(the art of calculation)

BY HENRY STICKER

DOVER PUBLICATIONS, INC.

This Dover edition, first published in 1955, is an unabridged republication, with minor corrections, of the work originally published by Essential Books in 1945 under the title The Art of Calculation. It is reprinted through special arrangement with Duell, Sloan and Pearce, Inc.

Copyright

Copyright © 1945 by Essential Books.

Copyright © 1955 by Dover Publications, Inc.

All rights reserved.

International Standard Book Number

ISBN-13: 978-0-486-20295-2

ISBN-10: 0-486-20295-X

Manufactured in the United States by Courier Corporation

20295X35

www.doverpublications.com

PREFACE

Arithmetic is a science, but calculation is an art. Science is knowledge—art is skill. You have all the knowledge you could possibly need to determine that 57 times 25 equals 1425, but if you are asked to multiply 57 by 25 and cannot do this mentally in just about one second, you are not adept at the art of calculation.

Genuine skill in the calculating art can be acquired by any person of ordinary intelligence, no matter what his schooling may have been. To develop such skill is the purpose of this book. Special forms of short, graded exercises, performed for the most part mentally, lead the student by easy steps to a point where he will possess really exceptional calculating ability.

For instance, if you will look at Exercise No. 371 on page 191, you will find that you are expected to perform mentally such multiplications as 696 times 858, 858 times 878, etc. These are not “trick” examples—the student who systematically performs the practice examples presented in this book will be able to do many kinds of examples of this degree of difficulty by his sheer ability to hold and manipulate figures in his head.

How is this skill developed? Essentially by developing number sense. Number sense consists in the ability to recognize the relations that exist between numbers considered as whole quantities, and to work with the thought of their broad relations always uppermost. Number sense is possessed by many people in all walks of life—particularly by accountants, bookkeepers, estimators, cashiers, storekeepers and the like. On the other hand, it is absent in many who have an excellent understanding of advanced mathematics. The engineering professions are full of those who require slide rules to perform calculations which the average billing clerk would do mentally.

To give an example of what is meant by number sense, suppose you were asked to multiply mentally 11625 by 12. If you felt at all competent to try to do so, you would probably (unless you are the exceptional case) proceed like this: 12 times 5 is 60, remember 0 and carry 6; 12 times 2 is 24, put 0 before the other 0 and carry 3, etc. In this way you would eventually arrive at the correct answer—if you did not get all mixed up in the meantime; but at best you would take a long time, because number sense would have played no part whatever in your awkward method of approaching this very simple little problem.

Suppose now that we introduce a little of this number sense—suppose that instead of dealing with plain figures, you were told to imagine that you had sold twelve machines on each of which you made a commission of $11.62. As soon as money enters into the matter you immediately see the whole picture in a different light. If you were asked approximately how much your commissions amounted to, you would figure quick as a flash that 11 times 12 is 132, and you would probably answer instantly that you had made something over $132. If you were then asked how much over $132, you would either figure that 62 are of one dollar, or else that this amount is equal to half a dollar plus of a dollar. You would not take long in determining that the excess over $132 comes to $7 and that therefore the total amount received would be$139 or $139.50.

Why not apply to numbers “in the raw” the same methods that you use when dealing with small amounts of dollars and cents? It is no more difficult to multiply 11 thousands by 12 than 11 dollars. If 11 dollars times 12 is 139 dollars, then 11 thousands times 12 is 139 thousands, or 139,500.

From this illustration you may correctly infer that the person with number sense works very largely from left to right instead of from right to left. Left-to-right calculation is of the essence of number sense. Countless practical people know this, yet the art of left-to-right calculation is never taught in the schools, and is, in fact, rarely mentioned in books of any kind.

Step-by-step instruction and practice in this neglected art of left-to-right calculation constitutes the greater part of the substance of this book. Methods of this kind are applied not only to multiplication but to all the fundamental operations. By means of such methods, for instance, you learn to add two columns of figures at a time, and you even get a little practice in three-column addition. You are also taught comparable methods of subtraction and division.

In addition to the exercises having to do with left-to-right calculation, there are many that are based on an extension of the multiplication table. You are taught by easy stages to use all the numbers up to 25 as direct multipliers—that is to say, you acquire a complete knowledge of the multiplication table up to 25 times 25.

The subject of fractions is treated with special reference to the addition and subtraction of the fractions that are most commonly met with in everyday work. The object here is to enable the student to memorize the answers to the kinds of problems that are ordinarily figured out over and over again.

The exercises dealing with decimals are designed to give the student a large workable fund of knowledge of the decimal equivalents of fractions. Memory work includes twelfths and sixteenths, and there is practice in the rapid calculation of thirty-seconds and twenty-fourths.

The final broad subject developed in this book is “short cuts.” These are of the highest value in developing a general understanding of numbers.

The subject matter of this book is limited to the four fundamental operations, with the inclusion of fractions and decimals. No attempt is made to consider the various fields of arithmetical application. Skill in calculation pure and simple is the only goal.

The exercises, nearly four hundred in number, are for the most part very short. Few should take more than ten minutes to do, and many will take less. As progress is by graded steps, the instruction is in small “doses.” The book, accordingly, can be used with profit whenever you happen to have a few free minutes. Its pocket size, moreover, makes it all the more suitable for odd-moment study.

Taken as a whole, this book will prove valuable to anybody engaged in work or study that requires any considerable amount of arithmetical calculation. It is especially recommended to heads of departments in industrial and commercial organizations, for general distribution to the members of their staffs.

CONTENTS

The Plan of This Book

ADDITION

Addition in General

Adding Single Columns by Pairs, starts on

Adding Single Columns by Trios, starts on

Mental Addition of Large Numbers, starts on

Two-Column Addition, starts on

SUBTRACTION

Subtraction in General

Left-to-Right Subtraction, starts on

MULTIPLICATION

Multiplication in General

Factoring, starts on

Direct Multiplication by Numbers Greater than 12, starts on

Multiplying Three Figures by One, starts on

Multiplying Two Figures by Two, starts on

Multiplying Three Figures by Two, starts on

Multiplying Three Figures by Three, starts on

DIVISION

Division in General

Direct Division by Numbers Greater than 12, starts on

Mental Division of Large Numbers, starts on

Division by Three Figures, starts on

Division by Two Figures, starts on

FRACTIONS

Fractions in General

Addition and Subtraction of Fractions, starts on

DECIMALS

Decimals in General

Decimal Equivalents of Fractions, starts on

SHORT CUTS

Horizontal Addition

Combined Addition and Subtraction

Multiplying by a Near Number

Aliquot Parts in Multiplication

Simplifying the Multiplier

Multiplication by Factoring

Factors between 11 and 19

Multiplying by 11

Multiplying by 21, 31, 41, etc

Squares of Numbers, starts on

Multiplying When Corresponding Orders Are Alike, starts on

Multiplying a Sum by a Difference

Multiplications Involving Fractions, starts on

Aliquot Parts in Division

Cubes of Numbers

Algebraic Multiplication

Table of Prime and Composite Numbers

ANSWERS

THE PLAN OF THIS BOOK

The subject matter here presented might have been divided into sections on addition, subtraction, multiplication, etc., in the manner usual to text-books on arithmetic. Because, however, of the special purpose of this book, no such division is made. The general plan is to have several branches proceed simultaneously. Progress is not from subject to subject but from less to more difficult calculation.

For each of the fundamental divisions of arithmetic there is a general introduction—for instance, Addition in General on page 3. In these introductions the special objects sought are described, as well as the methods by which these objects are attained. The student, therefore, always has a clear view of the ultimate aims of his studies and knows how the work immediately in hand fits into the general plan.

Wherever anything new is introduced, it is clearly explained and illustrated. Usually the exercises that go with each explanation are spread through many succeeding pages. In a large number of cases the exercise calls for work with the numbers in a certain list or table (for instance, Table I on page 7). The same lists of numbers are used for various kinds of calculation. This method of presentation makes possible the remarkably great number (about 15,000) of practice examples that are included.

ADDITION IN GENERAL

Two main objects are sought. The first is to add by single columns, grouping three successive numbers at a time; the second is to add two columns at a time:

Take the following sum:

26

43

84

72

96

27

42

35

68

64

37

97

By the first method, starting at the top of the units’ column, we would add these numbers thus: (sum of the first three figures) 13 (+ sum of the next three figures, 15) 28 (+ 15) 43 (+ 18) 61; write 1 and carry 6; (6 + 14) 20 (+ 18) 38 (+ 13) 51 (+ 18) 69; total, 691.

By the second method, starting at the top, we would add both columns simultaneously thus: (26 + 43) 69 (+ 84) 153 (+ 72) 225 (+ 96) 321 (+ 27) 348 (+ 42) 390 (+ 35) 425 (+ 68) 493 (+ 64) 557 (+ 37) 594 (+ 97) 691.

In actual practice, very rapid addition is possible by either method, and you will be left free to choose whichever you prefer. You should, however, learn both.

How do you proceed to learn these methods? You were taught—or should have been taught— at school that speed in addition is acquired by combining pairs of successive numbers that add up to 10. It is at this point that we start, because this is the simplest way in which grouped numbers can be added to a preceding sum. You are given short columns of numbers to be added by incidentally selecting such pairs of successive figures as make 10. In succeeding exercises the columns are lengthened, and you are also asked to group any pairs that add up to less than 10.

In the meantime, you will have been doing exercises in mentally adding all the numbers from 11 to 18 to all the numbers from 1 to 99. Since no pair of figures in a column can add to more than 18, this amount of practice will enable you to add any pair of successive figures in a column to a previous sum, and hence to add the entire column by taking two figures at a time.

You are similarly taught to add trios of numbers that make 10 or less than 10, and to add any number from 19 to 27 to any number from 1 to 99. With this practice you will be able to add any column by taking three figures at a time.

If you can quickly add any number from 1 to 27 to another number, you will not find it difficult to add numbers greater than 27 in the same manner. You are accordingly ready now to add two columns at a time. Exercises in this method are introduced, and these are gradually increased in difficulty.

Toward the end of the book there are some exercises in three-column addition—just enough to demonstrate that it will be possible for you to add this way if you wish to use this method.

There are examples in addition of still another kind. These are not included for practice in addition as such but have a special bearing on the art of multiplying mentally. We need not consider sums of this kind at this point.

You will note that in the exercises in one-column addition you are alternately instructed to add from the top down and to add from the bottom up. In practical work it is of course immaterial in which direction addition is performed. You should, however, be able to add with equal facility in either direction, and by alternating as suggested you will get the necessary practice.

Exercise No. 1

Pairs Adding to 10

Add the following columns by grouping pairs of numbers that make 10. Add from the top down.

Thus you would add the first column by saying to yourself : 7, 17, 22, 32.

Do not consciously repeat in your mind anything but the successive totals. That is to say, do not add this column thus: 7 + 10, 17, +5, 22, +10, 32.

For another illustration of the correct method, take the second example. This is added thus: 8, 18, 20, 30.

Write your answers in succession on a piece of paper and compare them with the correct answers on page 154. (A good plan is to place the edge of your paper immediately under the examples, write the answers along this edge, and fold it under as it becomes used up.)

1.

7

6

4

5

1

9

2.

8

9

1

2

3

7

3.

4

5

5

5

4

6

4.

5

2

8

4

1

9

5.

6

4

6

3

2

8

6.

5

5

3

6

4

8

7.

5

4

6

6

3

7

8.

3

2

7

3

1

2

9.

8

2

9

8

1

9

10.

6

9

1

5

4

6

11.

5

5

3

2

4

6

12.

9

6

4

8

1

7

13.

3

7

6

2

8

8

14.

1

9

9

1

5

4

15.

6

4

4

5

4

3

16.

6

3

7

2

2

5

17.

1

3

7

9

3

7

18.

7

6

2

8

5

5

19.

1

9

4

3

9

1

20.

1

5

5

9

4

6

21.

6

4

7

6

3

7

22.

3

4

6

4

6

3

23.

7

5

5

3

6

2

24.

4

9

1

3

2

8

Table I

Numbers from 1 to 99

Exercise No. 2

Mental Addition

Add 11 to each of the numbers in Table I above.

Use left-to-right addition, which is performed by first adding the tens of one number to the whole of another. In other words, starting with the number in the table you first add 10 and then 1. A few illustrations will be in order:

15 + 11: say 15, 25, 26;

22 + 11: say 22, 32, 33;

29 + 11: say 29, 39, 40;

99 + 11: say 99, 109, 110.

Work down the columns—not across the page. Write down your answers and compare them with those on page 154.

Exercise No. 3

Pairs Adding to 10

Group all pairs of successive numbers that make 10. Add from the bottom up.

1.

7

8

4

6

5

3

5

5

1

8

2

5

2.

6

4

5

2

4

5

4

1

2

8

7

3

3.

5

2

5

4

6

6

7

3

4

8

2

4

4.

9

7

6

4

8

8

9

1

1

7

5

5

5.

6

7

9

1

3

4

6

3

8

9

2

8

6.

3

1

6

4

4

1

8

2

9

6

4

7

7.

4

7

3

8

3

2

2

8

1

9

1

9

8.

8

2

9

1

5

3

8

5

5

2

6

5

9.

4

4

3

2

4

6

1

6

4

9

3

7

10.

6

5

7

3

4

2

8

9

1

3

2

1

11.

9

8

8

2

7

1

9

6

5

d

9

4

12.

3

7

6

6

1

2

7

6

4

5

5

6

13.

7

4

6

3

2

6

4

1

8

3

7

9

14.

3

7

8

2

8

5

5

8

2

7

1

9

15.

9

1

6

3

7

5

4

6

4

3

2

9

16.

1

8

7

5

5

6

7

3

5

4

4

6

17.

3

6

4

2

8

5

1

4

1

9

3

7

18.

6

9

1

7

7

3

2

1

5

2

9

1

Exercise No. 4

Mental Addition

Add 12 to the numbers in Table I on page 7.

To illustrate:

15 + 12:say 15, 25, 27;

22 + 12: say 22, 32, 34;

29 + 12:say 29, 39, 41;

99 + 12:say 99, 109, 111.

Exercise No. 5

Mental Addition

Add 13 to the numbers in Table I on page 7.

Exercise No. 6

Mental Addition

Add 14 to the numbers in Table I on page 7.

Exercise No. 7

Mental Addition

Add 15 to the numbers in Table I on page 7.

Exercise No. 8

Pairs Adding to 10 or Less

The grouping of pairs of successive numbers is now to be extended to include any that add to less than 10 as well as any that add to 10. That is to say, as you add each column watch to see whether any two successive numbers add to either 10 or less than 10, and if they do, make one addition of them to the preceding sum.

For this exercise use the columns of numbers in Exercise No. 1 and compare your answers with those for Exercise No. 1. Add from the top down.

To illustrate, the first column is added: 7, 17, 23, 32; the second: 8, 18, 23, 30; the third: 9, 19, 29.

Exercise No. 9

Mental Addition

Add 16 to each of the numbers in Table I on page 7.

Exercise No. 10

Mental Addition

Add 17 to each of the numbers in Table I on page 7.

Exercise No. 11

Pairs Adding to 10 or Less

Add the columns in Exercise No. 3 by grouping all pairs of successive numbers that add to 10 or less than 10. Add from the bottom up.

Exercise No. 12

Mental Addition

Add 18 to each of the numbers in Table I on page 7.

Exercise No. 13

Adding Single Columns by Pairs

Add the following by single columns, taking pairs of successive numbers at a time. Add from the top down. The first example would be added: 5, 14, 25, write 5 and carry 2; 2, 12, 27, 36; answer 365.

1.

43

62

78

81

14

87

2.

29

75

36

69

43

16

3.

58

33

65

98

72

45

4.

87

62

94

27

89

74

5.

16

91

33

56

29

32

6.

19

99

36

71

61

41

7.

48

21

68

29

18

25

8.

77

29

49

11

51

53

9.

36

49

94

59

22

27

10.

63

78

96

44

41

88

11.

33

39

43

51

55

36

12.

21

79

74

85

63

82

13.

34

43

27

53

17

57

14.

24

14

11

15

75

78

15.

16

44

49

54

49

99

16.

31

35

67

44

84

42

17.

28

63

21

34

52

56

18.

63

35

12

31

81

15

19.

32

65

16

67

73

55

20.

63

28

76

45

69

62

21.

85

56

75

37

73

24

22.

54

42

68

13

99

84

23.

14

27

54

85

59

69

24.

68

42

28

34

83

16

25.

69

28

45

37

71

91

Exercise No. 14

Mental Addition

Add 19 to each of the numbers in Table I on page 7.

Exercise No. 15

Adding Single Columns by Pairs

Add the following by single columns, taking pairs of successive numbers at a time. Add from the bottom up. The first example would be added: 11, 15, 27, 42, 49, 60, write 0 and carry 6; 6, 17, 24, 37, 43, 54, 62; answer, 620.

1.

27

64

32

85

46

29

78

64

31

43

75

46

2.

81

28

75

43

96

57

51

89

75

42

54

86

3.

92

92

29

86

54

18

68

62

11

86

53

65

4.

16

14

14

31

97

65

29

79

73

22

58

64

5.

29

27

25

25

32

19

76

51

12

84

33

19

6.

43

51

38

36

37

33

41

87

62

23

95

44

7.

58

54

62

49

47

36

34

52

98

73

34

27

8.

74

69

65

74

71

58

47

35

63

31

84

45

9.

91

85

91

76

85

82

69

58

37

74

42

95

10.

99

13

96

13

87

96

93

87

69

47

75

53

11.

19

12

26

18

24

24

18

15

98

36

85

49

12.

39

41

23

37

29

35

98

29

26

91

48

96

13.

51

55

52

34

48

56

46

31

53

37

13

59

14.

63

62

62

63

45

59

67

57

42

64

48

24

15.

84

99

75

73

74

56

82

78

68

53

59

57

Exercise No. 16

Mental Addition

Add 20 to each of the numbers in Table I on page 7.

Exercise No. 17

Adding Single Columns by Pairs

Add the following by single columns, taking pairs of successive numbers at a time. Add from the top down.

1.

51

30

96

24

25

75

48

49

93

80

13

58

88

86

20

99

59

65

2.

42

53

90

79

87

76

92

52

45

72

18

63

22

21

59

91

15

78

3.

41

73

32

12

62

11

44

84

70

40

61

67

56

16

98

55

27

54

4.

34

36

97

19

69

94

83

37

38

46

17

23

66

64

89

68

60

23

5.

33

81

28

39

43

10

85

47

29

14

95

10

82

31

77

74

35

84

6.

61

81

82

24

59

95

53

37

27

31

92

83

80

38

54

98

41

77

7.

34

90

86

85

16

58

64

47

23

45

44

65

72

68

42

40

89

99

8.

39

32

21

49

87

33

48

11

60

20

70

26

55

57

12

46

75

18

9.

36

25

97

96

52

30

63

94

35

62

51

91

88

43

19

14

56

42

10.

17

66

28

74

84

15

67

93

73

69

10

29

79

78

22

13

76

39

Exercise No. 18

Mental Addition

Add 21 to each of the numbers in Table I on page 7.

SUBTRACTION IN GENERAL

In keeping with the general object of this book, the succeeding exercises in subtraction are performed by left-to-right methods.

When subtraction is performed on paper there is no special advantage in working from left to right instead of from right to left. Paper practice in the former method, however, fits in with the broad purpose of developing number sense.

When it comes to doing subtraction mentally, the left-to-right method is natural and logical. Thus, if you had started the day with $17.43 in your pocket, and if you wanted to figure without paper and pencil how much you had left after spending $5.89, you would not be likely to start by subtracting 9 from 13. You would probably calculate that if you had spent the full $6, you would have $11.43 left, but that having spent 11 less than $6, the remainder comes to 11 more than $11.43, or $11.54.

In considering the specific aims of these exercises in subtraction, look first at the written examples. If you will glance at the first exercise that follows, and which is included merely to familiarize you with the idea of working from left to right, you will see that in every case the figures in the subtrahend (lower number) are smaller than those in the minuend. The examples are all of the type of

and you can determine the answers faster than you can write them down. If, however, you take the example

and try to write the answer in the same way, you will run into trouble when you reach the final figures at the right because 6 is greater than 5. What to do about cases of this kind is the subject of the instruction. The exercises take into account the possible variations that may occur in numbers of two and three places.

