40ths; to 60ths.

21. Reduce to 27ths; to 56ths;

22. Reduce to 10ths.

First reduce to 5ths.

to 48ths.

23. Reduce 2 to 15ths; to 20ths; to 16ths.

CASE I V.

238. To reduce a fraction to lowest terms.

A fraction is in lowest terms when its numerator and de

nominator are prime to each other.

1. Reduce to lowest terms.

SOLUTION. Divide each term by 3:

12÷3 4.

27÷3 9

SUGGESTION. Divide each term by 12, the greatest common factor of

of the two terms.