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148. Fractional units are named from the number of parts into which the unit is divided. Thus, & is read one-sixth; 1, one-seventh.

Fractions are read by naming the number and kind of fractional units. Thus, & is read five-sixths; z1, five twenty-firsts; , thirteen thirty-fifths. 149. Read the following: 3245 12

54971 Tlfo

11

13

23 100

35 140 8 29

29 235

35 816 385 986

684 5 100000 18 9 5 3 88296

4 1 5 6 84 30 77606

5 1 684 7 8 9 6 54

3 2 5 639

469 831

5190
1896

Express by figures:

1. Three elevenths. Five thirteenths. Eight twentyfirsts.

2. Forty-eight fiftieths. Twenty-seven eighty-fifths. 3. Sixty forty-eighths. Fifty-seven ninety-ninths. 4. Forty-two eighty-sevenths. Thirty-nine ninety-thirds.

5. Seventy-four one-hundredths. Ninety-seven one-hundred-fifths.

6. Fifty-two seventy-eighths. Thirty-six eighty-fourths. -7. Two hundred three-hundred-ninetieths. 8. Seven hundred seventy-one eight-hundred-sixtieths. 9. Two hundred forty-nine three-hundredths. 10. Five hundred sixty-six seven-hundred-fiftieths. 11. One hundred eleven two-hundredths. 12. Four thousand six hundred thirty five-thousandths.

Fractions are classified with reference to the relation of numerator and denominator thus:

150. A Proper Fraction is one in which the numerator is less than the denominator.

Thus, $, , , etc., are proper fractions.
The value of a proper fraction is therefore less than 1.

151. An Improper Fraction is a fraction in which the numerator equals or exceeds the denominator.

Thus, 4, 3, 14, are improper fractions.
The value of an improper fraction is therefore 1 or more than 1.

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

Thus, 23, 5), are mixed numbers.

Mixed numbers are read by naming the fraction after the whole number. Thus, 2 is read two and three-fourths.

Fractions may be regarded as expressing unexecuted division. Thus, 14 is equal to 16:8; 1 is read 15 = 3.

153. 1. Interpret the expression 5.

ANALYSIS.—5 represents 5 of 7 equal parts into which any thing is
divided. It also represents one-seventh of five, and 5 divided by 7.
It is read five-sevenths.
In like manner interpret:

2. . 5. 38 8. 184 11. 1.
3. 유. 6.31 9. 12 12.98
4. ਨ.
7.
10. 138

13. 368.

45
139

41.

REDUCTION.

CASE I.

154. To reduce fractions to larger, or higher terms.

1. In į of an apple how many fourths are there? How many eighths ? 2. How many sixths are there in }? How many

ninths? How are the terms of the fraction obtained from those of 1? from ?

3. How many eighths are there in 1? How many twelfthis?

4. How do the terms of the fraction s compare with the terms of the fraction į?

5. In what equivalent fraction can be expressed ?

6. How do the terms of the fraction į compare with those of 1?

7. How are the terms of the fraction 1c obtained from those of j?

8. How are the terms of the fraction { obtained from 1? 9. How are the terms of the fraction obtained from į? 10. What then may be done to the terms of a fraction without changing the value of the fraction ?

11. Change { to 24ths. 15. Change 1 to 36ths. 12. Change i to 16ths. 16. Change to 20ths. 13. Change to 24ths. 17. Change 4 to 14ths. 14. Change to 12ths. 18. Change to 18ths.

1

155. Reduction of Fractions is the process of changing their form without changing their value.

156. A fraction is expressed in Larger or Higher Terms when its numerator and denominator are expressed by larger numbers.

157. PRINCIPLE. - Multiplying both terms of a fraction by the same number, does not change the value of the fraction.

WRITTEN EXERCISES.

PROCESS.

1. Change to forty-fifths.

ANALYSIS.—Since there are 45 forty-fifths in 45:- 15 -- 3 1, in Is there are 3 forty-fifths; and in 15 there

are 7 times A's, or 4}; or, 7 X3=21

Since the denominator of the required frac15 X 3=45 tion is 3 times that of the given fraction, we

must multiply the terms of the fraction by 3.

RULE.— Multiply the terms of the fraction by such a number as will change the given denominator to the required denominator.

Reduce:

Reduce:

2. 13 to 50ths. 3. 17 to 60ths. 4. to 70ths. 5. 41 to 84ths. 6. 26 to 40ths. 7. 24 to 54ths. 8. 3 to 66ths. 9. 31 to 54ths. 10. 15 to 84ths.

11. áb to 120ths.
12. 31 to 64ths.
13. to 74ths.
14. 46 to 210ths.
15. fi to 240ths.
16. 4 to 225ths.
17. 54 to 348ths.
18. 3 to 558ths.
19. j 4 to 235ths.

CASE II.

158. To reduce fractions to smaller, or lower terms. 1. How many fourths are there in ? How many in ? 2. How many thirds are there in ? How many in ?

3. How does the number of eighths of any thing compare with the fourths? Thirds with sixths? Halves with eights?

4. How do the terms of the fraction compare with those of j? How with those of 16?

5. How do the terms of the fraction { compare with those of į? How with those of sz?

6. How are the terms of the fraction į obtained from those of the fraction ? How from those of 4.?

7. How are the terms of the faction { obtained from 3?

8. What then may be done to the terms of a fraction without changing the value of the fraction?

9. Express 1, 2, u, in smaller or lower terms.
10. Express , 38, 19, in smaller or lower terms.

to smaller or lower terms.
to smaller or lower terms.

18

11. Reduce 45, 54,
12. Reduce $0, 1920, 1803

24
482
4 5

60

159. A fraction is expressed in Smaller, or Lower Terms when its numerator and denominator are expressed in smaller numbers.

160. A fraction is expressed in the smallest, or Lowest Terms when its numerator and denominator have no common divisor.

161. PRINCIPLE.Dividing both terms of a fraction by the same number does not change the value of the fraction.

WRITTEN EXERCISES.

PROCESS.

162. 1. Change to an equivalent fraction expressed in its smallest, or lowest terms.

ANALYSIS. —Since the denominator of 4/32 418 2 the required fraction is to be smaller than 4 48 412 3

that of the given fraction, we may obtain

an equivalent fraction having smaller Or,

terms, by dividing the terms of the given 32

fraction by any exact divisor, as 4 (Prin.), 32 : 16 2

and the terms of the resulting fraction by 4. 48 48 : 16 3

We thus obtain the fraction }, whose terms

have no common divisor. The fraction is therefore in its smallest terms. Or,

Since fractions are in their smallest terms when their numerator and denominator have no common divisor, to reduce them to their smallest terms we may divide both terms by their greatest common divisor.

RULE.—Divide the numerator and denominator by any common divisor, and continue to divide thus until the terms have no common divisor, Or, Divide both terms by their greatest common divisor.

2. Reduce 18, 48, 4, $4, to their smallest terms. 3. Reduce tš; , , 134, to their smallest terms.

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