The examples in mental subtraction are performed by methods altogether different from those that apply to written work. There are two such methods, of which one has already been illustrated. We subtracted $5.89 from $17.43 by taking $6 from $17.43 and then adding to $11.43 the difference between $6 and $5.89, obtaining as our answer $11.43 + $.11, or $11.54. To do the same example mentally by the other method, we calculate that if you had started with $17 even, you would have $11.11 left; but you had $.43 more than $17 at the start, and therefore the actual remainder is $11.11 + $.43, or $11.54. One method is as good as the other. Examples are given that carry the practice in both methods as far as numbers involving hundreds of dollars and odd cents.

Incidentally, you should know that ordinary written subtraction is commonly performed by two entirely different methods—the borrow method and the carry method. The borrow method is taught almost exclusively in this country today, but in times past the carry method had similar acceptance.

Take the example

To do this by the borrow method you reason: 9 from 16 leaves 7, 6 from 14 leaves 8, 5 from 7 leaves 2; answer, 287. To do the same example by the carry method you would say: 9 from 16 leaves 7, 7 from 15 leaves 8, 6 from 8 leaves 2; answer, 287.

You should understand both these methods (neither of which has any clear advantage over the other), though you continue to use regularly whichever one comes most naturally to you. In the illustrations given in this book the borrow method is followed because it is the more familiar to the majority of people.

Exercise No. 19

Left-to-Right Subtraction

Perform the following subtractions by directly writing your answers from left to right.

1.

67

55

2.

48

14

3.

41

20

4.

78

22

5.

64

31

6.

98

20

7.

53

41

8.

65

52

9.

28

16

10.

66

45

11.

99

92

12.

69

35

13.

83

31

14.

32

21

15.

93

41

Exercise No. 20

Left-to-Right Subtraction

Directly write your answers from left to right.

To take the first example, you simply note that 6 is greater than 4, and therefore the 5 in the minuend becomes a 4: 2 from 4 leaves 2 (writing 2), 6 from 14 leaves 8 (writing 8); answer 28.

1.

54

26

2.

47

19

3.

51

39

4.

46

27

5.

52

37

6.

84

58

7.

37

18

8.

35

17

9.

72

24

10.

50

29

11.

83

44

12.

56

39

13.

71

45

14.

96

38

15.

77

49

16.

94

76

17.

45

16

18.

48

29

19.

68

39

20.

71

52

Exercise No. 21

Mental Addition

Add 22 to each of the numbers in Table I on page 7.

Exercise No. 22

Trios that Add to 10 or Less

This exercise introduces the idea of taking in three successive numbers at a glance. Every column contains four groups of three numbers each; each of these groups adds to 10 or less. Add by combining these groups. Add from the top down.

1.

27

21

11

45

41

13

65

12

12

25

11

11

2.

14

11

12

33

21

13

25

21

24

35

12

13

3.

64

21

13

44

42

22

43

32

33

78

11

11

4.

57

31

12

56

21

23

56

12

12

45

21

12

5.

34

31

11

54

42

13

52

31

22

44

31

14

6.

41

21

26

31

31

22

81

11

11

72

21

13

7.

62

32

12

61

21

23

52

21

16

44

12

14

8.

43

33

24

21

11

27

43

11

45

62

12

15

9.

21

11

15

12

11

14

33

11

23

24

21

25

10.

33

12

15

63

11

24

42

22

44

43

32

33

Exercise No. 23

Left-to-Right Subtraction

Sight practice with pairs of three-place numbers. No borrowings are involved. Work from left to right.

1.

754

233

2.

827

614

3.

468

235

4.

659

338

5.

746

415

6.

928

615

7.

675

423

8.

558

146

9.

649

437

10.

458

328

11.

727

605

12.

898

457

13.

753

321

14.

462

111

15.

941

720

Exercise No. 24

Mental Addition

Add 23 to each of the numbers in Table I on page 7.

Exercise No. 25

Mental Addition

Add 24 to each of the numbers in Table I on page 7.

Exercise No. 26

Adding Single Columns by Pairs

Take successive pairs at a time. Add from the top down.

1. $40.72

33.32

98.21

29.05

53.69

79.66

83.97

45.77

42.63

46.68

64.39

37.62

2. $35.51

56.28

43.90

49.44

84.57

99.61

24.25

16.23

80.17

82.67

86.93

91.76

3.

$27.13

96.92

22.07

38.71

58.94

34.88

60.26

65.14

18.19

89.30

41.75

50.95

4.

$47.15

10.20

36.09

59.73

55.70

85.54

31.78

11.12

52.48

87.81

74.01

25.60

5.

$79.45

85.30

70.46

83.73

69.97

34.21

64.81

20.72

60.26

31.57

59.86

58.35

6.

$77.52

54.05

61.65

76.29

74.43

38.10

87.37

63.25

32.93

22.98

89.84

91.23

7.

$48.68

49.99

14.78

11.12

90.55

17.18

15.50

56.47

67.06

19.16

41.40

56.15

8.

$88.09

44.80

75.03

36.53

95.96

62.39

82.01

26.13

33.28

42.71

94.66

10.34

Exercise No. 27

Left-to-Right Subtraction

In these examples, in the vertical pairs of figures at the extreme right the subtrahend is greater than the minuend, reducing by 1 the tens’ figure of the minuend.

Taking the first example, we note that the tens’ figure of the minuend will become a 4 instead of a 5; 5 from 7 leaves 2, 3 from 4 leaves 1, 9 from 14 leaves 5; answer 215.

1.

754

539

2.

863

448

3.

528

319

4.

642

313

5.

995

217

6.

422

313

7.

323

109

8.

676

428

9.

266

138

10.

583

346

11.

912

509

12.

365

259

13.

744

619

14.

390

265

15.

555

419

16.

983

779

17.

696

587

18.

472

329

19.

713

606

20.

626

318

21.

718

409

22.

683

246

23.

951

229

24.

648

539

25.

873

358

26.

715

506

27.

582

246

28.

246

139

29.

997

129

30.

737

318

Exercise No. 28

Mental Addition

Add 25 to each of the numbers in Table I on page 7.

Exercise No. 29

Mental Addition

Add 26 to each of the numbers in Table I on page 7.

Exercise No. 30

Mental Addition

Add 27 to each of the numbers in Table I on page 7.

Exercise No. 31

Trios that Add to 20 or Less

In the separate columns of the following examples the successive groups of three figures add to some number between 11 and 20. Add by combining these groups of three. Add from the top down.

The first example would be added: 16, 30, 41, 61, write 1 and carry 6; 6, 18, 30, 46, 62; answer 621.

1.

23

46

67

21

55

58

22

54

95

12

69

99

2.

31

46

46

12

24

97

13

73

86

23

57

77

3.

12

84

89

33

43

78

13

37

99

13

88

98

4.

24

64

74

35

45

95

14

45

75

25

65

86

5.

24

74

78

35

55

78

14

44

99

25

35

69

6.

33

36

98

11

25

89

13

77

75

23

56

69

7.

32

44

58

13

33

77

23

57

88

31

46

68

8.

24

67

69

36

47

87

13

48

69

14

99

98

9.

34

54

56

25

25

89

24

64

97

35

55

67

10.

24

75

85

35

56

86

14

55

56

25

36

77

Exercise No. 32

Left-to-Right Subtraction

In the type of example given here we see by inspection that the subtrahend has a larger figure than the minuend in the tens’ place, reducing by 1 the hundreds’ figure of the minuend. To take the first example: 5 from 6 leaves 1, 9 from 15 leaves 6, 3 from 4 leaves 1; answer 161.

Subtract from left to right.

1.

754

593

2.

648

356

3.

262

191

4.

548

357

5.

629

458

6.

856

792

7.

435

183

8.

468

271

9.

914

291

10.

765

481

11.

787

693

12.

547

160

13.

341

171

14.

112

51

15.

783

190

16.

486

291

17.

888

494

18.

489

194

19.

944

452

20.

842

161

Exercise No. 33

Left-to-Right Subtraction

In these examples the tens and the units are larger in the subtrahend than in the minuend, thus reducing by 1 both the hundreds and the tens of the minuend. Taking the first example: 2 from 6 leaves 4, 8 from 14 leaves 6, 9 from 14 leaves 5; answer, 465.

1.

754

289

2.

773

194

3.

413

249

4.

484

298

5.

342

189

6.

626

578

7.

787

298

8.

383

197

9.

867

379

10.

672

295

11.

918

589

12.

666

197

13.

586

298

14.

232

176

15.

515

299

16.

353

169

17.

428

179

18.

856

779

19.

481

192

20.

318

149

Exercise No. 34

Adding Single Columns by Pairs

Add the following by single columns, taking pairs of successive numbers at a time. Add from the bottom up.

1.

$14.44

38.42

72.09

61.90

63.26

56.78

73.76

62.58

91.28

31.41

71.15

50.82

22.78

33.34

25.12

92.49

58.43

75.64

2.

$80.54

33.20

13.40

55.95

10.17

75.79

77.52

39.51

83.85

87.19

59.57

24.23

94.70

61.90

50.05

82.98

93.63

20.67

3.

$74.43

67.27

18.02

21.60

25.98

96.45

89.84

11.12

64.48

19.92

22.53

65.99

66.75

11.54

74.45

55.62

95.37

72.71

4.

$43.93

32.06

94.34

97.86

30.29

36.47

70.66

35.07

81.68

49.37

69.16

57.84

53.69

96.17

36.03

30.35

39.51

48.15

5.

$22.78

69.33

48.14

17.81

44.88

40.18

19.02

63.95

89.16

99.08

87.83

77.52

22.78

40.18

66.75

53.45

60.39

71.09

6.

$94.70

34.61

27.10

68.47

76.13

31.05

26.30

37.86

46.65

20.67

92.49

21.60

56.25

31.82

62.77

69.33

51.85

48.15

7.

$66.75

90.72

80.11

73.29

56.25

74.45

35.58

24.38

39.51

84.36

82.98

92.13

49.12

94.70

52.05

34.57

64.61

27.10

8.

$79.53

71.09

54.96

59.15

50.91

57.42

43.93

32.23

85.64

28.41

55.01

16.46

50.91

98.55

74.79

21.65

90.72

80.06

Exercise No. 35

Left-to-Right Subtraction

This exercise illustrates a principle: if a figure in the subtrahend is the same as the one above it in the minuend, the effect on the minuend will depend on whether or not a borrowing has been necessary with the next figure to the right.

In the first example we see that because 9 is greater than 4, the 5 in the minuend becomes a 4, and since 5 is greater than this the 7 in the minuend becomes a 6. We perform the subtraction thus: 3 from 6 leaves 3, 5 from 14 leaves 9, 9 from 14 leaves 5; answer, 395.

1.

754

359

2.

655

358

3.

251

159

4.

546

247

5.

592

294

6.

862

667

7.

444

146

8.

968

569

9.

773

279

10.

763

266

11.

832

536

12.

233

139

13.

983

488

14.

572

278

15.

656

357

16.

395

197

17.

856

659

18.

645

248

19.

721

428

20.

941

249

21.

527

329

22.

863

569

23.

985

389

24.

267

168

25.

843

448

Exercise No. 36

Trios that Add to 27 or Less

The groups of three here add to numbers between 21 and 27. Add by combining these groups. Add from the top down.

1.

36

98

99

69

99

99

56

89

89

73

79

99

2.

63

79

89

86

89

89

33

99

99

67

97

97

3.

47

87

98

74

78

79

67

77

97

84

88

99

4.

65

78

98

87

87

99

54

89

99

77

87

88

5.

47

97

99

75

78

89

49

89

99

75

78

78

6.

55

88

89

77

78

98

65

89

89

87

98

98

7.

68

88

88

85

99

99

57

98

99

76

87

98

8.

56

87

99

78

88

89

96

97

98

78

78

88

9.

68

88

99

96

98

98

68

89

99

96

97

99

10.

56

98

98

78

89

99

66

78

89

84

88

89

Exercise No. 37

Left-to-Right Subtraction

In these examples another consideration arises: the tens’ figure in the minuend is 0; when 1 is borrowed to make possible the subtraction of the units, the tens in the minuend become 9 and the hundreds are also reduced by 1.

To illustrate with the first example: 3 from 6 leaves 3, 5 from 9 leaves 4, 7 from 14 leaves 7; answer, 347.

Subtract from left to right.

1.

704

357

2.

307

118

3.

806

457

4.

204

126

5.

404

297

6.

808

549

7.

706

517

8.

308

189

9.

302

236

10.

203

115

11.

800

585

12.

501

323

13.

300

122

14.

805

796

15.

601

374

16.

902

793

17.

500

386

18.

408

159

19.

700

466

20.

207

178

21.

807

509

22.

603

319

23.

200

162

24.

600

224

25.

300

171

Exercise No. 38

Adding Single Columns by Pairs

Take pairs of successive numbers at a time. Add from the bottom up.

1.

$5759.37

2186.62

4491.67

3848.60

6874.79

1831.04

1080.33

6461.73

9823.34

2.

$7856.21

2477.50

5843.84

3993.36

4751.85

9213.53

3363.26

9994.90

9617.89

3.

$6525.49

5214.44

8788.76

1115.81

2740.32

4569.82

9528.30

7271.70

8983.55

4.

$4142.97

4629.22

2089.83

9766.48

3367.72

9849.04

1623.26

4308.52

5354.34

4244.07

6874.79

6118.91

5.

$6675.01

3508.07

5624.21

6039.10

7677.25

6393.03

6257.59

3646.51

9678.28

7170.27

3229.30

4569.73

6.

$1916.46

2009.03

6538.82

8788.80

7531.01

8635.19

5096.58

1185.13

1714.55

4015.81

6422.37

9947.94

Exercise No. 39

Mental Subtraction

Use the method of making the subtrahend a round number. Subtract $1 from the minuend and add to this the difference between $1 and the given subtrahend.

Taking the first example: $1 from $5.18 leaves $4.18; $.83 from $1 leaves $.17; $4.18 + $.17 = $4.35.

1. $5.18 - $.83

2. $6.42 - $.83

3. $1.89 - $.95

4. $2.47 - $.99

5. $7.48 - $.56

6. $8.29 - $.66

7. $3.18 - $.87

8. $7.27 - $.43

9. $4.19 - $.49

10. $3.53 - $.77

11. $3.22 - $.93

12. $7.37 - $.61

13. $4.56 - $.97

14. $6.87 - $.91

15. $2.21 - $.65

16. $4.86 - $.97

17. $3.32 - $.64

18. $7.75 - $.83

19. $4.12 - $.63

20. $6.23 - $.26

Exercise No. 40

Adding Single Columns by Trios

Do the addition examples in Exercise No. 13 on page 11 by grouping three numbers at a time.

Taking the first example there presented, the following illustrates the method of adding: 13 (+12) 25, write 5 and carry 2; 2 (+17) 19, (+17) 36; answer, 365. Do not consciously repeat to yourself the individual amounts that you are adding, but only the successive total. Add from the top down.

Exercise No. 41

Adding Single Columns by Pairs

1.

$7489.99

2897.66

7828.17

3519.16

2237.61

7170.27

5950.95

1209.63

8152.92

5354.14

7725.75

6101.98

5429.30

4414.57

7812.07

5056.24

2593.26

4569.35

2.

$8356.24

4860.39

8084.05

2303.32

1891.45

4015.94

5843.08

9326.73

3646.51

5520.33

3104.60

4953.91

6772.76

5910.18

7170.06

9564.22

2075.27

9236.74

3.

$2165.38

1034.96

8788.86

2922.64

4142.44

9062.57

9849.04

4768.79

1185.13

6772.76

1348.37

6039.62

1780.84

9134.96

8788.86

7755.63

4033.03

8932.58

4.

$8799.55

4437.14

9793.08

4223.59

3218.94

9564.65

6296.78

4569.35

7006.68

7976.92

3612.97

8765.77

5960.54

5546.31

4347.04

9570.06

6935.05

6774.27

5.

$1319.16

5781.63

5266.88

3926.73

9156.24

2227.49

1207.54

7729.30

6772.11

9036.17

8909.50

2930.51

9964.75

7188.86

4147.61

1457.10

3218.94

4913.26

6.

$8348.84

2538.82

2861.41

9809.50

5834.43

5340.33

5446.31

5115.71

8521.65

8074.89

2124.56

1507.23

2279.76

2858.34

8085.37

4884.44

8168.39

7273.93

Exercise No. 42

Mental Subtraction

Perform the subtractions in Exercise No. 39 by using the method of making a round number of the minuend. That is, reduce the minuend to the next lower number of even dollars. Subtract the subtrahend from this and then add the excess of cents in the minuend.

Taking the first example ($5.18 - $.83): $.83 from $5 leaves $4.17; $4.17 + 18 = $4.35.

Exercise No. 43

Mental Subtraction

Perform the following subtractions mentally. Raise the subtrahend to the next larger number of even dollars.

1. $2.79 - $1.86

2. $3.17 - $1.97

3. $9.50 - $6.69

4. $2.56 - $1.91

5. $4.77 - $2.81

6. $9.78 - $3.94

7. $7.44 - $4.49

8. $4.37 - $2.72

9. $5.22 - $2.98

10. $6.04 - $5.33

11. $5.53 - $3.64

12. $2.62 - $1.89

13. $3.05 - $1.82

14. $8.28 - $6.65

15. $8.10 - $6.39

16. $5.15 - $2.67

17. $4.47 - $2.61

18. $7.93 - $5.99

19. $5.40 - $2.95

20. $3.23 - $1.60

Exercise No. 44

Mental Subtraction

Do the examples in Exercise No. 43 by lowering the minuend to the next smaller number of even dollars.

MULTIPLICATION IN GENERAL

Multiplication is the heart’s core of the art of calculation. In itself it constitutes an art about which a large volume might be written.

The multiplication exercises in this book have three main objects in view—first, to enable the student to use all numbers up to 25 as direct multipliers in written work; second, to teach him to multiply mentally any number up to 1000 by any other number up to 1000; third, to drill him in various short-cut methods that apply to particular cases.

The use of numbers up to 25 as direct multipliers may be illustrated by this example:

In Method A, which is here shown for comparison, the usual procedure is followed. In Method B the calculation is performed thus: 8 × 23 = 184, write 4 and carry 18; 4 × 23 = 92, 92 + 18 = 110, write 0 and carry 11; 6 × 23 = 138, 138 + 11 = 149, write 9 and carry 14; 7 × 23 = 161, 161 + 14 = 175. Multiplication by 19 is done in the same way, and the partial products added.

To multiply in the manner described it is of course necessary to acquire a knowledge of the multiplication table up to 25 × 25. Instruction in this direction is given by very easy steps. There are several types of exercises leading to the same end.

Exercises in mental multiplication are similarly graded. You start by multiplying two figures by one, then two by two, then three by one, three by two, and finally three by three.

The subject of short cuts is highly specialized and need not detain us for the present.

Exercise No. 45

Mental Multiplication

Multiply by 2 the numbers in Table I on page 7. Proceed from left to right. A few examples of the method calculating will suffice.

32 × 2 : 30 × 2 = 60, 2 × 2 = 4, 60 + 4 = 64

45 × 2 : 40 × 2 = 80, 5 × 2 = 10, 80 + 10 = 90

49 × 2 : 40 × 2 = 80, 9 × 2 = 18, 80 + 18 = 98

99 × 2 : 90 × 2 = 180, 9 × 2 = 18, 180 + 18 = 198

Exercise No. 46

Mental Multiplication

Multiply mentally by 3 the numbers in Table I on page 7.

Exercise No. 47

Mental Multiplication

Multiply mentally by 4 the numbers in Table I on page 7.

Exercise No. 48

Adding Single Columns by Pairs

Take pairs of successive numbers at a time. Add from the bottom up.

1.

$227976.55

491368.39

476170.02

804501.33

920950.63

512573.15

2.

$364631.71

291241.97

620314.57

378990.83

267278.30

586721.69

3.

$693505.74

822427.23

186620.98

871060.54

118577.94

996475.17

4.

$430413.93

525632.59

198886.28

651653.40

964295.81

480444.80

5.

