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REDUCTION OF FRACTIONS.

Reduction changes the terms of a fraction without changing its value. The change is to Higher Terms, to Lower Terms, or to Lowest Terms.

Reduction of Fractions to Higher Terms. Is not $i = $3? May we not multiply the terms of 1 by 2 and thus obtain 3 ?

PRINCIPLE.

Multiplying both terms of a fraction by the same number does not change the value of the fraction. (See p. 68.)

Note.—The pupil should perceive that fractions are, by their very nature, subject to the principles of division.

EXERCISES.

1. Change i to twentieths. Process. Analysis.

Explanation. 20 : 4 = 5 1 20

The division shows that the terms of 1 =

the fraction must be multiplied by 5 to 20

change fourths to twentieths. Multiply5

ing both 3 and 4 by 5 we have já.

20
5

15

4

RULE.

Divide the required denominator by the given denominator, and multiply both terms of the fraction by the quotient.

2. Reduce :

1. 14 to 60ths.
2. 13 to 80ths.
3. li to 40ths.

4. Té to 1000ths.
5. z1 to 270ths.
6. 4 to 150ths.

7. 15 to 90ths. 8. 71 to 110ths. 9. ğ to 99ths. 10. to 49ths. 11. I to 60ths. 12. to 24ths. 13. 1 to 70ths.

14. 14 to 74ths. 15. 21 to 42nds. 16. 18 to 38ths. 17. 6 to 30tbs. 18. 41 to 100ths. 19. i to 10ths. 20. 18 to 12ths.

Reduction of Fractions to Lower Terms. Is not $2

$? May we not divide both terms of 4 by 2 and obtain 1 ?

PRINCIPLE.

Dividing both terms of a fraction by the same number does not change the value of a fraction. (See page 68.)

EXERCISES.

1. Reduce 1 to eighths. Process.

Explanation. 16 ; 8 2

The division of 16 by 8 shows that both terms of the

fraction must be divided by 2 to change sixteenths to 1 I 2 g

eighths. Dividing both 12 and 16 by 2, we have .

RULE.

Divide the given denominator by the required denominator, and divide both terms of the fraction by the quotient. 2. Reduce : 1. 34 to 15ths.

6. 328 to 16ths. 2. 16 to 9ths.

7. 379 to 100ths. 3. 4 to 10ths.

8. 144 to 12ths.

1728 4. 128 to 12ths.

9. 438 to 9ths. 5. 21 to 4ths.

276

10. 1776 to 949ths.

1898

49 63

Reduction of Fractions to Lowest Terms. Reduction to lowest terms requires the terms of the fraction to be divided by their greatest common factor (G. C. D.).

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5280

0

Explanation. The G. C. D. of 1760 and 5280 is 1760. Dividing both terms of the fraction by 1760 we obtain }, the lowest terms. (See Principle, p. 95.)

176 0)1760 176 015280

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RULE.

Divide both terms of the given fraction by their G. C. D.

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Continued division by a common factor will secure lower or lowest terms. 4)1760

4) 14:42 = 11/18 1838 = The terms are the lowest when they are prime to each other. 2. Reduce to lowest terms :

(1.) (2.) (3.) (4.) (5.) (6.)
22 276
796

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Reduction of Integers and Mixed Numbers.
We have learned that 1, 3, 4, or , etc., equal one.
How many halves in one whole thing ?
How many thirds? Sixths ? Tenths ?
How many thirds in two? In 2?

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EXERCISES. 1. Reduce 8p to fourths. Process.

Explanation. 8 32

Since 1 4,8= 3; and 8 +=+ 8f= 3 + f= 36

RULE. Multiply the integer by the denominator, to the product add the numerator, and write the sum over the denominator. 2. Reduce : 1. 64 to fourths.

11. 15 to fifths. 2. 24 to thirds.

12. 13ļ to sixths. 3. 121 to halves. 13. 18ť to elevenths. 4. 94 to sevenths. 14. 5 to ninths. 5. 161 to fourths. 15. 54 to eighteenths. 6. 135 to eighths. 16. 272 / to elevenths. 7. 314271 to twenty-firsts. 17. 2784 to ninths. 8. 67312 to twelfths. 18. 946 to thirteenths. 9. 70211 to elevenths. 19. 6153 to fifths.

10. 1221; to fifteenths. 20. 241 A to twenty-firsts. Have your results been proper or improper fractions ? ?

3. Reduce to improper fractions the following:
1. 98

6. 22316 11. 2104 16. 1542
2. 173.
7. 134.
12. 16.25

17. 108.05
3. 2811 8. 5044.

13. 6231

18. 5117 4. 273. 9. 1147 14. 1594. 19. 4019. 5. 494. 10. 31211.

15. 6715 20. 86114.

3

Reduction of Improper Fractions. How many dollars in $? In $38 ? How many units in 16? In 13? In 37? In 48? In 51 ? What kind of numbers are your results ?

4

EXERCISES.

1. Reduce 594 to an integer and 595. a mixed number. Process.

Explanation. 594 = 72 Since 594 indicates the division of 504 by 7, we 595 = 721

divide and obtain the integer 72.

RULE.

Perform the division indicated.

2 5

1. 68 2. 57 3. 721

20

13

50

13. 87

36

2. Reduce the following improper fractions :

10. 49

19. 3.
13•

28. 1956
11. 4.
93
20. 351

29. 1947
31:

17.
12. 117
21. 82,7

30. 22 6 2 5
20

.

2 2 5 4. 2492

22. 9.23 28•

31. $843

5 •
14. $1.
81

23. 384
2

32. 2 4 8 40
37:

1 6 200
15. 655
24. 600

33. 29428

2
28
32.

1 76
7. 157.
16. 543
25. 6.8.20

31. 32 700
32

238 20
8. 5.
65
17. 144

26. 16315. 35. 11 8 0 4 10

36.
9. 15. 18. 112¢. 27. 3182. 36. 1976 8 3 2

15

138 2 40).

.

5. 77

20. 6. 93

25:

1 72

104

184807

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