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20. Find the totals in the following list of the naval forces of the United States in 1917 :

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21. Find the totals in the following list showing the extent of a recent Naval Appropriation Bill :

TYPE

NUMBER

Cost

10

6 10 50 9

$ 187,134,310
131,601,186
69,719,000
76,265,700
15,169,707

co

Battleships.
Battle cruisers .
Scout cruisers
Destroyers
Fleet submarines
Coast submarines :

800-ton type

3,918,906 44,346,610

Smaller type Neff submarine Fuel ships Ammunition ships Hospital ship Repair ship Transport Destroyer tenders Submarine tender Gunboat

Totals

3 55 1 3 2 1 1 1 2 1

250,000 4,794,816 4,929,300 2,350,000 2,714,650 2,663,299 4,395,884 1,835,523 1,827,222

CHAPTER III

FRACTIONS AND FACTORS

A Fraction is an expression denoting one or more equal parts of a unit. It is an indicated division, usually written in the form

The number below the line is called the denominator. It shows into how many parts the whole number has been divided.

The number above the line is called the numerator. It shows how many parts have been taken to form the fraction.

The numerator and denominator are called the terms of the fraction.

A proper fraction is one whose numerator is less than its denominator. Thus, i, j, and 4 are proper fractions.

An improper fraction is one whose numerator is equal to, or greater than, its denominator. Thus, 4, {, and 44 are improper fractions.

A mired number is a whole number with a fraction. Thus, 13, 23, and 34 are mixed numbers.

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Mixed Numbers Expressed as Fractions Rule. - To express a mixed number as a fraction, multiply the whole number by the denominator, add the numerator, and write the result over the denominator.

=

EXAMPLE. Express 3 as a fraction.
8 x3 + 5 = 29, the numerator of the fraction required.
The denominator is 8.
Then 3 =

ORAL DRILL
Express as fractions the following mixed numbers :
1. 11

7. 116
13. 45
19. 63

25. 8 2. 1} 8. 176 14.51

26. 91

20. 73 3. 14 9. 116 15. 5

27. 93 21. 81

28. 9 4. 13 10. 113

22. 8}

16. 54 5. 15 11. 45 17. 53

23. 84 29. 107 6. 11 12. 44 18. 55

24. 8} 30. 1016

Exercise Express as fractions the following mixed numbers : 1. 473

7. 743

13. 973 19.734 25. 924 2. 684

8. 593 14. 703 20. 87 26. 94 3. 963 9.764 15. 275 21. 823 27. 693 4. 783 10. 383

16. 49 22. 80 28. 53 5. 563 11. 823 17. 963 23. 966 29. 57 6. 645

12.653 18. 804 24.738 30.667

Improper Fractions Expressed as Mixed Numbers Rule. - To express an improper fraction as

whole or mixed number, divide the numerator by the denominator.

EXAMPLE. Express 22 as a whole or mixed number.

34

1.22

30 18

25. jo

36

30

ORAL DRILL Express as whole or mixed numbers the following fractions :

7. *

13.30 19. 2.30 8. 18

20.

26. 3.26

21. 3
10. 18

16. 12
17.

29.
6. !
12.30

24. 36

14. 3.0

36 14 36

9.

20

15. 30

8

27. ja 28. in

4. 20

20

30

36

8

22. 36

4

5. 20

11. 3

30

23.36

36 18

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Exercise Express as whole or mixed numbers the following fractions : 1. 122

19. 256 32

25.

75 2. 1242 8. 128

20.253

6 56

85 3. 1388

9. 128

15. 259 64

35

21. 360 27.380 4. 128 10. 16.466 22. 3 6 0

28. 3672

15 5. 1,28 11.25

23. 360

32 6. 12 12.25 18. 256

24. 360
35

30. 3670

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The Greatest Common Divisor of two or more numbers is the greatest number that is exactly contained in all of them. It is also called the highest common factor.

Thus, 12 is the greatest common divisor of 24 and 36.

Greatest Common Divisor is commonly abbreviated G. C. D. or H. C. F.

The principal use of greatest common divisor is its application in reducing fractions to their lowest terms.

Prime Numbers. If numbers have no common factor other than 1, they are said to be prime to each other.

The greatest common divisor of two or more numbers can frequently be found by inspection, but when this is impossible, observe the following rule. Rule. - (1) Separate the numbers into prime factors,

. and (2) find the product of the prime factors common to the numbers.

EXAMPLE. Find the G. C. D. of 54, 72, and 108.

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Exercise Find the G.C. D. of the following: 1. 4, 16

11. 10, 35, 85 2. 12, 78

12. 14, 21, 77 3. 15, 75

13. 18, 72, 81 4. 17, 51

14. 24, 60, 96 5. 19, 171

15. 32, 48, 96 6. 21, 105

16. 36, 54, 90 7. 23, 207

17. 39, 52, 91 8. 6, 8, 42

18. 63, 84, 126 9. 8, 28, 84 19. 33, 77, 121 10. 9, 33, 81 20.30, 60, 250

21. 32, 48, 128 22. 51, 63, 132 23. 36, 84, 132 24. 60, 144, 576 25. 38, 190, 760 26. 32, 160, 808 27. 33, 132, 143 28. 50, 125, 250 29. 64, 128, 256 30. 64, 160, 640

Factors

A Factor of a number is a number which is exactly contained in that number. Thus, 2, 3, 4, and 6 are all factors of 12.

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