$605465.38

599320.95

810064.74

112279.76

431275.17

890890.55

6.

$694235.68

483929.91

841653.40

344518.66

624133.37

364698.97

Exercise No. 49

Mental Subtraction

Raise the subtrahend to the next larger number of even dollars.

1. $19.03 - $.50

2. $26.52 - $.86

3. $24.27 - $.32

4. $15.58 - $.80

5. $42.35 - $.59

6. $39.29 - $.91

7. $16.53 - $.79

8. $43.12 - $.17

9. $61.70 - $.94

10. $72.04 - $.85

11. $67.30 - $.73

12. $60.54 - $.69

13. $94.20 - $.48

14. $81.64 - $.74

15. $76.34 - $.66

16. $62.41 - $.89

Exercise No. 50

Mental Multiplication

Multiply mentally by 5 the numbers in Table I on page 7.

Exercise No. 51

Mental Subtraction

Do the examples in Exercise No. 49 by reducing the minuend to the next smaller number of even dollars.

Exercise No. 52

Mental Multiplication

Multiply mentally by 6 the numbers in Table I on page 7.

Exercise No. 53

Mental Multiplication

Multiply mentally by 7 the numbers in Table I on page 7.

Exercise No. 54

Adding Single Columns by Pairs

Take pairs of successive numbers at a time. Add from the top down.

1.

$806054.65

681097.85

451866.93

431248.39

298291.24

322157.61

700177.25

714913.58

746789.23

569055.36

534011.98

281472.87

2.

$386942.35

933492.59

209507.09

751706.02

882750.78

305181.62

733115.33

379499.64

663265.52

444684.16

227976.86

77730.32

3.

$243130.39

158010.21

519794.95

893672.07

870485.02

834913.40

287919.76

697537.73

225942.35

435756.84

996168.05

164864.14

4.

$559663.93

882067.60

265254.65

332750.44

380353.71

462925.62

583492.78

411711.98

230882.09

911270.45

180190.66

744732.86

Exercise No. 55

Mental Subtraction

Raise the subtrahend to the next larger number of even dollars.

1. $24.31 - $4.55

2. $26.36 - $7.50

3. $49 13 - $4.62

4. $34.37 - $7.98

5. $43.12 - $1.70

6. $14.06 - $7.86

7. $15.10 - $2.88

8. $26.52 - $6.89

9. $96.15 - $8.88

10. $87.04 - $2.53

11. $79.19 - $7.58

12. $59.42 - $3.82

13. $99.05 - $1.90

14. $77.24 - $3.55

15. $67.60 - $5.97

16. $72.07 - $3.87

Exercise No. 56

Mental Multiplication

Multiply mentally by 8 the numbers in Table I on page 7.

Exercise No. 57

Adding Single Columns by Trios

Do the examples in Exercise No. 15 on page 12 by taking three successive numbers at a time. Add from the top down.

Exercise No. 58

Mental Subtraction

Do the examples in Exercise No. 55 by lowering the minuend to the next smaller number of even dollars.

Exercise No. 59

Addition of Partial Products

The type of exercise here presented has a bearing on mental multiplication. Thus the first example represents, in inverted position, the partial products we get when we multiply 15 by 53.

When partial products of this kind occur in mental multiplication you are of necessity compelled to retain them in your mind. Hence to develop your ability to do this kind of memory work, you are asked to read each example once and then write it three times on paper before you perform the mental addition.

Complete the mental addition before writing the answer. Work from left to right. Thus in doing the first example you would say to yourself: 750, 790, 795. In doing the second you would say: 620, 680, 682.

1.

750

45

2.

620

62

3.

470

94

4.

740

74

5.

520

78

6.

880

44

7.

720

90

8.

880

66

9.

960

1

10.

840

72

11.

850

51

12.

540

81

13.

570

95

14.

220

88

15.

910

52

16.

680

4

17.

980

28

18.

280

84

19.

640

96

20.

690

92

21.

760

95

22.

810

54

23.

750

15

24.

910

78

25.

580

87

Exercise No. 60

Mental Multiplication

Multiply mentally by 9 the numbers in Table I on page 7.

Exercise No. 61

Mental Multiplication

Multiply mentally by 11 the numbers in Table I.

Exercise No. 62

Adding Single Columns by Pairs

Add from the bottom up.

1.

$698504.99

845643.09

761979.28

401349.83

740614.80

553930.31

896554.52

975160.67

417337.75

882110.35

116448.16

477406.66

801415.93

340939.01

380272.36

656958.68

882152.17

401304.99

2.

$457012.91

820823.58

622529.46

715303.47

159363.96

380272.36

268195.94

789234.17

773286.20

425922.98

669001.18

502733.07

906396.55

301831.05

820889.23

548620.61

874185.10

761944.26

3.

$662533.75

380277.80

847236.82

735356.57

236569.58

862061.88

178735.81

464385.34

425919.44

789249.94

395497.48

194426.67

129066.25

464347.56

316085.34

499498.27

776980.14

518437.35

4.

$473105.74

141593.51

111290.63

897350.27

379128.68

966221.52

644107.29

104004.99

266722.95

987983.35

183216.70

295788.92

336353.75

578389.73

740638.09

236540.02

159383.58

729128.36

Exercise No. 63

Mental Subtraction

Raise the subtrahend to the next larger number of even dollars.

1. $83.37 - $35.72

2. $68.20 - $61.99

3. $97.48 - $17.87

4. $64.41 - $29.67

5. $25.33 - $10.65

6. $79.58 - $51.84

7. $48.54 - $20.61

8. $52.17 - $30.32

9. $91.28 - $36.82

10. $76.42 - $62.59

11. $55.30 - $18.81

12. $95.12 - $90.66

13. $65.40 - $14.93

14. $37.35 - $28.82

15. $49.01 - $21.85

16. $81.03 - $41.16

Exercise No. 64

Continuous Addition Drill

Count by 3’s to 75.

Count by 4’s to 100.

Count by 6’s to 150.

Count by 7’s to 175.

Count by 8’s to 200.

Count by 9’s to 225.

Count by 11’s to 275.

Count by 12’s to 300.

Repeat this exercise three times.

Exercise No. 65

Mental Subtraction

Do the examples in Exercise No. 63 by lowering the minuend to the next smaller number of even dollars.

Exercise No. 66

Mental Addition

Read each of these examples once, write it three times and then add it mentally from left to right.

Be careful to think of the upper number in each case as something in the thousands and not as so many hundreds. Thus in the first example the upper number should be called one thousand seven hundred forty, not seventeen hundred forty. It is easier to think of comparatively small numbers as hundreds rather than as thousands plus hundreds, but this method of naming leads to trouble when dealing with larger numbers, and it is best to follow one uniform system.

1.

1740

87

2.

1650

55

3.

1080

90

4.

1280

96

5.

2430

81

6.

2560

64

7.

3690

82

8.

1120

80

9.

1450

87

10.

1140

95

11.

1320

88

12.

1350

78

13.

1340

67

14.

1320

88

15.

1920

96

16.

2340

78

17.

3680

92

18.

1080

84

19.

1950

65

20.

2520

72

Exercise No. 67

Mental Subtraction

Raise the subtrahend to the next larger number of even dollars.

1. $855.30 - $8.32

2. $844.16 - $7.29

3. $671.46 - $4.47

4. $834.06 - $4.09

5. $642.02 - $7.80

6. $836.11 - $8.68

7. $862.21 - $4.45

8. $532.13 - $4.41

9. $426.22 - $7.78

10. $912.25 - $5.33

11. $453.31 - $5.60

12. $594.10 - $7.23

13. $415.37 - $7.91

14. $520.39 - $9.76

15. $542.17 - $8.55

16. $673.29 - $9.44

Exercise No. 68

Adding Single Columns by Trios

Do the examples in Exercise No. 17 on page 15 by grouping three successive numbers at a time. Add from the top down.

Exercise No. 69

Mental Subtraction

Do the examples in Exercise No. 67 by reducing the minuend to the next smaller number of even dollars.

Table II

Numbers for Multiplication Table Drill

Exercise No. 70

Multiplication Table Drill

Use Table II on this page. Multiply the numbers in Column A successively by 2, 3, 4, 5, 6, 7, 8, 9, 10,11, and 12. Repeat this exercise three times.

Exercise No. 71

Mental Subtraction

Raise the subtrahend to the next larger number of even dollars, and raise this amount in turn to an even $100. Thus, taking the first example: $100 from $365.42 leaves $265.42; $265.42 + $11 (difference between $100 and $89) equals $276.42; $276.42 + $.27 = $276.69.

1. $365.42 - $88.73

2. $950.49 - $94.98

3. $723.67 - $40.77

4. $614.15 - $93.79

5. $858.51 - $84.72

6. $928.36 - $36.82

7. $413.54 - $86.61

8. $342.21 - $96.62

9. $459.48 - $87.55

10. $553.18 - $81.64

11. $416.07 - $29.19

12. $426.22 - $95.78

13. $912.25 - $33.63

14. $753.46 - $56.57

15. $831.05 - $60.85

16. $743.16 - $68.29

Exercise No. 72

Adding Single Columns by Trios

Do the examples in Exercise No. 22 on page 20 by grouping three successive numbers at a time. Add from the bottom up.

Table III

Numbers to Be Multiplied

1. 111315

2. 111417

3. 121416

4. 121518

5. 541316

6. 171922

7. 182123

8. 897254

9. 248963

10. 258163

11. 222572

12. 541418

13. 192389

14. 151924

15. 212481

Exercise No. 73

Written Multiplication

Multiply the numbers in Table III by 6789.

Exercise No. 74

Mental Addition

Read each of the following examples once, write it three times and then add it mentally from left to right.

Think of the upper number in each case as being in the thousands and not the hundreds.

The first example would be added: 1280, 1480, 1536. In other words, take the first number as a whole, and then add to it successively the hundreds, tens and units of the second number.

1.

1280

256

2.

4410

196

3.

1960

686

4.

1380

115

5.

4620

693

6.

3060

170

7.

6510

837

8.

4150

664

9.

4080

204

10.

1110

185

11.

6480

144

12.

1450

174

13.

1640

246

14.

3350

268

15.

5150

44

16.

3510

51

17.

3040

04

18.

8080

528

19.

1240

72

20.

2250

405

Exercise No. 75

Mental Subtraction

Do the examples in Exercise No. 71 on page 49 by lowering the minuend. Reduce it to the next smaller number of even dollars. Taking the first example: $300 — $88.73 leaves $211.27; $211.27 + $65 = $276.27; $276.27 + $.42 = $276.69.

Exercise No. 76

Adding Single Columns by Trios

Do the examples in Exercise No. 26 on page 23 by grouping three successive numbers at a time. Add from the top down.

Exercise No. 77

Mental Multiplication

Multiply mentally by 12 the numbers in Table I on page 7.

Exercise No. 78

Adding Single Columns by Trios

Do the examples in Exercise No. 34 on page 28 by grouping three successive numbers at a time.

Exercise No. 79

Mental Subtraction

Raise the subtrahend to the next larger number of even hundreds of dollars.

1. $950.49 - $498.65

2. $646.43 - $456.57

3. $520.39 - $176.42

4. $821.13 - $468.54

5. $769.14 - $580.93

6. $831.05 - $685.34

7. $821.45 - $529.48

8. $862.39 - $197.76

9. $318.32 - $181.64

10. $636.09 - $549.95

11. $714.10 - $273.65

12. $821.45 - $599.97

13. $416.07 - $219.44

14. $640.02 - $493.79

15. $746.14 - $159.93

16. $752.30 - $183.81

Exercise No. 80

Mental Addition

Read each of the following examples once, write it three times and then add it mentally from left to right. The first example would be added: 16530, 17030, 17081.

1.

16530

551

2.

12930

431

3.

24920

623

4.

22080

552

5.

37150

743

6.

33650

673

7.

51780

863

8.

44460

741

9.

67340

962

10.

61810

883

11.

19360

242

12.

12160

152

13.

76960

962

14.

32670

363

15.

25380

282

16.

12690

141

17.

15320

766

18.

19620

654

19.

21720

543

20.

46650

933

21.

44160

736

Exercise No. 81

Written Multiplication

Multiply by 1112 each of the numbers in Table III on page 49. Wherever there occurs in the multiplicand a pair of figures that may be considered as 11 or 12, make one multiplication of this instead of two, and accordingly write down two figures in the partial product. Taking the first example:

111315 is successively multiplied (from right to left) by 12 and 11 thus: 5 × 12 = 60, write 0 and carry 6; 1 × 12 = 12, 12 + 6 = 18, write 8 and carry 1; 3 × 12 = 36, 36 + 1 = 37, write 7 and carry 3; 11 × 12 = 132, 132+3 = 135, write 35 and carry 1; 1 × 12 = 12, 12 + 1 = 13, write 13. Multiplication by 11 is carried out in the same way.

In doing these examples be watchful about placing the second partial product two places to the left of the first.

Exercise No. 82

Adding Single Columns by Trios

Do the examples in Exercise No. 38 on page 32 by grouping three successive numbers at a time. Add from the bottom up.

Exercise No. 83

Mental Subtraction

Do the examples in Exercise No. 79 on page 51 by lowering the minuend to the next smaller number of even hundreds of dollars.

Exercise No. 84

Mental Addition

Read each of the following examples once, write it three times and then add it mentally from left to right.

Add in turn the thousands, hundreds, tens and units to the upper number. In doing the first example you should say to yourself something like the following: 18360 + 1224, 19360; 19360 + 224, 19560; 19560 + 24, 19584.

1.

18360

1224

2.

21630

2163

3.

24960

3328

4.

18820

5646

5.

16260

1084

6.

19530

1953

7.

21360

2848

8.

16420

4926

9.

18640

6524

10.

10290

2401

11.

13530

3608

12.

16860

5058

13.

29240

1462

14.

33680

2526

15.

28590

4765

16.

13230

3969

17.

26520

1326

18.

28840

2163

19.

24960

4160

20.

28290

5658

21.

14120

2118

Exercise No. 85

Continuous Addition Drill

Count by 4’s to 100.

Count by 6’s to 150.

Count by 7’s to 175.

Count by 8’s to 200.

Count by 9’s to 225.

Count by ll’s to 275.

Count by 12’s to 300.

Count by 13’s to 325.

Repeat this exercise three times.

Exercise No. 86

Adding Single Columns by Trios

Do the examples in Exercise No. 41 on page 34 by grouping three successive numbers at a time. Add from the top down.

Exercise No. 87

Factoring

When numbers are multiplied together, they are considered factors of the resulting product. Thus 2 and 3 are factors of 6, and 3 and 5 are factors of 15.

Factoring a number is the process of resolving the number into the factors that will produce the number when multiplied together. Thus 36 may be factored as 2 × 18, or as 3 × 12, or as 4 × 9, or as 6 × 6.*

Any number that can be resolved into factors is called a composite number.

A prime number is one that has no factors besides itself and 1. Thus, 1, 2, 3, 5, 7, 11, 13, etc. are prime numbers.

* If it were required to give the prime factors of 36, these would be 2 × 2 × 3 × 3, but factoring into prime numbers has nothing to do with the purposes of this book.

On the pages starting with 146 will be found a table which analyzes all prime and composite numbers up to 625. You will be taught gradually to familiarize yourself with this entire table. The purpose of this is to help you to recognize quickly the character of these numbers—to enable you to multiply rapidly the factors that produce any of them, or to separate any of them into such factors.

Of special importance in this table are the numbers printed in italic type, since these can be produced by two factors each of which is 25 or less.

It is quite commonly appreciated that very small numbers have a definite individuality which grows out of the many associations built up around them in our minds. The individual character of higher numbers becomes similarly apparent and unforgettable when we single them out for particular attention.

For the first exercise in factoring read the first two columns of the table on page 146, and then write these from memory (or calculation) in the same form.

In studying the table note that each composite number is factored by first taking the smaller factors in the order of their size, and that the combinations are not repeated. Thus the separate ways of factoring 48 are given as 2 × 24, 3 × 16, 4 × 12 and 6 × 8. These combinations are not repeated as 8 × 6, 12 × 4, 16 × 3, and 24 × 2.

Exercise No. 88

Multiplication Table Drill

Use Table II on page 48.

Multiply the numbers in Column A successively by 3, 4, 6, 7, 8, 9, 11, 12 and 13.

Repeat this exercise three times.

This exercise takes us the first step beyond the customary limits of the multiplication table, which ordinarily goes no farther than 12 × 12. Succeeding examples will enable you to memorize the products of all pairs of numbers up to 25 × 25.

No multiplication table, as such, is presented in this book, because learning the products of higher factors by sheer power of memory is extremely difficult. On the other hand, when you are put over and over again to the necessity of figuring out these higher combinations for yourself, they soon come to stick firmly in the mind.

Exercise No. 89

Mental Addition

Read each of the following examples once, write it three times, and then add it mentally from left to right. The first example would be added: 165300, 170300, 170810.

1.

165300

5510

2.

129300

4310

3.

249200

6230

4.

220800

5520

5.

371500

7430

6.

336500

6730

7.

517800

8630

8.

444600

7410

9.

673400

9620

10.

618100

8830

11.

193600

2420

12.

121600

1520

13.

769600

9620

14.

326700

3630

15.

253800

2820

16.

126900

1410

17.

153200

7660

18.

196200

6540

19.

217200

5430

20.

456500

9330

21.

441600

7360

Exercise No. 90

Mental Multiplication

Multiply mentally by 13 the numbers in Table I on page 7.

In working with numbers from 80 upward, immediately name 1000 as the first part of the product. Thus 83 × 13 is 1040, (+39) 1079; 97 × 13 is 1170, 1261.

Exercise No. 91

Adding Single Columns by Trios

Do the examples in Exercise No. 48 on page 39 by grouping three successive numbers at a time. Add from the bottom up.

Exercise No. 92

Factoring

Read the table on page 146 from 31 to 72 inclusive, and then write it in the same form.

Exercise No. 93

Mental Addition

Read each of the following examples once, write it three times and then add it mentally from left to right.

Add in turn the tens of thousands, thousands, hundreds and tens to the upper number. The first example would be added: 183600, 193600, 195600, 195840.

1.

183600

12240

2.

216300

21630

3.

249600

33280

4.

188200

56460

5.

162600

10840

6.

195300

19530

7.

213600

28480

8.

164200

49260

9.

186400

65240

10.

102900

24010

11.

135300

36080

12.

168600

50580

13.

292400

14620

14.

336800

25260

15.

285900

47650

16.

132300

39690

17.

265200

13260

18.

288400

21630

19.

249600

41600

20.

282900

56580

21.

141200

21180

Exercise No. 94

Written Multiplication

Multiply by 1213 each of the numbers in Table III on page 49. Wherever there occurs in the multiplicand a pair of figures that may be considered as 11, 12 or 13, make one multiplication of this instead of two, and write two figures in the partial product. Thus, taking the first example, we successively multiply 15, 13 and 11 by 13 and again by 12. The partial products are accordingly written in two lines instead of the customary four.

Exercise No. 95

Adding Single Columns by Trios

Do the examples in Exercise No. 54 on page 41 by grouping three successive numbers at a time. Add from the top down.

Exercise No. 96

Factoring

Factor the numbers from 54 to 92 inclusive in the form shown in the table on page 146.

Exercise No. 97

Mental Addition

Read each of the following examples once, write it three times and then add it mentally from left to right.

Add the whole of the second number to the first before considering the third. Repeat to yourself several times the sum of the first and second if you find this necessary.

The third example would be added: 36300, 39300, 39930; (repeat 39930, 39930); 39930, 40030, 40051.

1.

10100

1010

101

2.

22200

2220

222

3.

36300

3630

121

4.

52400

5240

262

5.

70500

7050

141

6.

90600

1510

302

7.

19100

9950

382

8.

20200

1010

101

9.

33300

2220

222

10.

48400

3630

121

11.

65500

5240

262

12.

84600

7050

141

13.

18100

7240

181

14.

38200

9050

905

15.

20200

4040

202

16.

42400

6360

424

17.

66600

8880

666

18.

40400

4040

404

19.

33600

3360

336

20.

88800

8880

222

21.

30300

9090

303

Exercise No. 98

Continuous Addition Drill

Count by 6’s to 150.

Count by 7’s to 175.

Count by 8’s to 200.

Count by 9’s to 225.

Count by 11’s to 275.

Count by 12’s to 300.

Count by 13’s to 325.

Count by 14’s to 350.

Repeat this exercise three times.

Exercise No. 99

Adding Single Columns by Trios

Do the examples in Exercise No. 62 on page 44 by grouping three successive numbers at a time. Add from the bottom up.

Exercise No. 100

Factoring

Factor the numbers from 73 to 111 inclusive in the form shown in the table on page 146.

Exercise No. 101

Mental Addition

Read each of the following examples once, write it three times and then add it mentally from left to right.

The first example would be added: 26200, 33200, 34000, 34060; 34060, 36060, 36156.

1.

26200

7860

2096

2.

48400

9680

1210

3.

69900

9320

1398

4.

12100

9680

1089

5.

26400

9240

1056

6.

42900

8580

1144

7.

61600

9240

1078

8.

82500

9900

1155

9.

88000

8800

1056

10.

93500

9350

1122

11.

98000

9800

1188

12.

73200

9760

1098

13.

93100

9310

1064

14.

97600

9760

1220

15.

71000

7100

1065

16.

46600

9320

1398

17.

57700

5770

2308

18.

68800

6880

2064

19.

79900

7990

3196

20.

24600

9840

1107

21.

70200

9320

1170

Exercise No. 102

Multiplication Table Drill

Use Table II on page 48.

Multiply the numbers in Column A successively by 4, 6, 7, 8, 9, 11, 12, 13 and 14.

Repeat this exercise three times.

Exercise No. 103

Two-Column Addition

You are now ready to start adding two columns at a time. Take Exercise No. 13 on page 11. Add from the top down.

Two-column addition is simply an application of the left-to-right methods which you have already learned. To illustrate with the first example:

43

62

78

81

14

87

This would be added: 43, 103, 105, 175, 183, 263, 264, 274, 278, 358, 365. These are the actual steps, but with practice you will read this as 105, 183, 264, 278, 365.

Exercise No. 104

Factoring

Factor the numbers from 93 to 129 inclusive in the form shown in the table on pages 146 and 147.

Exercise No. 105

Mental Addition

Read each of the following examples once, write it three times, and then add it mentally from left to right.

1.

112700

3220

161

2.

136800

5130

342

3.

162900

2400

181

4.

105700

1510

302

5.

128800

3220

161

6.

153900

5130

342

7.

151200

5040

756

8.

183400

7860

262

9.

176400

5040

252

10.

209600

7860

524

11.

104800

5240

524

12.

103200

6880

860

13.

114100

6520

978

14.

112800

7050

423

15.

126000

7560

756

16.

111000

9250

740

17.

104400

8700

870

18.

135900

9060

302

19.

112800

9870

141

20.

130500

8700

435

21.

136800

6800

684

Exercise No. 106

Mental Multiplication

Multiply mentally by 14 the numbers in Table I on page 7.

Exercise No. 107

Two-Column Addition

Do the examples in Exercise No. 17 on page 15 by adding two columns at a time.Add from the bottom up.

Exercise No. 108

Factoring

Factor the numbers from 112 to 145 inclusive in the form shown in the table on pages 146 and 147.

Exercise No. 109

Mental Addition

Read each of the following examples once, write it three times, and then add it mentally from left to right.

1.

121000

14520

484

2.

217600

10880

544

3.

253800

14100

846

4.

116000

11600

464

5.

145200

14520

726

6.

224800

10880

816

7.

171500

24010

343

8.

211800

10590

706

9.

344700

22980

383

10.

129200

16150

323

11.

166500

19980

666

12.

290400

14520

363

13.

335700

18650

746

14.

272400

18160

454

15.

324800

23200

928

16.

124200

20700

828

17.

317800

18160

454

18.

371200

23200

924

19.

395500

34200

565

20.

210000

36750

525

21.

540800

33800

676

Exercise No. 110

Written Multiplication

Multiply by 1314 the numbers in Table III on page 49.

Exercise No. 111

Two-Column Addition

Do the examples in Exercise No. 26 on page 23 by adding two columns at a time. Add from the top down.

Exercise No. 112

Factoring

Factor the numbers from 130 to 162 inclusive in the form shown in the table on page 147.

Exercise No. 113

Mental Addition

Read each of the following examples once, write it three times, and then add it mentally from left to right.

1.

123200

39800

1232

2.

187800

37560

1878

3.

254400

44520

2544

4.

323000

51680

3230

5.

393600

59040

3936

6.

466200

26640

4662

7.

616200

41160

1392

8.

121200

48480

2424

9.

184800

55440

3080

10.

250400

25040

3956

11.

318000

31800

4452

12.

387600

38760

1292

13.

439200

43920

1312

14.

532800

53280

1998

15.

608400

60840

2704

16.

139200

34800

1392

17.

143400

28680

1434

18.

218700

36350

2187

19.

294800

44220

2948

20.

373500

52290

3735

21.

454200

60560

4542

Exercise No. 114

Continuous Addition Drill

Count by 7’s to 175.

Count by 8’s to 200.

Count by 9’s to 225.

Count by 11’s to 275.

Count by 12’s to 300.

Count by 13’s to 325.

Count by 14’s to 350.

Count by 15’s to 375.

Repeat this exercise three times.

Exercise No. 115

Two-Column Addition

Do the examples in Exercise No. 34 on page 28 by adding two columns at a time. Add from the bottom up.

Exercise No. 116

Multiplication Table Drill

Use Table II on page 48.

Multiply the numbers in Column B successively by 6, 7, 8, 9, 11, 12, 13, 14 and 15.

Repeat this exercise three times.

Exercise No. 117

Factoring

Factor the numbers from 146 to 179 inclusive in the form shown in the table on page 147.

Exercise No. 118

Two-Column Addition

Do the examples in Exercise No. 38 on page 32 by adding two columns at a time. Add from the top down.

It slows up addition by two columns to keep repeating the number of hundreds as you go along. A good plan is to keep tally of the number of hundreds with a pencil. In all addition of long columns write numbers to be carried either at the head of the next column or beneath the figures in the total as you set them down. When looking for errors in addition, add in the opposite direction from that in which the addition was originally performed.

Exercise No. 119

Mental Multiplication

Multiply mentally by 15 the numbers in Table I on page 7.

Exercise No. 120

Two-Column Addition

Do the examples in Exercise No. 41 on page 34 by adding two columns at a time. Add from the bottom up.

Exercise No. 121

Factoring

Factor the numbers from 163 to 194 inclusive in the form shown in the table on page 147.

Exercise No. 122

Two-Column Addition

Do the examples in Exercise No. 48 on page 39 by adding two columns at a time. Add from the top down.

Exercise No. 123

Written Multiplication

Multiply by 1415 the numbers in Table III on page 49.

Exercise No. 124

Two-Column Addition

Do the examples in Exercise No. 54 on page 41 by adding two columns at a time. Add from the bottom up.

Exercise No. 125

Factoring

Factor the numbers from 180 to 209 inclusive in the form shown in the table on page 147.

Exercise No. 126

Two-Column Addition

Do the examples in Exercise No. 62 on page 44 by adding two columns at a time. Add from the top down.

Exercise No. 127

Continuous Addition Drill

Count by 8’s to 200.

Count by 9’s to 225.

Count by 11’s to 275.

Count by 12’s to 300.

Count by 13’s to 325.

Count by 14’s to 350.

Count by 15’s to 375.

Count by 16’s to 400.

Repeat this exercise three times.

Exercise No. 128

Three-Column Addition

With the practice you have had in two-column addition you should now be able to add three columns at a time. Try this with the examples in Exercise No. 38 on page 32. No additional exercises in three-column addition are given, but you can of course practice it on your own account if you so desire.

Exercise No. 129

Multiplication Table Drill

Use Table II on page 48.

Multiply the numbers in Column C successively by 7, 8, 9, 11, 12, 13, 14, 15 and 16.

Repeat this exercise three times.

Exercise No. 130

Factoring

Factor the numbers from 195 to 224 inclusive in the form shown in the table on pages 147 and 148.

Exercise No. 131

Mental Multiplication

Multiply mentally by 16 the numbers in Table I on page 7.

Exercise No. 132

Written Multiplication

Multiply by 1516 the numbers in Table III on page 49.

Exercise No. 133

Factoring

Factor the numbers from 210 to 239 inclusive in the form shown in the table on pages 147 and 148.

DIVISION IN GENERAL

Division is multiplication in reverse. As you improve in multiplication you automatically develop your skill at division. For this reason it has been considered unnecessary to include any exercises in long division.

Exercises, however, are given in mental division, in order to round out your general calculating ability. These exercises are of the following types:

First you use the numbers from 2 to 25 as direct divisors, securing quotients from 1 to 99. Then you divide by the numbers from 2 to 9, finding answers of three places. Again, you divide by three-place numbers to arrive at quotients of one figure plus a remainder; the remainder is included so that the answer cannot be guessed but must be calculated accurately. Finally, you divide by numbers of two places and get results of two places. As division is somewhat more complicated, the exercises in division are not carried so far as those in multiplication.

Exercise No. 134

Mental Division

Divide mentally by 2 the answers to Exercise No. 45 as given on pages 161 and 162. Compare your answers with Table I on page 7.

Exercise No. 135

Continuous Addition Drill

Count by 9’s to 225.

Count by 11’s to 275.

Count by 12’s to 300.

Count by 13’s to 325.

Count by 14’s to 350.

Count by 15’s to 375.

Count by 16’s to 400.

Count by 17’s to 425.

Repeat this exercise three times.

Exercise No. 136

Mental Division

Divide mentally by 3 the answers to Exercise No. 46 as given on page 162. Compare your answers with Table I on page 7.

Exercise No. 137

Multiplication Table Drill

Use Table II on page 48.

Multiply mentally the numbers in Column D by 8, 9, 11, 12, 13, 14, 15, 16 and 17.

Repeat this exercise three times.

Exercise No. 138

Factoring

Factor the numbers from 225 to 254 inclusive in the form shown in the table on page 148.

Exercise No. 139

Mental Division

Divide mentally by 4 the answers to Exercise No. 47 as given on page 162. Compare your answers with Table I on page 7.

Exercise No. 140

Mental Multiplication

Multiply mentally by 17 the numbers in Table I on page 7.

Exercise No. 141

Written Multiplication

Multiply by 1617 the numbers in Table III on page 49. Make a single multiplication of pairs of figures in the multiplicand up to 17.

Exercise No. 142

Factoring

Factor the numbers from 240 to 269 inclusive in the form shown in the Table on page 148.

Exercise No. 143

Mental Division

Divide mentally by 5 the answers to Exercise No. 50 as given on page 163. Compare your answers with Table I on page 7.

Exercise No. 144

Continuous Addition Drill

Count by 11’s to 275.

Count by 12’s to 300.

Count by 13’s to 325.

Count by 14’s to 350.

Count by 15’s to 375.

Count by 16’s to 400.

Count by 17’s to 425.

Count by 18’s to 450.

Repeat this exercise three times.

Exercise No. 145

Multiplication Table Drill

Use Table II on page 48.

Multiply mentally the numbers in Column E by 9, 11, 12, 13, 14, 15, 16, 17 and 18.

Repeat this exercise three times.

Exercise No. 146

Factoring

Factor the numbers from 255 to 284 inclusive in the form shown in the table on page 148.

Exercise No. 147

Mental Division

Divide mentally by 6 the answers to Exercise No. 52 as given on page 163. Compare your answers with Table I on page 7.

Exercise No. 148

Mental Multiplication

Multiply mentally by 18 the numbers in Table I on page 7.

Exercise No. 149

Written Multiplication

Multiply by 1718 the numbers in Table III on page 49. Make a single multiplication of pairs of figures in the multiplicand up to 18.

Exercise No. 150

Factoring

Factor the numbers from 270 to 299 inclusive in the form shown in the table on pages 148.

Exercise No. 151

Mental Division

Divide mentally by 7 the answers to Exercise No. 53 as given on pages 163 and 164. Compare your answers with Table I on page 7.

Exercise No. 152

Continuous Addition Drill

Count by 12’s to 300.

Count by 13’s to 325.

Count by 14’s to 350.

Count by 15’s to 375.

Count by 16’s to 400.

Count by 17’s to 425.

Count by 18’s to 450.

Count by 19’s to 475.

Repeat this exercise three times.

Exercise No. 153

Multiplication Table Drill

Use Table II on page 48.

Multiply mentally the numbers in Column F by 11, 12, 13, 14, 15, 16, 17, 18 and 19.

Repeat this exercise three times.

Exercise No. 154

Factoring

Factor the numbers from 285 to 312 inclusive in the form shown in the table on page 148.

Exercise No. 155

Mental Division

Divide mentally by 8 the answers to Exercise No. 56 as given on page 164. Compare your answers with Table I on page 7.

Exercise No. 156

Mental Multiplication

Multiply mentally by 19 the numbers in Table I on page 7.

Exercise No. 157

Factoring

Factor the numbers from 300 to 328 inclusive in the form shown in the table on page 148.

Exercise No. 158

Mental Division

Divide mentally by 9 the answers to Exercise No. 60 as given on page 164. Compare your answers with Table I on page 7.

Exercise No. 159

Written Multiplication

Multiply by 1819 the numbers in Table III on page 49. Make a single multiplication of pairs of figures in the multiplicand up to 19.

Exercise No. 160

Factoring

Factor the numbers from 313 to 343 inclusive in the form shown in the table on page 149.

Exercise No. 161

Mental Division

Divide mentally by 11 the answers to Exercise No. 61 as given on page 165. Compare your answers with Table I on page 7.

Exercise No. 162

Multiplication Table Drill

Use Table II on page 48.

Multiply mentally the numbers in Column G by 12, 13, 14, 15, 16, 17, 18, 19 and 20.

Exercise No. 163

Factoring

Factor the numbers from 329 to 359 inclusive in the form shown in the table on pages 148 and 149.

Exercise No. 164

Mental Division

Divide mentally by 12 the answers to Exercise No. 77 as given on page 166. Compare your answers with Table I on page 7.

Exercise No. 165

Mental Multiplication

Multiply mentally by 20 the numbers in Table I on page 7.

Exercise No. 166

Written Multiplication

Multiply by 1920 the numbers in Table III on page 49. Make a single multiplication of pairs of figures in the multiplicand up to 20.

Exercise No. 167

Factoring

Factor the numbers from 344 to 372 inclusive in the form shown in the table on page 149.

Exercise No. 168

Mental Division

Divide mentally by 13 the answers to Exercise No. 90 as given on page 167. Compare your answers with Table I on page 7.

Exercise No. 169

Continuous Addition Drill

Count by 13’s to 325.

Count by 14’s to 350.

Count by 15’s to 375.

Count by 16’s to 400.

Count by 17’s to 425.

Count by 18’s to 450.

Count by 19’s to 475.

Count by 21’s to 525.

Exercise No. 170

Multiplication Table Drill

Use Table II on page 48.

Multiply mentally the numbers in Column H by 12, 13, 14, 15, 16, 17, 18, 19, 20 and 21.

Exercise No. 171

Factoring

Factor the numbers from 360 to 386 inclusive in the form shown in the table on page 149.

Exercise No. 172

Mental Multiplication

Multiply mentally by 21 the numbers in Table I on page 7.

Exercise No. 173

Written Multiplication

Multiply by 2021 the numbers in Table III on page 49. Make a single multiplication of pairs of figures in the multiplicand up to 21.

ExerciseNo. 174

Factoring

Factor the numbers from 373 to 399 inclusive in the form shown in the table on pages 149 and 150.

Exercise No. 175

Mental Division

Divide mentally by 14 the answers to Exercise No. 106 as given on page 168. Compare your answers with Table I on page 7.

Exercise No. 176

Continuous Addition Drill

Count by 14’s to 350.

Count by 15’s to 375.

Count by 16’s to 400.

Count by 17’s to 425.

Count by 18’s to 450.

Count by 19’s to 475.

Count by 21’s to 525.

Count by 22’s to 550.

Repeat this exercise three times.

Exercise No. 177

Multiplication Table Drill

Use Table II on page 48.

Multiply mentally the numbers in Column J by 13, 14, 15, 16, 17, 18, 19, 20, 21 and 22.

Exercise No. 178

Factoring

Factor the numbers from 387 to 413 inclusive in the form shown in the table on pages 149 and 150.

Exercise No. 179

Mental Multiplication

Multiply mentally by 22 the numbers in Table I on page 7.

Exercise No. 180

Written Multiplication

Multiply by 2122 the numbers in Table III on page 49. Make a single multiplication of pairs of figures in the multiplicand up to 22.

Exercise No. 181

Factoring

Factor the numbers from 400 to 427 inclusive in the form shown in the table on page 150 -

Exercise No. 182

Mental Division

Divide mentally by 15 the answers to Exercise No. 119 as given on page 169. Compare your answers with Table I on page 7.

Exercise No. 183

Continuous Addition Drill

Count by 15’s to 375.

Count by 16’s to 400.

Count by 17’s to 425.

Count by 18’s to 450.

Count by 19’s to 475.

Count by 21’s to 525.

Count by 22’s to 550.

Count by 23’s to 575.

Repeat this exercise three times.

Exercise No. 184

Multiplication Table Drill

Use Table II on page 48.

Multiply mentally the numbers in Column K by 14, 15, 16, 17, 18, 19, 20, 21, 22 and 23.

Exercise No. 185

Factoring

Factor the numbers from 414 to 440 inclusive in the form shown in the table on page 150.

Exercise No. 186

Mental Multiplication

Multiply mentally by 23 the numbers in Table I on page 7.

Exercise No. 187

Written Multiplication

Multiply by 2223 the numbers in Table III on page 49. Make a single multiplication of pairs of figures in the multiplicand up to 23.

Exercise No. 188

Factoring

Factor the numbers from 428 to 455 inclusive in the form shown in the table on page 150.

Exercise No. 189

Mental Division

Divide mentally by 16 the answers to Exercise No. 131 as given on pages 169 and 170. Compare your answers with Table I on page 7.

Exercise No. 190

Continuous Addition Drill

Count by 16’s to 400.

Count by 17’s to 425.

Count by 18’s to 450.

Count by 19’s to 475.

Count by 21’s to 525.

Count by 22’s to 550.

Count by 23’s to 575.

Count by 24’s to 600.

Repeat this exercise three times.

Exercise No. 191

Multiplication Table Drill

Use Table II on page 48.

Multiply mentally the numbers in Column L by 15, 16, 17, 18, 19, 20, 21, 22, 23 and 24.

Exercise No. 192

Factoring

Factor the numbers from 441 to 467 inclusive in the form shown in the table on pages 150 and 151.

Exercise No. 193

Mental Multiplication

Multiply mentally by 24 the numbers in Table I on page 7.

Exercise No. 194

Written Multiplication

Multiply by 2324 the numbers in Table III on page 49. Make a single multiplication of pairs of figures in the multiplicand up to 24.

Exercise No. 195

Factoring

Factor the numbers from 456 to 479 inclusive in the form shown in the table on pages 150 and 151.

Exercise No. 196

Mental Division

Divide mentally by 17 the answers to Exercise No. 140 as given on page 170. Compare your answers with Table I on page 7.

Exercise No. 197

Continuous Addition Drill

Count by 17’s to 425.

Count by 18’s to 450.

Count by 19’s to 475.

Count by 21’s to 525.

Count by 22’s to 550.

Count by 23’s to 575.

Count by 24’s to 600.

Count by 25’s to 625.

Repeat this exercise three times.

Exercise No. 198

Multiplication Table Drill

Use Table II on page 48.

Multiply mentally the numbers in Column M by 16, 17, 18, 19, 20, 21, 22, 23, 24 and 25.

Exercise No. 199

Factoring

Factor the numbers from 468 to 491 inclusive in the form shown in the table on page 151.

Exercise No. 200

Mental Multiplication

Multiply mentally by 25 the numbers in Table I on page 7.

Exercise No. 201

Written Multiplication

Multiply by 2425 the numbers in Table III on page 49. Make a single multiplication of pairs of figures in the multiplicand up to 25.

Exercise No. 202

Factoring

Factor the numbers from 480 to 503 inclusive in the form shown in the table on page 151.

Exercise No. 203

Mental Division

Divide mentally by 18 the answers to Exercise No. 148 as given on page 170 and 171. Compare your answers with Table I on page 7.

Exercise No. 204

Mental Multiplication

Multiply mentally by 20 the numbers in Table I on page 7.

Exercise No. 205

Continuous Addition Drill

Count by 18’s to 450.

Count by 19’s to 475.

Count by 21’s to 525.

Count by 22’s to 550.

Count by 23’s to 575.

Count by 24’s to 600.

Count by 25’s to 625.

Repeat this exercise three times.

Exercise No. 206

Factoring

Factor the numbers from 492 to 515 inclusive in the form shown in the table on page 151.

Exercise No. 207

Continuous Addition Drill

Count by 19’s to 475.

Count by 21’s to 525.

Count by 22’s to 550.

Count by 23’s to 575.

Count by 24’s to 600.

Count by 25’s to 625.

Repeat this exercise three times.

Exercise No. 208

Mental Multiplication

Multiply mentally by 30 the numbers in Table I on page 7.

Exercise No. 209

Factoring

Factor the numbers from 504 to 527 inclusive in the form shown in the table on page 151.

Exercise No. 210

Mental Division

Divide mentally by 19 the answers to Exercise No. 149 as given on page 171. Compare your answers with Table I on page 7.

Exercise No. 211

Continuous Addition Drill

Count by 21’s to 525.

Count by 22’s to 550.

Count by 23’s to 575.

Count by 24’s to 600.

Count by 25’s to 625.

Repeat this exercise three times.

Exercise No. 212

Mental Multiplication

Multiply mentally by 40 the numbers in Table I on page 7.

Exercise No. 213

Factoring

Factor the numbers from 516 to 539 inclusive in the form shown in the table on page 151.

Exercise No. 214

Continuous Addition Drill

Count by 22’s to 550.

Count by 23’s to 575.

Count by 24’s to 600.

Count by 25’s to 625.

Repeat this exercise three times.

Exercise No. 215

Mental Multiplication

Multiply mentally by 50 the numbers in Table I on page 7.

Exercise No. 216

Factoring

Factor the numbers from 528 to 551 inclusive in the form shown in the table on pages 151 and 152.

Exercise No. 217

Continuous Addition Drill

Count by 23’s to 575.

Count by 24’s to 600.

Count by 25’s to 625.

Repeat this exercise three times.

Exercise No. 218

Mental Division

Divide mentally by 20 the answers to Exercise No. 165 as given on page 172. Compare your answers with Table I on page 7.

Exercise No. 219

Mental Multiplication

Multiply mentally by 60 the numbers in Table I on page 7.

Exercise No. 220

Factoring

Factor the numbers from 540 to 564 inclusive in the form shown in the table on page 152.

Exercise No. 221

Continuous Addition Drill

Count by 24’s to 600.

Count by 25’s to 625.

Repeat this exercise three times.

Exercise No. 222

Mental Multiplication

Multiply mentally by 70 the numbers in Table I on page 7.

Exercise No. 223

Factoring

Factor the numbers from 552 to 576 inclusive in the form shown in the table on page 152.

Exercise No. 224

Mental Division

Divide mentally by 21 the answers to Exercise No. 172 as given on page 172. Compare your answers with Table I on page 7.

Exercise No. 225

Continuous Addition Drill

Count by 25’s to 625.

Repeat this exercise three times.

Exercise No. 226

Mental Multiplication

Multiply mentally by 80 the numbers in Table I on page 7.

Exercise No. 227

Factoring

Factor the numbers from 565 to 592 inclusive in the form shown in the table on page 152.

Exercise No. 228

Mental Multiplication

Multiply mentally by 90 the numbers in Table I on page 7.

Exercise No. 229

Multiplying Three Figures by One

We are now ready to start the mental multiplication of numbers of three places by numbers of one place. Work from left to right. Immediately name the first partial product as hundreds or thousands. Thus, taking the fourth example, this would be calculated as 800, 900, 902. The fifth example would be figured as 1000, 1120, 1124.

When dealing with numbers in the thousands be sure to consider the thousands as such and not as so many hundreds. If you wish, however, you may shorten the terminology. You may, for instance, think of one thousand one hundred twenty-six simply as one, one twenty-six, or as one, one two six.

Perform mentally the following multiplications.

1. 121 × 2

2. 232 × 2

3. 343 × 2

4. 451 × 2

5. 562 × 2

6. 623 × 2

7. 731 × 2

8. 842 × 2

9. 953 × 2

10. 161 × 2

11. 222 × 2

12. 333 × 2

13. 441 × 2

14. 552 × 2

15. 663 × 2

16. 721 × 2

17. 832 × 2

18. 943 × 2

19. 151 × 2

20. 262 × 2

Exercise No. 230

Factoring

Factor the numbers from 577 to 605 inclusive in the form shown in the table on page 152.

Exercise No. 231

Mental Division

Divide mentally by 22 the answers to Exercise No. 179 as given on page 173. Compare your answers with Table I on page 7.

Exercise No. 232

Mental Multiplication

Multiply mentally by 110 the numbers in Table I on page 7.

Exercise No. 233

Multiplying Three Figures by One

Perform mentally the following multiplications.

1. 131 × 3

2. 242 × 3

3. 353 × 3

4. 464 × 3

5. 571 × 3

6. 632 × 3

7. 743 × 3

8. 854 × 3

9. 961 × 3

10. 172 × 3

11. 233 × 3

12. 344 × 3

13. 451 × 3

14. 562 × 3

15. 673 × 3

16. 734 × 3

17. 841 × 3

18. 952 × 3

19. 163 × 3

20. 274 × 3

Exercise No. 234

Factoring

Factor the numbers from 593 to 625 inclusive in the form shown in the table on pages 152 and 153.

Exercise No. 235

Mental Division

Divide mentally by 23 the answers to Exercise No. 186 as given on pages 173 and 174. Compare your answers with Table I on page 7.

Exercise No. 236

Mental Multiplication

Multiply mentally by 120 the numbers in Table I on page 7.

Exercise No. 237

Multiplying Three Figures by One

Perform mentally the following multiplications.

1. 141 × 4

2. 252 × 4

3. 363 × 4

4. 474 × 4

5. 585 × 4

6. 641 × 4

7. 752 × 4

8. 863 × 4

9. 974 × 4

10. 185 × 4

11. 241 × 4

12. 352 × 4

13. 463 × 4

14. 574 × 4

15. 685 × 4

16. 741 × 4

17. 852 × 4

18. 963 × 4

19. 174 × 4

20. 285 × 4

Exercise No. 238

Mental Division

Divide mentally by 24 the answers to Exercise No. 193 as given on page 174. Compare your answers with Table I on page 7.

Exercise No. 239

Mental Multiplication

Multiply mentally by 130 the numbers in Table I on page 7.

Exercise No. 240

Multiplying Three Figures by One

Perform mentally the following multiplications.

1. 151 × 5

2. 262 × 5

3. 373 × 5

4. 484 × 5

5. 595 × 5

6. 656 × 5

7. 761 × 5

8. 872 × 5

9. 983 × 5

10. 194 × 5

11. 255 × 5

12. 366 × 5

13. 471 × 5

14. 582 × 5

15. 693 × 5

16. 754 × 5

17. 865 × 5

18. 976 × 5

19. 181 × 5

20. 292 × 5

Exercise No. 241

Mental Division

Divide mentally by 25 the answers to Exercise No. 200 as given on pages 174 and 175. Compare your answers with Table I on page 7.

Exercise No. 242

Mental Multiplication

Multiply mentally by 140 the numbers in Table I on page 7.

Exercise No. 243

Multiplying Three Figures by One

Perform mentally the following multiplications.

1. 141 × 6

2. 252 × 6

3. 363 × 6

4. 474 × 6

5. 585 × 6

6. 696 × 6

7. 747 × 6

8. 851 × 6

9. 962 × 6

10. 173 × 6

11. 284 × 6

12. 395 × 6

13. 446 × 6

14. 557 × 6

15. 661 × 6

16. 772 × 6

17. 883 × 6

18. 994 × 6

19. 145 × 6

20. 256 × 6

Exercise No. 244

Mental Multiplication

Multiply mentally by 150 the numbers in Table I on page 7.

Exercise No. 245

Multiplying Three Figures by One

Perform mentally the following multiplications.

1. 131 × 7

2. 242 × 7

3. 353 × 7

4. 464 × 7

5. 575 × 7

6. 686 × 7

7. 797 × 7

8. 838 × 7

9. 941 × 7

10. 152 × 7

11. 263 × 7

12. 374 × 7

13. 485 × 7

14. 596 × 7

15. 637 × 7

16. 748 × 7

17. 851 × 7

18. 962 × 7

19. 173 × 7

20. 284 × 7

Exercise No. 246

Mental Multiplication

Multiply mentally by 160 the numbers in Table I on page 7.

Exercise No. 247

Multiplying Three Figures by One

Perform mentally the following multiplications.

1. 141 × 8

2. 252 × 8

3. 363 × 8

4. 474 × 8

5. 585 × 8

6. 696 × 8

7. 747 × 8

8. 858 × 8

9. 969 × 8

10. 171 × 8

11. 282 × 8

12. 393 × 8

13. 444 × 8

14. 555 × 8

15. 666 × 8

16. 777 × 8

17. 888 × 8

18. 999 × 8

19. 741 × 8

20. 652 × 8

FRACTIONS IN GENERAL

The multiplication or the division of fractions will present no difficulty to the student of these pages since it is simply a matter of combining operations in which he is well practised.

What needs more particular attention is the addition and subtraction of the kinds of fractions most commonly encountered in practical work in office, shop and home. The average person would immediately reach for a pencil if asked the sum of and or the difference between 1 and . Yet a little practice with calculations of this kind makes it very easy to perform them mentally.

The succeeding examples in addition and subtraction of fractions are based on the possible combinations of two fractions of the-orders of halves, quarters, eighths, sixteenths, thirds, sixths, twelfths, fifths and tenths.

These exercises are to stimulate memory and rapid thinking. No instructions are given as to how to perform them because it is assumed that the student is familiar with the reduction of fractions to a common denominator.

Exercise No. 248

Reduction of Fractions

1. Reduce to eighths:

2. Reduce to sixteenths:

3. Reduce to sixths:

4. Reduce to twelfths:

5. Reduce to twenty-fourths:

6. Reduce to tenths:

7. Reduce to twentieths:

8. Reduce to fortieths:

9. Reduce to fifteenths:

10. Reduce to thirtieths:

Exercise No. 249

Mental Multiplication

Multiply mentally by 170 the numbers in Table I on page 7.

Exercise No. 250

Addition of Fractions

Add the following mentally.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

Exercise No. 251

Multiplying Three Figures by One

1. 152 × 9

2. 263 × 9

3. 374 × 9

4. 485 × 9

5. 96 × 9

6. 647 × 9

7. 758 × 9

8. 869 × 9

9. 973 × 9

10. 184 × 9

11. 295 × 9

12. 346 × 9

13. 457 × 9

14. 568 × 9

15. 679 × 9

16. 784 × 9

17. 895 × 9

18. 946 × 9

19. 157 × 9

20. 268 × 9

Exercise No. 252

Mental Division

Divide mentally by 2 the answers to Exercise No. 229 as given on page 175.

Exercise No. 253

Addition of Fractions

Do the last thirty examples in Exercise No. 250 on the preceding page, and also add the following.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 254

Mental Multiplication

Multiply mentally by 180 the numbers in Table I on page 7.

Exercise No. 255

Mental Division

Divide mentally by 3 the answers to Exercise No. 233 as given on page 175. Compare your answers with Exercise No. 233.

Exercise No. 256

Addition of Fractions

Review the last twenty examples in Exercise No. 250 on page 97 and those in Exercise No. 253 on page 98 . Also add the following.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 257

Mental Multiplication

Multiply mentally by 190 the numbers in Table I on page 7.

Exercise No. 258

Mental Division

Divide mentally by 4 the answers to Exercise No. 237 as given on page 175.

Exercise No. 259

Addition of Fractions

Review the last ten examples in Exercise No. 250 on page 97 , as well as those in Exercise No. 253 on page 98 and Exercise No. 256 on page 98 . Also add the following.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 260

Mental Multiplication

Multiply mentally by 200 the numbers in Table I on page 7 .

Exercise No. 261

Addition of Fractions

Review the examples in Exercise No. 253 on page 98 , No. 256 on page 98 and No. 259 above. Also add the following.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 262

Mental Division

Divide mentally by 5 the answers to Exercise No. 240 as given on page 175.

Exercise No. 263

Subtraction of Fractions

Perform mentally the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

Exercise No. 264

Mental Multiplication

Multiply mentally by 210 the numbers in Table I on page 7.

Exercise No. 265

Subtraction of Fractions

Review the last twenty examples in Exercise No. 263 above, and also perform the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 266

Mental Division

Divide mentally by 6 the answers to Exercise No. 243 as given on page 175.

Exercise No. 267

Addition of Fractions

Review the examples in Exercise No. 256 on page 98 , No. 259 on page 99 and No. 261 on page 99 . Also perform the following additions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 268

Mental Multiplication

Multiply mentally by 220 the numbers in Table I on page 7.

Exercise No. 269

Subtraction of Fractions

Review the last ten examples in Exercise No. 263 on page 100 and No. 265 on page 100. Also perform the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 270

Mental Division

Divide mentally by 7 the answers to Exercise No. 245 as given on page 176.

Exercise No. 271

Addition of Fractions

Review the examples in Exercise No. 259 on page 99 , No. 261on page 99 and No. 267 on page 101. Also perform the following additions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 272

Mental Multiplication

Multiply mentally by 230 the numbers in Table I on page 7.

Exercise No. 273

Subtraction of Fractions

Review the examples in Exercise No. 265 on page 100 and No. 269 on page 101. Also perform the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 274

Mental Division

Divide mentally by 8 the answers to Exercise No. 247 as given on page 176.

Exercise No. 275

Addition of Fractions

Review the examples in Exercise No. 261 on page 99 , No. 267 on page 101 and No. 271 on this page. Also perform the following additions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 276

Mental Multiplication

Multiply mentally by 240 the numbers in Table I on page 7.

Exercise No. 277

Subtraction of Fractions

Review the examples in Exercise No. 269 on page 101 and No. 273 on page 102. Also perform the following.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 278

Mental Division

Divide mentally by 9 the answers to Exercise No. 251 as given on page 176.

Exercise No. 279

Addition of Fractions

Review the examples in Exercise No. 267 on page 101, No. 271 on page 102 and No. 275 on this page. Also perform the following additions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 280

Mental Multiplication

Multiply mentally by 250 the numbers in Table I on page 7.

Exercise No. 281

Subtraction of Fractions

Review the examples in Exercise No. 273 on page 102 and No. 277 on page 103.Also perform the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 282

Mental Division

Divide mentally the following. Express remainders as such instead of as fractions.

1. 328 ÷ 121

2. 593 ÷ 232

3. 794 ÷ 343

4. 1249 ÷ 451

5. 1580 ÷ 562

6. 1835 ÷ 623

7. 1774 ÷ 731

8. 1786 ÷ 842

9. 2114 ÷ 953

10. 439 ÷ 161

11. 406 ÷ 131

12. 776 ÷ 242

13. 1164 ÷ 353

14. 1574 ÷ 464

15. 1998 ÷ 571

16. 690 ÷ 141

17. 1208 ÷ 252

18. 1704 ÷ 363

19. 2178 ÷ 474

20. 2620 ÷ 585

Exercise No. 283

Addition of Fractions

Review the examples in Exercise No. 271 on page 102, No. 275 on page 103 and No. 279 on page 103. Also perform the following additions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 284

Multiplying Two Figures by Two

With this exercise we start the general multiplication of two numbers of two places each. You have had some experience with such numbers in using the numbers up to 25 as direct multipliers. In the succeeding exercises, however, the multipliers are greater than 25 and the operation is performed differently.

Multiply the whole of the multiplicand by the first figure of the multiplier; next multiply the whole of the multiplicand by the second figure of the multiplier; and finally add the two partial products.

When you multiply the first figure of the multiplicand by the first figure of the multiplier you will get a number of either three places, as in the first example (where 20 × 40 produces 800), or four places, as in the second example (where 2 × 5 produces 10). Add to this first result as you work along from left to right. Similarly, when you multiply the first figure of the multiplicand by the second figure of the multiplier, you will get a number of either two or three places.

Repeat to yourself the original example and the partial products as often as you find necessary. The need for such repetitions will grow less as you become more practised.

Taking the first example: repeat, 41 × 26, 41 × 26, 41 × 26. 40 × 20 is 800, 1 × 2 is 2, 820. (say 1 × 2 rather than 1 × 20 because the former method is simpler when dealing with large numbers. When you think of the 2 as following the 8 it of course becomes a 20 in the product.) Repeat 820, 820, 820. 40 × 6 is 240, 1 × 6 is 6, 246. Repeat 820 + 246, 820 + 246, 820 + 246. Add: 1020, 1060, 1066.

The second example is performed: 1000, 1020; 350, 357. 1020 + 357, 1320, 1370, 1377.

Most of the examples in this exercise are very simple and there can be no objection to your shortening the method given, which is a general method applicable to increasingly larger numbers. Thus in the examples illustrated you should be able to note at a glance that the first partial products are 820 and 1020.

1. 41 × 26

2. 51 × 27

3. 61 × 28

4. 71 × 29

5. 81 × 31

6. 91 × 32

7. 31 × 33

8. 41 × 34

9. 51 × 26

10. 61 × 27

11. 71 × 28

12. 81 × 29

13. 91 × 31

14. 31 × 32

15. 41 × 33

16. 51 × 34

17. 61 × 26

18. 71 × 27

19. 81 × 28

20. 91 × 29

Exercise No. 285

Subtraction of Fractions

Review the examples in Exercise No. 277 on page 103 and No. 281 on page 104. Also perform the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 286

Mental Division

Divide mentally the following.

1. 445 ÷ 222

2. 695 ÷ 333

3. 1258 ÷ 441

4. 1655 ÷ 552

5. 1700 ÷ 663

6. 2274 ÷ 632

7. 2747 ÷ 743

8. 3242 ÷ 854

9. 3747 ÷ 961

10. 533 ÷ 172

11. 2830 ÷ 641

12. 3233 ÷ 752

13. 3624 ÷ 863

14. 3989 ÷ 974

15. 902 ÷ 185

16. 845 ÷ 151

17. 1440 ÷ 262

18. 2013 ÷ 373

19. 2564 ÷ 484

20. 3094 ÷ 595

Exercise No. 287

Addition of Fractions

Review the examples in Exercise No. 275 on page 103, No. 279 on page 103 and No. 283 on page 104.Also perform the following additions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 288

Multiplying Two Figures by Two

In doing exercises of this type always use the second number as the multiplier. Using the first example to illustrate, find 30 times 42 and then 5 times 42; do not work the other way around by finding 40 times 35 and then 2 times 35. This caution is given because of the special way in which the exercises are graded.

1. 42 × 35

2. 52 × 36

3. 62 × 37

4. 72 × 38

5. 82 × 39

6. 92 × 41

7. 32 × 42

8. 42 × 43

9. 52 × 35

10. 62 × 36

11. 72 × 37

12. 82 × 38

13. 92 × 39

14. 32 × 41

15. 42 × 42

16. 52 × 43

17. 62 × 34

18. 72 × 35

19. 82 × 36

20. 92 × 37

Exercise No. 289

Subtraction of Fractions

Review the examples in Exercise No. 277 on page 103 and No. 281 on page 104. Also perform the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 290

Mental Division

1. 1479 ÷ 721

2. 2435 ÷ 832

3. 2036 ÷943

4. 387 ÷ 151

5. 623 ÷ 262

6. 745 ÷ 233

7. 1134 ÷ 344

8. 1523 ÷ 451

9. 1966 ÷ 562

10. 2421 ÷ 673

11. 1156 ÷ 211

12. 1643 ÷ 352

13. 2128 ÷ 463

14. 2581 ÷ 574

15. 3012 ÷ 685

16. 3347 ÷ 656

17. 4498 ÷ 761

18. 4924 ÷ 872

19. 5547 ÷ 983

20. 1067 ÷ 194

Exercise No. 291

Addition of Fractions

Review the examples in Exercise No. 279 on page 103, No. 283 on page 104 and No. 287 on page 107. Also perform the following additions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 292

Mental Multiplication

Multiply mentally the following.

1. 43 × 44

2. 53 × 45

3. 63 × 46

4. 73 × 47

5. 83 × 48

6. 93 × 49

7. 33 × 51

8. 43 × 52

9. 53 × 44

10. 63 × 45

11. 73 × 46

12. 83 × 47

13. 93 × 48

14. 33 × 49

15. 43 × 51

16. 53 × 52

17. 63 × 44

18. 78 × 45

19. 83 × 46

20. 93 × 47

Exercise No. 293

Subtraction of Fractions

Review the examples in Exercise No. 281 on page 104 and No. 289 on page 108. Also do the following.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 294

Mental Division

Divide mentally the following.

1. 444 ÷ 131

2. 795 ÷ 242

3. 1154 ÷ 353

4. 1424 ÷ 464

5. 1767 ÷ 571

6. 3186 ÷ 740

7. 3493 ÷ 852

8. 4716 ÷ 963

9. 815 ÷ 174

10. 1348 ÷ 285

11. 1421 ÷ 255

12. 2118 ÷ 366

13. 2676 ÷ 471

14. 3375 ÷ 582

15. 3573 ÷ 693

16. 971 ÷ 141

17. 1712 ÷ 252

18. 2255 ÷ 363

19. 2955 ÷ 474

20. 3820 ÷ 585

Exercise No. 295

Addition of Fractions

Review the examples in Exercise No. 279 on page 103, No. 283 on page 104 and No. 292 on page 108. Also perform the following additions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 296

Mental Multiplication

Multiply mentally the following.

1. 44 × 53

2. 54 × 54

3. 64 × 55

4. 74 × 56

5. 84 × 57

6. 94 × 58

7. 34 × 59

8. 44 × 61

9. 54 × 53

10. 64 × 54

11. 74 × 55

12. 84 × 56

13. 94 × 57

14. 34 × 58

15. 44 × 59

16. 59 × 61

17. 64 × 53

18. 74 × 54

19. 84 × 55

20. 94 × 56

Exercise No. 297

Subtraction of Fractions

Review the examples in Exercise No. 289 on page 108 and No. 293 on page 109. Also perform the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 298

Mental Division

Divide mentally the following.

1. 3989 ÷ 754

2. 4967 ÷ 865

3. 5192 ÷ 976

4. 1002 ÷ 181

5. 1566 ÷ 292

6. 4486 ÷ 696

7. 4632 ÷ 747

8. 5206 ÷ 851

9. 6381 ÷ 962

10. 1153 ÷ 173

11. 982 ÷ 131

12. 1829 ÷ 242

13. 2706 ÷ 353

14. 3433 ÷ 464

15. 4089 ÷ 575

16. 1200 ÷ 141

17. 2141 ÷ 252

18. 3084 ÷ 363

19. 4152 ÷ 474

20. 5101 ÷ 585

Exercise No. 299

Addition of Fractions

Review the examples in Exercise No. 283 on page 104, No. 292 on page 108 and No. 295 on page 109. Also perform the following additions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 300

Mental Multiplication

Multiply mentally the following.

1. 45 × 62

2. 55 × 63

3. 65 × 64

4. 75 × 65

5. 85 × 66

6. 95 × 67

7. 35 × 68

8. 45 × 69

9. 55 × 62

10. 65 × 63

11. 75 × 64

12. 85 × 65

13. 95 × 66

14. 35 × 67

15. 45 × 68

16. 55 × 69

17. 65 × 62

18. 75 × 63

19. 85 × 64

20. 95 × 65

Exercise No. 301

Subtraction of Fractions

Review the examples in Exercise No. 293 on page 109 and No. 297 on page 110. Also perform the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 302

Mental Division

Divide mentally the following.

1. 1714 ÷ 284

2. 2399 ÷ 395

3. 2714 ÷ 446

4. 3507 ÷ 557

5. 4617 ÷ 661

6. 5303 ÷ 686

7. 5886 ÷ 797

8. 6665 ÷ 838

9. 7233 ÷ 941

10. 1084 ÷ 152

11. 5757 ÷ 696

12. 6588 ÷ 747

13. 7189 ÷ 858

14. 8238 ÷ 969

15. 1385 ÷ 171

16. 1493 ÷ 152

17. 2502 ÷ 263

18. 3440 ÷ 374

19. 4450 ÷ 485

20. 5423 ÷ 596

Exercise No. 303

Addition of Fractions

Review the examples in Exercise No. 292 on page 108, No. 295 on page 109 and No. 299 on page 111. Also perform the following additions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 304

Mental Multiplication

Multiply mentally the following.

1. 46 × 71

2. 56 × 72

3. 66 × 73

4. 76 × 74

5. 86 × 75

6. 96 × 76

7. 36 × 77

8. 46 × 78

9. 56 × 71

10. 66 × 72

11. 76 × 73

12. 86 × 74

13. 96 × 75

14. 36 × 76

15. 46 × 77

16. 56 × 78

17. 66 × 71

18. 76 × 72

19. 86 × 73

20. 96 × 74

Exercise No. 305

Subtraction of Fractions

Review the examples in Exercise No. 297 on page 110 and No. 301 on page 111. Also perform the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 306

Mental Division

Divide mentally the following.

1. 5338 ÷ 772

2. 5393 ÷ 883

3. 6001 ÷ 994

4. 908 ÷ 145

5. 1576 ÷ 256

6. 1859 ÷ 263

7. 2736 ÷ 374

8. 3606 ÷ 485

9. 4518 ÷ 596

10. 4711 ÷ 637

11. 2284 ÷ 282

12. 3183 ÷ 393

13. 3956 ÷ 444

14. 4795 ÷ 555

15. 5954 ÷ 666

16. 5887 ÷ 647

17. 7123 ÷ 758

18. 8221 ÷ 869

19. 9257 ÷ 973

20. 1721 ÷ 184

Exercise No. 307

Addition of Fractions

Review the examples in Exercise No. 295 on page 109, No. 297 on page 110 and No. 303 on page 112. Also perform the following additions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 308

Mental Multiplication

Perform mentally the following multiplications.

1. 47 × 79

2. 57 × 81

3. 67 × 82

4. 77 × 83

5. 87 × 84

6. 97 × 85

7. 37 × 86

8. 47 × 87

9. 57 × 79

10. 67 × 81

11. 77 × 82

12. 87 × 83

13. 97 × 84

14. 37 × 85

15. 47 × 86

16. 57 × 87

17. 67 × 79

18. 77 × 81

19. 87 × 82

20. 97 × 83

Exercise No. 309

Subtraction of Fractions

Review the examples in Exercise No. 301 on page 111 and No. 305 on page 112. Also perform the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 310

Mental Division

Divide mentally the following.

1. 5365 ÷ 748

2. 6599 ÷ 851

3. 7445 ÷ 962

4. 1243 ÷ 173

5. 2220 ÷ 284

6. 6293 ÷ 777

7. 7548 ÷ 888

8. 8304 ÷ 999

9. 6075 ÷ 741

10. 5241 ÷ 652

11. 2682 ÷ 295

12. 3411 ÷ 346

13. 4471 ÷ 457

14. 5667 ÷ 568

15. 6720 ÷ 679

16. 7831 ÷ 784

17. 8917 ÷ 895

18. 9441 ÷ 946

19. 1563 ÷ 157

20. 2627 ÷ 268

Exercise No. 311

Addition of Fractions

Review the examples in Exercise No. 297 on page 110, No. 303 on page 112 and No. 307 on page 113. Also add the following.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 312

Mental Multiplication

Multiply mentally the following.

1. 48 × 88

2. 58 × 89

3. 68 × 91

4. 78 × 92

5. 88 × 93

6. 98 × 94

7. 38 × 95

8. 48 × 96

9. 58 × 88

10. 68 × 89

11. 78 × 91

12. 88 × 92

13. 98 × 93

14. 38 × 94

15. 48 × 95

16. 58 × 96

17. 68 × 88

18. 78 × 89

19. 88 × 91

20. 98 × 92

Exercise No. 313

Subtraction of Fractions

Review the examples in Exercise No. 305 on page 112 and No. 309 on page 114. Also perform the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 314

Addition of Fractions

Review the examples in Exercise No. 303 on page 112, No. 307 on page 113 and No. 311 on page 114. Also perform the following additions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 315

Mental Multiplication

Multiply the following mentally.

1. 49 × 95

2. 59 × 96

3. 69 × 97

4. 79 × 98

5. 89 × 99

6. 99 × 95

7. 39 × 96

8. 49 × 97

9. 59 × 98

10. 69 × 99

11. 79 × 95

12. 89 × 96

13. 99 × 97

14. 39 × 98

15. 49 × 99

16. 59 × 95

17. 69 × 96

18. 79 × 97

19. 89 × 98

20. 99 × 99

Exercise No. 316

Subtraction of Fractions

Review the examples in Exercise No. 309 on page 114 and No. 313 on page 115. Also perform the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 317

Addition of Fractions

Review the examples in Exercise No. 307 on page 113,No. 311 on page 114 and No. 314 on page 115. Also perform the following additions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 318

Subtraction of Fractions

Review the examples in Exercise No. 313 on page 115 and No. 316 on this page. Also perform the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 319

Mental Division

Divide the following mentally.

1. 1066 ÷ 26

2. 1377 ÷ 27

3. 1708 ÷ 28

4. 2059 ÷ 29

5. 2511 ÷ 31

6. 2912 ÷ 32

7. 1023 ÷ 33

8. 1394 ÷ 34

9. 1326 ÷ 26

10. 1647 ÷ 27

11. 1988 ÷ 28

12. 2349 ÷ 29

13. 2821 ÷ 31

14. 992 ÷ 32

15. 1353 ÷ 33

16. 1734 ÷ 34

17. 1586 ÷ 26

18. 1917 ÷ 27

19. 2268 ÷ 28

20. 2639 ÷ 29

Exercise No. 320

Addition of Fractions

Review the examples in Exercise No. 311 on page 114, No. 314 on page 115 and No. 315 on page 115. Also perform the following additions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 321

Subtraction of Fractions

Review the examples in Exercise No. 314 on page 115, No. 316 on page 116 and No. 320 above. Also perform the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 322

Mental Division

Divide the following mentally.

1. 1470 ÷ 35

2. 1872 ÷ 36

3. 2294 ÷ 37

4. 2736 ÷ 38

5. 3198 ÷ 39

6. 3772 ÷ 41

7. 1344 ÷ 42

8. 1806 ÷ 43

9. 1820 ÷ 35

10. 2232 ÷ 36

11. 2664 ÷ 37

12. 3116 ÷ 38

13. 3588 ÷ 39

14. 1312 ÷ 41

15. 1764 ÷ 42

16. 2236 ÷ 43

17. 2108 ÷ 34

18. 2520 ÷ 35

19. 2952 ÷ 36

20. 3404 ÷ 37

Exercise No. 323

Addition of Fractions

Review the examples in Exercise No. 314 on page 115, No. 317 on page 116 and No. 320 on page 117. Also perform the following additions.

1.

2.

3.

4.

5.

6.

Exercise No. 324

Subtraction of Fractions

Review the examples in Exercise No. 318 on page 116 and No. 321 on page 117. Also perform the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 325

Mental Division

Divide the following mentally.

1. 1892 ÷ 44

2. 2385 ÷ 45

3. 2898 ÷ 46

4. 3431 ÷ 47

5. 3984 ÷ 48

6. 4557 ÷ 49

7. 1683 ÷ 51

8. 2236 ÷ 52

9. 2332 ÷ 44

10. 2835 ÷ 45

11. 3358 ÷ 46

12. 3901 ÷ 47

13. 4464 ÷ 48

14. 1617 ÷ 49

15. 2193 ÷ 51

16. 2756 ÷ 52

17. 2772 ÷ 44

18. 3285 ÷ 45

19. 3818 ÷ 46

20. 4371 ÷ 47

Exercise No. 326

Addition of Fractions

Review the examples in Exercise No. 317 on page 116, No. 320 on page 117 and No. 323 on this page.

Exercise No. 327

Subtraction of Fractions

Review the examples in Exercise No. 321 on page 117 and No. 324 on page 118. Also perform the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 328

Mental Division

Divide the following mentally.

1. 2332 ÷ 53

2. 2916 ÷ 54

3. 3520 ÷ 55

4. 4144 ÷ 56

5. 4788 ÷ 57

6. 5452 ÷ 58

7. 2006 ÷ 59

8. 2684 ÷ 61

9. 2862 ÷ 53

10. 3456 ÷ 54

11. 4070 ÷ 55

12. 4704 ÷ 56

13. 5358 ÷ 57

14. 1972 ÷ 58

15. 2596 ÷ 59

16. 3294 ÷ 61

17. 3392 ÷ 53

18. 3996 ÷ 54

19. 4620 ÷ 55

20. 5264 ÷ 56

Exercise No. 329

Addition of Fractions

Review the examples in Exercise No. 320 on page 117 and 323 on page 118.

Exercise No. 330

Subtraction of Fractions

Review the examples in Exercise No. 321 on page 117 and No. 324 on page 118. Also perform the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 331

Mental Division

Divide the following mentally.

1. 2790 ÷ 62

2. 3465 ÷ 63

3. 4160 ÷ 64

4. 4875 ÷ 65

5. 5610 ÷ 66

6. 6365 ÷ 67

7. 2380 ÷ 68

8. 3105 ÷ 69

9. 3410 ÷ 62

10. 4095 ÷ 63

11. 4800 ÷ 64

12. 5525 ÷ 65

13. 6270 ÷ 66

14. 2345 ÷ 67

15. 3060 ÷ 68

16. 3795 ÷ 69

17. 4030 ÷ 62

18. 4725 ÷ 63

19. 5440 ÷ 64

20. 6175 ÷ 65

Exercise No. 332

Mental Division

Divide the following mentally.

1. 3266 ÷ 71

2. 4032 ÷ 72

3. 4818 ÷ 73

4. 5624 ÷ 74

5. 6450 ÷ 75

6. 7296 ÷ 76

7. 2772 ÷ 77

8. 3588 ÷ 78

9. 3976 ÷ 71

10. 4752 ÷ 72

11. 5548 ÷ 73

12. 6364 ÷ 74

13. 7200 ÷ 75

14. 2736 ÷ 76

15. 3542 ÷ 77

16. 4368 ÷ 78

17. 4686 ÷ 71

18. 5472 ÷ 72

19. 6278 ÷ 73

20. 7104 ÷ 74

Exercise No. 333

Subtraction of Fractions

Review the examples in Exercise No. 324 on page 118 and No. 330 on page 119. Also perform the following subtractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise No. 334

Mental Division

Divide the following mentally.

1. 3713 ÷ 79

2. 4617 ÷ 81

3. 5494 ÷ 82

4. 6391 ÷ 83

5. 7308 ÷ 84

6. 8245 ÷ 85

7. 3182 ÷ 86

8. 4089 ÷ 87

9. 4503 ÷ 79

10. 5427 ÷ 81

11. 6314 ÷ 82

12. 7221 ÷ 83

13. 8148 ÷ 84

14. 3145 ÷ 85

15. 4042 ÷ 86

16. 4959 ÷ 87

17. 5293 ÷ 79

18. 6237 ÷ 81

19. 7134 ÷ 82

20. 8051 ÷ 83

Exercise No. 335

Subtraction of Fractions

Review the examples in Exercise No. 330 on page 119 and No. 333 on page 120. Also perform the following subtractions.

1.

2.

3.

4.

5.

6.

Exercise No. 336

Mental Division

Divide the following mentally.

1. 4224 ÷ 88

2. 5162 ÷ 89

3. 6188 ÷ 91

4. 7176 ÷ 92

5. 8184 ÷ 93

6. 9212 ÷ 94

7. 3610 ÷ 95

8. 4608 ÷ 96

9. 5104 ÷ 88

10. 6052 ÷ 89

11. 7098 ÷ 91

12. 8096 ÷ 92

13. 9114 ÷ 93

14. 3572 ÷ 94

15. 4560 ÷ 95

16. 5568 ÷ 96

17. 5984 ÷ 88

18. 6942 ÷ 89

19. 8008 ÷ 91

20. 9016 ÷ 92

Exercise No. 337

Mental Division

Divide the following mentally.

1. 4655 ÷ 95

2. 5664 ÷ 96

3. 6693 ÷ 97

4. 7742 ÷ 98

5. 8811 ÷ 99

6. 9405 ÷ 95

7. 3744 ÷ 96

8. 4753 ÷ 97

9. 5782 ÷ 98

10. 6831 ÷ 99

11. 7505 ÷ 95

12. 8544 ÷ 96

13. 9603 ÷ 97

14. 3822 ÷ 98

15. 4851 ÷ 99

16. 5605 ÷ 95

17. 6624 ÷ 96

18. 7663 ÷ 97

19. 8722 ÷ 98

20. 9801 ÷ 99

DECIMALS IN GENERAL

For the purposes of this book our interest in decimals centers in the equivalence of value between certain decimals and common fractions. Decimal parts of a number that may be represented as simple fractions of that number are known as aliquot parts of it. Thus, 12, 25 and 33 are aliquot parts of 100, being respectively equal to , and of 100.

A knowledge of aliquot parts simplifies many arithmetical calculations. Thus if it be required to multiply 7928 by 25, the simplest way is to annex two 0’s to 7928, making it 792800, and then divide by 4, since 25 is of 100. The answer, which may easily be figured mentally, comes to 198200.

Again, if we wanted to know the cost of 25 gross of penholders at 66per dozen, we would figure that 1 gross costs $ × 12, or $8, and that 25 gross therefore cost $200.

Everybody with any degree of arithmetical training or experience is familiar with the equivalent decimal values for halves, quarters, eighths, thirds, sixths, fifths, tenths, twentieths, twenty-fifths and fiftieths. It is not difficult to extend the list of memorized values so as to include sixteenths and twelfths, and with this knowledge to make rapid calculations of values in thirty-seconds and twenty-fourths.

The succeeding exercises in decimals are designed toward this end. The student is drilled in representing the values of various fractions as decimals of an increasingly higher number of places. No tables are given because values are more quickly learned by repeated calculation than by any effort at mere memorization.

Exercise No. 338

Two-Place Decimal Values

Express the following fractions as decimals of two places. Use fractional terminations where necessary. Thus, expressed as a two-place decimal becomes .33.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

Repeat this exercise three times.

Exercise No. 339

Multiplying Three Figures by Two

Multiply mentally the following.

No new principles are involved in multiplications of this type. The student is simply asked to apply the methods which he has already learned to larger numbers.

1. 111 × 26

2. 222 × 27

3. 331 × 28

4. 442 × 29

5. 551 × 31

6. 612 × 32

7. 721 × 33

8. 832 × 34

9. 941 × 26

10. 152 × 27

Exercise No. 340

Two-Place Decimal Values

Review the examples in Exercise No. 338 above.

Express the following as decimals of two places.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

Repeat this exercise three times.

Exercise No. 341

Multiplying Three Figures by Two

Multiply mentally the following.

1. 121 × 35

2. 232 × 36

3. 343 × 37

4. 451 × 38

5. 562 × 39

6. 623 × 41

7. 731 × 42

8. 842 × 43

9. 953 × 35

10. 161 × 36

SHORT CUTS

There are a number of devices for shortening the work of calculation in specific cases, though most of the methods usually included under this head have only a limited practical value because they are applicable only in highly special cases. A few methods, like horizontal addition and combined addition and subtraction have first-class utility. A variety of short cuts of varying degrees of value are given in the following pages without any attempt to classify them. The student should become familiar with all of them because there is always benefit in viewing numbers from as many angles as possible.

Exercise No. 342

Horizontal Addition

The term horizontal addition is applied to the adding of numbers that are not arranged in column form. There is often an unnecessary waste of time in arranging numbers in the form of columns. This is particularly true when the numbers to be added are on bills, invoices, etc. Values on such papers may be totalled by writing down each partial sum as it is arrived at, and then making a final addition.

Consider the first of the following examples. The sum of the units is 37, the sum of the tens is 45, etc. The sums of the various orders are successively set down in the form shown below, and then added.

The process might of course be shortened somewhat by adding two orders at a time.

Add the following.

1. $32 + $183 + $54 + $3486 + $569 + $9375 + $85 + $4103

2. $875 + $284 + $37 + $5200 + $398 + $62 + $74 + $2168 + $720

3. 763 + 827 + 49 + 5283 + 768 + 2175

4. 1536 + 8973 + 5178 + 926 + 8259 + 36 + 867

5. 9365 + 8375 + 1473 + 826 + 4123 + 15378

6. 986 + 325 + 7261 + 5820 + 569 + 8371

7. 6275 + 5183 + 985 + 3267 + 75 + 1528

8. 1738 + 9168 + 8273 + 5298 + 9 + 6832 + 65

9. $783.52 + $41.27 + $837.45 + $9681.73 + $48.26 + $912.78 + $91.75 + $683.12 + $41.83 + $591.87 + $291.83 + $758.32 + $58.67

10. 46235 + 8976 + 5807 + 98397 + 68325 + 892 + 5140 + 6839 + 326 + 2125

Exercise No. 343

Multiplying Three Figures by Two

Multiply mentally the following.

1. 131 × 44

2. 242 × 45

3. 353 × 46

4. 464 × 47

5. 571 × 48

6. 632 × 49

7. 743 × 51

8. 854 × 52

9. 961 × 44

10. 172 × 45

Exercise No. 344

Four-Place Decimal Values

Review the examples in Exercises No. 338 and 340 on page 123.

Express the fractions listed in Exercise No. 340 as decimals of four places. This is done by simply writing the value as parts of 100 of the terminal fractions of the proper two-place decimals. Thus, which is .06 as a two-place decimal, becomes .0625 as a decimal of four places. Again, is .08 or .0833.

Exercise No. 345

Multiplying Three Figures by Two

Multiply mentally the following.

1. 141 × 53

2. 252 × 54

3. 363 × 55

4. 474 × 56

5. 585 × 57

6. 641 × 58

7. 752 × 59

8. 863 × 61

9. 974 × 53

10. 185 × 54

Exercise No. 346

Combined Addition and Subtraction

It sometimes becomes necessary to subtract the sum of several numbers from a single number. If the numbers to be added are arranged in column form, this may be done at what amounts to one operation by a very simple process.

The numbers may be arranged either as a sum with a missing addend, as in the examples given for practice, or else with the minuend written at the top with underscoring and the difference written at the bottom, as in the examples shown for illustration.

The so-called carry method of subtraction is used. The sum of each successive column is subtracted from the corresponding figure of the minuend plus as many tens as may be necessary to make the subtraction possible. The number of tens thus used is then added to the next column.

To illustrate: from 122808 take the sum of 35635, and 68921.

The sum of 5 and 1 is subtracted from 8; write 2 and carry 0. Subtract 5 from 10; write 5 and carry 1 because 1 ten was used to make the subtraction possible. With 1 to carry, the next column adds to 16; subtract this from 18 and again carry 1. The next column adds to 14; subtract this from 22 and carry 2 because 2 tens were needed to make the subtraction possible in this case. Carrying 2 and subtracting from 12 gives the final necessary figure, 1.

The method of carrying may be made still more clear by taking an example that involves larger numbers; from 3744 subtract the sum of 366, 466, 566, 666, 766, 266 and 466.

The sum of the first column, 42, is subtracted from 44 because 44 is the next higher number ending in 4 from which a subtraction can be made; 4 is carried. The sum of the second column, 46, is subtracted from 54 because 54 is the next higher number ending in 4 from which a subtraction can be made; 5 is carried. The sum of the hundreds’ column subtracted from 39 leaves 1.

In the following examples fill in in each case the missing number that will make all the numbers add to the total shown.

1.

2.

3.

4.

5.

6.

7.

8.

Exercise No. 347

Multiplying Three Figures by Two

Multiply mentally the following.

1. 151 × 62

2. 262 × 63

3. 373 × 64

4. 484 × 65

5. 595 × 66

6. 656 × 67

7. 761 × 68

8. 872 × 69

9. 983 × 62

10. 194 × 63

Exercise No. 348

Five-Place Decimal Values

Review the examples in Exercises No. 338 and 340 on page 123 and No. 344 on page 126.

Express the following fractions as decimals of five places.

To find values in thirty-seconds, add .0312 to the next lower value in sixteenths, etc. The calculation is clearer in the mind if both sixteenths and thirty-seconds are first thought of as decimals of four places. Changing the four-place answer to five places is the work of an instant.

To find values in twenty-fourths, add .0416 to the next lower value in twelfths, etc. In writing answers, drop final , and raise final to make the last figure a 7.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

Exercise No. 349

Multiplying Three Figures by Two

Multiply mentally the following.

1. 141 × 71

2. 252 × 72

3. 363 × 73

4. 474 × 74

5. 585 × 75

6. 696 × 76

7. 747 × 77

8. 851 × 78

9. 962 × 71

10. 173 × 72

Exercise No. 350

Multiplying by a Near Number

It sometimes happens that a multiplier is a little more or a little less than 100, 1000, 10000, etc. In cases of this kind it is quickest to multiply by the round number and then add or subtract the necessary difference. For example, multiply $385.20 by 998. We multiply the dollar value by 1000 and subtract from this product twice $385.20, thus:

Multiply the following. The student should be able to do most of these mentally.

1. $425 × 999

2. $865 × 98

3. $735.25 × 998

4. $258.30 × 104

5. $827.58 × 1003

6. $516 × 1.02

7. $989 × 992

8. $99 × 97

9. $1005 × 1002

Exercise No. 351

Multiplying Three Figures by Two

Multiply mentally the following.

1. 131 × 79

2. 242 × 81

3. 353 × 82

4. 464 × 83

5. 575 × 84

6. 686 × 85

7. 797 × 86

8. 838 × 87

9. 941 × 79

10. 152 × 81

Exercise No. 352

Review of Decimals

Review the examples in Exercise No. 340 on page 123, No. 344 on page 126 and No. 348 on page 129.

Exercise No. 353

Multiplying Three Figures by Two

Multiply mentally the following.

1. 141 × 88

2. 252 × 89

3. 363 × 91

4. 474 × 92

5. 585 × 93

6. 696 × 94

7. 747 × 95

8. 858 × 96

9. 969 × 88

10. 171 × 89

Exercise No. 354

Aliquot Parts in Multiplication

Reference has already been made to the fact that multiplication may be simplified by considering one of the factors as an aliquot part of some number ending in two or more 0’s. Thus, 628 × 25 would be solved by multiplying 628 by 100 and dividing by 4; the answer comes to 15700. Again, multiplying 56 × 75 would be done most quickly by taking of 56 and then multiplying by 100.

Perform the following multiplications by the method of aliquot parts.

1. $35 × 15

2. $42 × 18

3. $24 × 16

4. $18 × 45

5. $72 × 75

6. $36 × 25

7. $52 × 250

8. $42 × 350

9. $150 × 48

10. $64 × 25

11. $35 × 18

12. $28 × 450

13. $36 × 33§

14. $72 × 16§

15. $96 × 12§

Exercise No. 355

Multiplying Three Figures by Two

Multiply mentally the following. Do not use short cuts.

1. 152 × 95

2. 263 × 96

3. 374 × 97

4. 485 × 98

5. 596 × 99

6. 647 × 95

7. 758 × 96

8. 869 × 97

9. 973 × 98

10. 194 × 99

Exercise No. 356

Review of Decimals

Review the examples in Exercise No. 344 on page 126 and No. 348 on page 129.

Exercise No. 357

Multiplying Three Figures by Three

Multiply mentally the following. Add together the first two partial products before determining the third.

1. 111 × 101

2. 222 × 111

3. 331 × 121

4. 442 × 131

5. 551 × 141

6. 612 × 151

7. 721 × 161

8. 832 × 171

9. 941 × 181

10. 152 × 191

Exercise No. 358

Simplifying the Multiplier

Sometimes a multiplier is of such a nature that one part of it may be taken as an exact multiple of another. In such cases an operation is eliminated by making a single multiplication of the first-found partial product instead of two multiplications of the original multiplicand. In the example at the left above, the 18 in the multiplier is equal to 3 times the 6. We therefore multiply the first partial product by 3 instead of multiplying the original multiplicand by 18. In the example at the right, 56 being equal to 8 times 7, we multiply first by 8, placing the result in the proper position, and then multiply this partial product by 7.

Multiply the following by this method.

1. $385.85 × 642

2. $742.50 × 328

3. $82615 × 729

4. $4265.25 × 255

5. $9541.12 × 546

6. $172.48 × 763

7. $2153.28 × 18624

8. $530.75 × 16412

Exercise No. 359

Multiplying Three Figures by Three

Multiply mentally the following.

1. 121 × 202

2. 232 × 212

3. 343 × 222

4. 451 × 232

5. 562 × 242

6. 623 × 252

7. 731 × 262

8. 842 × 272

9. 953 × 282

10. 161 × 292

Exercise No. 360

Review of Decimals

Review the examples in Exercise No. 348 on page 129.

Exercise No. 361

Multiplying Three Figures by Three

Multiply mentally the following.

1. 131 × 303

2. 242 × 313

3. 353 × 323

4. 464 × 333

5. 571 × 343

6. 632 × 353

7. 743 × 363

8. 854 × 373

9. 961 × 383

10. 172 × 393

Exercise No. 362

Multiplication by Factoring

When a multiplier can be taken as the product of two factors, it may be quicker to make separate multiplications by each of these factors than to proceed in the ordinary manner. Take the example 632 × 156. In the illustrations below, the one at the left shows the ordinary method. At the right the multiplier is split up into the factors 13 and 12; the multiplicand is multiplied by 13 and the result is then multiplied by 12.

Multiply the following by this method.

1. 759 × 182

2. 684 × 169

3. 327 × 228

4. 656 × 285

5. 309 × 289

6. 728 × 324

7. 542 × 221

8. 327 × 224

9. 986 × 196

Exercise No. 363

Multiplying Three Figures by Three

Multiply mentally the following.

1. 141 × 404

2. 252 × 414

3. 363 × 424

4. 474 × 434

5. 585 × 444

6. 641 × 454

7. 752 × 464

8. 863 × 474

9. 974 × 484

10. 185 × 494

Exercise No. 364

Factors Between 11 and 19

A quick way to calculate the product of two numbers between 11 and 19 is to add the units of one number to the whole of the other, annex 0 and add the product of the units of both numbers. Thus, to multiply 16 × 18: 16 and 8 are 24; call this 240 and add 48, making 288. The same result would be reached by adding 6 to 18.

Multiply by this method:

1. 14 × 15

2. 18 × 19

3. 15 × 17

4. 15 × 16

5. 13 × 15

6. 13 × 19

7. 16 × 17

8. 14 × 16

9. 19 × 19

Exercise No. 365

Multiplying Three Figures by Three

Multiply mentally the following.

1. 151 × 505

2. 262 × 515

3. 373 × 525

4. 484 × 535

5. 595 × 545

6. 656 × 555

7. 761 × 565

8. 872 × 575

9. 983 × 585

10. 194 × 595

Exercise No. 366

Multiplying by 11

When the multiplicand consists of two figures the sum of which is less than 10, the product is found by writing the two figures of the multiplicand with their sum between them. Thus, to multiply 62 by 11 we write 6 and 2 with the sum of 6 and 2 between these figures, obtaining 682.

To multiply larger numbers by 11, apply the following rule. Beginning at the right, write the units’ figure of the multiplicand, then successively the units plus the tens, the tens plus the hundreds, the hundreds plus the thousands, etc., carrying wherever necessary, and ending with the highest order of the multiplicand, or the highest order plus the carrying figure. Thus, to multiply 4762 by 11: write 2; add 2 and 6 and write 8; add 6 and 7, write 3 and carry 1; add 7 and 4, increase it by the 1 carried, write 2 and carry 1; add this 1 to 4 and write 5. Answer, 52382.

Multiply the following by this method.

1. $5136 × 11

2. $72638 × 11

3. $514832 × 11

4. $37281.05 × 11

5. $41268.45 × 11

6. $3275.75 × 11

7. $48263.25 × 11

8. $94873.30 × 11

Exercise No. 367

Multiplying Three Figures by Three

Multiply mentally the following.

1. 141 × 606

2. 252 × 616

3. 363 × 626

4. 474 × 636

5. 585 × 646

6. 696 × 656

7. 747 × 666

8. 851 × 676

9. 962 × 686

10. 173 × 696

Exercise No. 368

Multiplying by 21, 31, 41, etc.

Setting down the product from right to left, write the units’ figure of the multiplicand, then multiply each order of the multiplicand by the tens’ figure of the multiplier, increasing the result in each case by the next higher order of the multiplicand and any necessary carrying figure.

Example, multiply 387 by 41; write 7; multiply 7 by 4, add the 8 of the multiplicand, making 36, write 6 and carry 3; multiply 8 by 4, add the 3 of the multiplicand and the carried 3, making 38, write 8 and carry 3; multiply 3 by 4 and add the carried 3 making 15, write 15. Answer, 15867.

Multiply by this method:

1. $2735.50 × 51

2. $1824.75 × 81

3. $5104.30 × 31

4. $6238.65 × 21

5. $7415.40 × 61

6. $8291.25 × 91

7. $2134.15 × 71

8. $5827.80 × 41

Exercise No. 369

Multiplying Three Figures by Three

Multiply mentally the following.

1. 131 × 707

2. 242 × 717

3. 353 × 727

4. 464 × 737

5. 575 × 747

6. 686 × 757

7. 797 × 767

8. 838 × 777

9. 941 × 787

10. 152 × 797

Exercise No. 370

Squares of Numbers

The square of a number is the number multiplied by itself. Squares may be determined quickly if the given number is considered to be the sum of two numbers. In algebra such a sum would ordinarily be taken as a + b and its square would be a^2 + 2 ab + b^2. In regular arithmetical cases a becomes the tens of the number and b the units. Thus, 25 is 20 + 5, and 146 is 140 + 6. The algebraic formula for the square of the sum of two numbers is expressed as the square of the first plus twice the product of the first by the second plus the square of the second. Thus, 25 squared is 20 × 20 (400) plus 2 × 20 × 5 (200) plus 5 × 5 (25); the total is 625.

In computing squares by this principle you may immediately annex the square of the second to the square of the first, and then add twice the product of the first by the second. Thus in squaring 25 you would immediately say 425, and then add to this 2 × 20 × 5 (200), making 625. In squaring 146 you immediately say 19636 and add to this 2 × 140 × 6 (1680), making 21316. Always allow two places for the square of the second. Thus in squaring 61 the first partial product is 3601, to which 120 is added to make 3721.

In squaring numbers on paper the following method will be found rapid where large numbers are involved. Set the given number down twice as if for regular multiplication. Assuming that it is considered to consist of tens and units, multiply units by units, write units in the result and carry the tens. Add the two given tens together, multiply this sum by the given units, add the carried figure, write tens in the result and carry hundreds. Multiply tens by tens, add the carried figure and write the result.

In the first illustrative example at the left, 7 × 7 = 49, write 9 and carry 4; 6 + 6 = 12, 12 × 7 = 84, 84 + 4 = 88, write 8 and carry 8; 6 × 6 = 36, 36 + 8 = 44.

In the second example, 4 × 4 = 16, write 6 and carry 1; 13 + 13 = 26, 26 × 4 = 104, 104 + 1 = 105, write 5 and carry 10; 13 × 13 = 169, 169 + 10 = 179, write 179.

The third example is worked somewhat differently because here the parts of the number are considered to be 1600 and 13. 13 × 13 = 169, write 69 (two figures) and carry 1; 16 + 16 = 32, 32 × 13 = 416, 416 + 1 = 417, write 17 and carry 4; 16 × 16 = 256, 256 + 4 = 260, write 260.

Find the squares of the following numbers. Do all the examples first by the first method, then by the second method.

1. 74

2. 93

3. 82

4. 64

5. 38

6. 112

7. 124

8. 146

9. 168

10. 197

11. 1112

12. 1213

13. 1314

14. 1516

15. 1719

Exercise No. 371

Multiplying Three Figures by Three

Multiply mentally the following.

1. 141 × 808

2. 252 × 818

3. 363 × 828

4. 474 × 838

5. 585 × 848

6. 696 × 858

7. 747 × 868

8. 858 × 878

9. 969 × 888

10. 171 × 898

Exercise No. 372

Multiplying When Units Are Alike

The following method is a variation of that explained in connection with the squaring of numbers.

In the illustration at the left, 7 × 7 = 49, write 9 and carry 4; 6 + 4 = 10, 10 × 7 = 70, 70 + 4 = 74, write 4 and carry 7; 4 × 6 = 24, 24 + 7 = 31, write 31.

In the illustration at the right, 13 × 13 = 169, write 69 and carry 1; 6 + 9 = 15, 15 × 13 = 195, 195 + 1 = 196, write 96 and carry 1; 6 × 9 = 54, 54 + 1 = 55, write 55.

Perform the following multiplications by this method.

1. 136 × 56

2. 159 × 79

3. 172 × 92

4. 195 × 115

5. 234 × 174

6. 217 × 197

7. 516 × 816

8. 714 × 314

9. 217 × 917

Exercise No. 373

Multiplying Three Figures by Three

1. 152 × 909

2. 263 × 919

3. 374 × 929

4. 485 × 939

5. 596 × 949

6. 647 × 959

7. 758 × 969

8. 869 × 979

9. 973 × 989

10. 184 × 999

Exercise No. 374

Multiplying When Tens or Hundreds Are Alike

This is a variation of the method explained in Exercise No. 372 above.

In the example on page 139,3 × 9 = 27, write 7 and carry 2; 3 + 9 = 12, 12 × 8 = 96, 96 + 2 = 98, write 8 and carry 9; 8 × 8 = 64, 64 + 9 = 73, write 73.

In the example on page 139, 17 × 14 = 238, write 38 and carry 2; 17 + 14 = 31,31 × 7 = 217, 217 + 2 = 219, write 19 and carry 2; 7 × 7 = 49, 49 + 2 = 51, write 51.

Multiply the following by this method.

1. 92 × 93

2. 62 × 65

3. 84 × 87

4. 92 × 97

5. 213 × 215

6. 321 × 312

7. 416 × 418

8. 509 × 519

9. 913 × 917

Exercise No. 375

Square of Numbers Ending in 5

If a number to be squared consists of tens and units, and if the units are 5, then twice the product of the first part by the second is equal to the given number of tens. Thus, in 25 × 25, 20 × 5 × 2 is equal to 20 × 10; in 35 × 35, 30 × 5 × 2 is equal to 30 × 10. Accordingly when dealing with numbers of this type we may at once annex 25 to the product of the given tens multiplied by one more than the given tens. That is to say, 25 × 25 = 625, in which the 6 represents 3 × 2; 35 × 35 = 1225 in which the 12 represents 4 × 3; 45 × 45 = 2025, in which the 20 represents 5 × 4, etc.

Find the squares of the following numbers by this method.

1. 45

2. 55

3. 65

4. 75

5. 85

6. 95

7. 115

8. 135

9. 155

10. 175

11. 195

12. 315

13. 335

14. 355

15. 375

Exercise No. 376

Multiplying Like Tens with Units Making 10

The principle explained above applies to any case in which the tens are alike and the sum of the units is 10. Thus the product of 46 × 44 is 2024. We arrive at this by multiplying 4 × 5, making 20, and writing after this the product of 4 × 6 or 24.

Multiply in this manner the following.

1. 23 × 27

2. 41 × 49

3. 36 × 34

4. 103 × 107

5. 112 × 118

6. 154 × 156

7. 178 × 172

8. 169 × 161

9. 192 × 198

Exercise No. 377

Squaring Numbers Ending in 25

When a number ends in 25, like 725 for instance, we may take it as the sum of two numbers of which one represents hundreds and the other tens and units. In such cases twice the product of the first part by the second is equal to 50 times the first part. The result of this multiplication is a certain number of thousands.

To find the square of 725 we first write 0625 after the square of 7, making 490625. To this we add as many thousands as are represented by 7 × 5. 490625 + 35000 = 525625.

Another method of finding these squares is by setting the numbers down as in the following illustration.

At once write 625 as the square of 25. Multiply 7 by 5, write 5 and carry 3; multiply 7 by 7, add 3, write 52.

Find the square of the following numbers by both of the foregoing methods.

1. 525

2. 625

3. 825

4. 1025

5. 1225

6. 1325

7. 1625

8. 1725

9. 1825

10. 1925

Exercise No. 378

Multiplying a Sum by a Difference

The algebraic product of a + b and a — b is a^2 — b^2. When numbers to be multiplied can be expressed as the sum of and the difference between two numbers, the product equals the square of the first minus the square of the second. Thus 63 × 57 may be expressed as 60 + 3 multiplied by 60 — 3. The product equals 60 × 60 minus 3 × 3. This comes to 3600 — 9 or 3591.

There is no limit to the combinations of numbers for which this principle would hold true, but for practical purposes we may be satisfied to recognize those in which the units add to 10 and the tens have a difference of 1.

Multiply the following by this method.

1. 72 × 68

2. 83 × 77

3. 94 × 86

4. 101 × 119

5. 123 × 137

6. 146 × 154

7. 152 × 168

8. 173 × 187

9. 182 × 198

Exercise No. 379

Multiplying Mixed Numbers with Like Integers

When integers are alike in mixed numbers, as in 9 × 9, their product is found by multiplying one integer by the other plus the sum of the two fractions; to this partial product add that obtained by multiplying together the two fractions.

In the illustrative example at the left, 9 is multiplied by 9 + + , or 10. The product of this is 90, and to 90 is added the product of and , or

In the second example 8 is multiplied by 8 + + , or 9, producing 76. To this is added the product of × , or , making a total of 77

Multiply the following.

1. 9 × 9

2. 10 × 10

3. 12 × 12

4. 18 × 18

5. 3 × 3

6. 60 × 60

7. 40 × 40

8. 25 × 25

9. 5 × 5

10. 8 × 8

11. 6 × 6

12. 12 × 12

Exercise No. 380

Multiplying by a Number Nearly Whole

Sometimes a multiplier lacks a single fractional unit of being a whole number. Examples would be 5, 6 and 7, which respectively lack , and of being 6, 7 and 8. In cases of this kind raise the multiplier to the next larger whole number, and after multiplying the multiplicand by this number, subtract from the product the necessary fractional part of the multiplicand. Thus, to multiply 64 by 3 , we multiply 64 by 4, obtaining 256, and from this we subtract of 64, or 8, arriving at a final result of 248.

Multiply by this method the following.

1. 48 × 5

2. 75 × 10

3. 136 × 6

4. 250 × 3

5. 522 × 4

6. 672 × 8

7. 180 × 7

8. 720 × 2

9. 342 × 9

Exercise No. 381

Aliquot Parts in Division

The method of aliquot parts is as applicable to division as it is to multiplication. In ordinary cases we determine how many times the given divisor is contained exactly in some multiple of 10. We multiply the given dividend by the result of such division, and point off the product decimally in such a way as to express division by the proper multiple of 10. Thus, to divide 1840 by 25, we obtain a multiplier of 4 by dividing 25 into 100. Multiplying 1840 by 4 we get 7360, and dividing this decimally by 100 we obtain 73.60

Another method of using aliquot parts is illustrated by the example shown above. The problem is to divide 6375 by 7. We note that 7 lacks one-third of itself of being 10. We therefore add one-third of itself to 6375 and divide the resulting sum decimally by 10.

Divide by the foregoing methods:

1. 580 ÷ 25

2. 750 ÷ 16

3. 450 ÷ 12

4. 875 ÷ 250

5. 640÷ 125

6. 435÷ 33

7. 1527 ÷ 150

8. 918 ÷ 15

9. 582÷7

Exercise No. 382

Cubes of Numbers

The algebraic formula for the cube of the sum of two numbers, a and b, is a^3 + 3a^2b + 3ab^2 + b^3. This may be expressed as the cube of the first plus three times the square of the first multiplied by the second, plus three times the first multiplied by the square of the second plus the cube of the second.

By applying this formula it is not difficult to calculate mentally the cubes of numbers of two places. Suppose, for instance, that we want to find the cube of 26. We immediately annex the cube of 6 (216) to the cube of 2 (8), obtaining 8216. (Always allow three places for the cube of the second.) Multiplying 3 × 400 (square of 20) × 6, we get 7200, which, added to 8216, makes 15416. Multiplying 3 × 20 × 36 (square of 6) we obtain 2160, which, added to 15416 gives 17576 as the cube of 26.

Cubes may be readily written down from right to left by using a different method.

All the necessary writing is shown on p.144 at the left. The method of making the calculation is analyzed at the right. The cube of 6 is 216, write 6 and carry 21. The square of 6 (36) multiplied by 2 (72) multiplied by 3 (216) plus 21 comes to 237, write 7 and carry 23. The product of 6 times the square of 2 (24) multiplied by 3 (72) plus 23 comes to 95, write 5 and carry 9. The cube of 2 is 8, which, added to 9, makes 17.

Before attempting the examples which follow the student ought to make himself thoroughly familiar with the cubes of the numbers from 1 to 9, so that he will not have to slow up to make such computations in the course of the example.

Find the cubes of the following numbers by both of the foregoing methods.

1. 14

2. 27

3. 33

4. 46

5. 59

6. 62

7. 65

8. 71

9. 73

10. 84

11. 86

12. 88

13. 95

14. 97

15. 99

Exercise No. 383

Algebraic Multiplication

Arithmetical products may be directly written down from right to left by using the method of cross-multiplication employed in algebra. A certain pattern is followed in multiplying each figure by every other figure. The operations are best explained by illustration.

In the example at the left, 7 × 6 = 42, write 2 and carry 4; 4 plus 4 × 6 (28) plus 2 × 7 comes to 42, write 2 and carry 4; 4 plus 4 × 2 is 12, write 12. (It is best to start each part of the calculation with the carried number, which otherwise might not be easy to remember.)

In the second example, multiply 5 × 8; then 4 × 8 and 7 × 5; then 3 × 8, 6 × 5 and 4 × 7; then 3 × 7 and 6 × 4; finally 3 × 6. Carry as may be necessary.

THE ART OF CALCULATION

Table IV

Prime and Composite Numbers

ANSWERS

The references at the head of each section are to the numbers of the exercises.

No. 1

1. 32

2. 30

3. 29

4. 29

5. 29

6. 31

7. 31

8. 18

9. 37

10. 31

11. 25

12. 35

13. 34

14. 29

15. 26

16. 25

17. 30

18. 33

19. 27

20. 30

21. 33

22. 26

23. 28

24. 27

No. 2

12

68

24

80

36

92

48

104

67

23

79

35

91

19

75

31

87

43

99

55

18

74

30

86

42

98

26

82

38

94

50

106

62

25

81

37

93

49

105

33

89

45

101

57

13

69

32

88

44

100

56

40

96

52

108

64

20

76

39

95

51

107

63

47

103

59

15

71

27

83

46

102

58

14

70

54

110

66

22

78

34

90

53

109

65

21

77

61

17

73

29

85

41

97

60

16

72

28

84

No. 3

1. 59

2. 51

3. 56

4. 70

5. 62

6. 55

7. 57

8. 59

9. 53

10. 51

11. 69

12. 58

13. 60

14. 65

15. 59

16. 61

17. 53

18. 53

No. 4

13

69

25

81

37

93

49

105

68

24

80

36

92

20

76

32

88

44

100

56

19

75

31

87

43

99

27

83

39

95

51

107

63

26

82

38

94

50

106

34

90

46

102

58

14

70

33

89

45

101

57

41

97

53

109

65

21

77

40

96

52

108

64

48

104

60

16

72

28

84

47

103

59

15

71

55

111

67

23

79

35

91

54

110

66

22

78

62

18

74

30

86

42

98

61

17

73

29

85

No. 5

14

70

26

82

38

94

50

106

69

25

81

37

93

21

77

33

89

45

101

57

20

76

32

88

44

100

28

84

40

96

52

108

64

27

83

39

95

51

107

35

91

47

103

59

15

71

34

90

46

102

58

42

98

54

110

66

22

78

41

97

53

109

65

49

105

61

17

73

29

85

48

104

60

16

72

56

112

68

24

80

36

92

55

111

67

23

79

63

19

75

31

87

43

99

62

18

74

30

86

No. 6

15

71

27

83

39

95

51

107

70

26

82

38

94

22

78

34

90

46

102

58

21

77

33

89

45

101

29

85

41

97

53

109

65

28

84

40

96

52

108

36

92

48

104

60

16

72

35

91

47

103

59

43

99

55

111

67

23

79

42

98

54

110

66

50

106

62

18

74

30

86

49

105

61

17

73

57

113

69

25

81

37

93

56

112

68

24

80

64

20

76

32

88

44

100

63

19

75

31

87

No. 7

16

72

28

84

40

96

52

108

71

27

83

39

95

23

79

35

91

47

103

59

22

78

34

90

46

102

30

86

42

98

54

110

66

29

85

41

97

53

109

37

93

49

105

61

17

73

36

92

48

104

60

44

100

56

112

68

24

80

43

99

55

111

67

51

107

63

19

75

31

87

50

106

62

18

74

58

114

70

26

82

38

94

57

113

69

25

81

65

21

77

33

89

45

101

64

20

76

32

88

No. 8

(Same as No.l)

No. 9

17

73

29

85

41

97

53

109

72

28

84

40

96

24

80

36

92

48

104

60

23

79

35

91

47

103

31

87

43

99

55

111

67

30

86

42

98

54

110

38

94

50

106

62

18

74

37

93

49

105

61

45

101

57

113

69

25

81

44

100

56

112

68

52

108

64

20

76

32

88

51

107

63

19

75

59

115

71

27

83

39

95

58

114

70

26

82

66

22

78

34

90

46

102

65

21

77

33

89

No. 10

18

74

30

86

42

98

54

110

73

29

85

41

97

25

81

37

93

49

105

61

24

80

36

92

48

104

32

88

44

100

56

112

68

31

87

43

99

55

111

39

95

51

107

63

19

75

38

94

50

106

62

46

102

58

114

70

26

82

45

101

57

113

69

53

109

65

21

77

33

89

52

108

64

20

76

60

116

72

28

84

40

96

59

115

71

27

83

67

23

79

35

91

47

103

66

22

78

34

90

No. 11

(Same as No. 3)

No. 12

19

75

31

87

43

99

55

111

74

30

86

42

98

26

82

38

94

50

106

62

25

81

37

93

49

105

33

89

45

101

57

113

69

32

88

44

100

56

112

40

96

52

108

64

20

76

39

95

51

107

63

47

103

59

115

71

27

83

46

102

58

114

70

54

110

66

22

78

34

90

53

109

65

21

77

61

117

73

29

85

41

97

60

116

72

28

84

68

24

80

36

92

48

104

67

23

79

35

91

No. 13

1. 365

2. 268

3. 371

4. 433

5. 257

6. 327

7. 209

8. 270

9. 287

10. 410

11. 257

12. 404

13. 231

14. 217

15. 311

16. 303

17. 254

18. 237

19. 308

20. 343

21. 350

22. 360

23. 308

24. 271

25. 341

No. 14

20

76

32

88

44

100

56

112

75

31

87

43

99

27

83

39

95

51

107

63

26

82

38

94

50

106

34

90

46

102

58

114

70

33

89

45

101

57

113

41

97

53

109

65

21

77

40

96

52

108

64

48

104

60

116

72

28

84

47

103

59

115

71

55

1ll

67

23

79

35

91

54

110

66

22

78

62

118

74

30

86

42

98

61

117

73

29

85

69

25

81

37

93

49

105

68

24

80

36

92

No. 15

1. 620

2. 777

3. 716

4. 562

5. 432

6. 590

7. 624

8. 716

9. 885

10. 828

11. 424

12. 592

13. 535

14. 656

15. 858

No. 16

21

77

33

89

45

101

57

113

76

32

88

44

100

28

84

40

96

52

108

64

27

83

39

95

51

107

35

91

47

103

59

115

71

34

90

46

102

58

114

42

98

54

110

66

22

78

41

97

53

109

65

49

105

61

117

73

29

85

48

104

60

116

72

56

112

68

24

80

36

92

55

111

67

23

79

63

119

75

31

87

43

99

62

118

74

30

86

70

26

82

38

94

50

106

69

25

81

37

93

No. 17

1. 1059

2. 1055

3. 903

4. 963

5. 897

6. 1113

7. 1067

8. 759

9. 994

10. 932

No. 18

22

78

34

90

46

102

58

114

77

33

89

45

101

29

85

41

97

53

109

65

28

84

40

96

52

108

36

92

48

104

60

116

72

35

91

47

103

59

115

43

99

55

111

67

23

79

42

98

54

110

66

50

106

62

118

74

30

86

49

105

61

117

73

57

113

69

25

81

37

93

56

112

68

24

80

64

120

76

32

88

44

100

63

119

75

31

87

71

27

83

39

95

51

107

70

26

82

38

94

No. 19

1. 12

2. 34

3. 21

4. 56

5. 33

6. 78

7. 12

8. 13

9. 12

10. 21

11. 7

12. 34

13. 52

14. 11

15. 52

No. 20

1. 28

2. 28

3. 12

4. 19

5. 15

6. 26

7. 19

8. 18

9. 48

10. 21

11. 39

12. 17

13. 26

14. 58

15. 28

16. 18

17. 29

18. 19

19. 29

No. 21

23

79

35

91

47

103

59

115

78

34

90

46

102

30

86

42

98

54

110

66

29

85

41

97

53

109

37

93

49

105

61

117

73

36

92

48

104

60

116

44

100

56

112

68

24

80

43

99

55

111

67

51

107

63

119

75

31

87

50

106

62

118

74

58

114

70

26

82

38

94

57

113

69

25

81

65

121

77

33

89

45

101

64

120

76

32

88

72

28

84

40

96

52

108

71

27

83

39

95

No. 22

1. 294

2. 234

3. 414

4. 358

5. 379

6. 381

7. 370

8. 347

9. 221

10. 374

No. 23

1. 521

2. 213

3. 233

4. 321

5. 331

6. 313

7. 252

8. 412

9. 212

10. 130

11. 122

12. 441

13. 432

14. 351

15. 221

No. 24

24

80

36

92

48

104

60

116

79

35

91

47

103

31

87

43

99

55

111

67

30

86

42

98

54

110

38

94

50

106

62

118

74

37

93

49

105

61

117

45

101

57

113

69

25

81

44

100

56

112

68

52

108

64

120

76

32

88

51

107

63

119

75

59

115

71

27

83

39

95

58

114

70

26

82

66

122

78

34

90

46

102

65

121

77

33

89

73

29

85

41

97

53

109

72

28

84

40

96

No. 25

25

81

37

93

49

105

61

117

80

36

92

48

104

32

88

44

100

56

112

68

31

87

43

99

55

111

39

95

51

107

63

119

75

38

94

50

106

62

118

46

102

58

114

70

26

82

45

101

57

113

69

53

109

65

121

77

33

89

52

108

64

120

76

60

116

72

28

84

40

96

59

115

71

27

83

67

123

79

35

91

47

103

66

122

78

34

90

74

30

86

42

98

54

110

73

29

85

41

97

No. 26

1. $655.71

2. $751.32

3. $604.24

4. $577.21

5. $718.69

6. $769.64

7. $488.04

8. $691.93

No. 27

1. 215

2. 415

3. 209

4. 329

5. 778

6. 109

7. 214

8. 248

9. 128

10. 237

11. 403

12. 106

13. 125

14. 125

15. 136

16. 204

17. 109

18. 143

19. 107

20. 308

21. 309

22. 437

23. 722

24. 109

25. 515

26. 209

27. 336

28. 107

29. 868

30. 419

No. 28

26

82

38

94

50

106

62

118

81

37

93

49

105

33

89

45

101

57

113

69

32

88

44

100

56

112

40

96

52

108

64

120

76

39

95

51

107

63

119

47

103

59

115

71

27

83

46

102

58

114

70

54

110

66

122

78

34

90

53

109

65

121

77

61

117

73

29

85

41

97

60

116

72

28

84

68

124

80

36

92

48

104

67

123

79

35

91

75

31

87

43

99

55

111

74

30

86

42

98

No. 29

27

83

39

95

51

107

63

119

82

38

94

50

106

34

90

46

102

58

114

70

33

89

45

101

57

113

41

97

53

109

65

121

77

40

96

52

108

54

120

48

104

60

116

72

28

84

47

103

59

115

71

55

111

67

123

79

35

91

64

110

66

122

78

62

118

74

30

86

42

98

61

117

73

29

85

69

125

81

37

93

49

105

68

124

80

36

92

76

32

88

44

100

56

112

75

31

86

43

99

No. 30

28

84

40

96

52

108

64

120

83

39

95

51

107

35

91

47

103

59

115

71

34

90

46

102

58

114

42

98

54

110

66

112

78

41

97

53

109

65

121

49

105

61

117

73

29

85

48

104

60

116

72

56

112

68

124

80

36

92

55

111

67

123

79

63

119

75

31

87

43

99

62

118

74

30

86

70

126

82

38

94

50

106

69

125

81

37

93

79

33

89

45

101

57

113

76

32

88

44

100

No. 31

1. 621

2. 585

3. 687

4. 647

5. 630

6. 605

7. 570

8. 671

9. 625

10. 624

No. 32

1. 161

2. 292

3. 71

4. 191

5. 171

6. 64

7. 252

8. 197

9. 623

10. 284

11. 94

12. 387

13. 170

14. 61

15. 593

16. 195

17. 394

18. 295

19. 492

20. 681

No. 33

1. 465

2. 579

3. 164

4. 186

5. 153

6. 48

7. 489

8. 186

9. 488

10. 377

11. 329

12. 469

13. 288

14. 56

15. 216

16. 184

17. 249

18. 77

19. 289

20. 169

No. 34

1. $995.69

2. $1044.85

3. $954.07

4. $1002.63

5. $994.32

6. $897.80

7. $1122.66

8. $1051.42

No. 35

1. 395

2. 297

3. 92

4. 299

5. 298

6. 195

7. 298

8. 399

9. 494

10. 497

11. 296

12. 94

13. 495

14. 294

15. 299

16. 198

17. 197

18. 397

19. 293

20. 692

21. 198

22. 294

23. 596

24. 99

25. 395

No. 36

1. 985

2. 987

3. 975

4. 1008

5. 953

6. 1011

7. 1042

8. 1032

9. 1095

10. 1012

No. 37

1. 347

2. 189

3. 349

4. 78

5. 107

6. 259

7. 189

8. 119

9. 66

10. 88

11. 215

12. 178

13. 178

14. 9

15. 227

16. 109

17. 114

18. 249

19. 234

20. 29

21. 298

22. 284

23. 38

24. 376

25. 129

No. 38

1. $42357.49

2. $57112.34

3. $54738.19

4. $62369.15

5. $70468.35

6. $63801.69

No. 39

1. $4.35

2. $5.59

3. $ .94

4. $1.48

5. $6.92

6. $7.63

7. $2.31

8. $6.84

9. $3.70

10. $2.76

11. $2.29

12. $6.76

13. $3.59

14. $5.96

15. $1.56

16. $3.89

17. $2.68

18. $6.92

19. $3.49

20. $5.97

No. 40

(Same as No. 13)

No. 41

1. $95513.02

2. $102635.78

3. $98506.46

4. $117398.69

5. $95153.78

6. $99073.91

No. 42

(Same as No. 39)

No. 43

1. $ .93

2. $1.20

3. $2.81

4. $ .65

5. $1.96

6. $5.84

7. $2.95

8. $1.65

9. $2.24

10. $ .71

11. $1.89

12. $ .73

13. $1.23

14. $1.63

15. $1.71

16. $2.48

17. $1.86

18. $1.94

19. $2.45

20. $1.63

No. 44

(Same as No. 43)

No. 45

2

114

26

138

50

162

74

186

112

24

136

48

160

16

128

40

152

64

176

88

14

126

38

150

62

174

30

142

54

166

78

190

102

28

140

52

164

76

188

44

156

68

180

92

4

116

42

154

66

178

90

58

170

82

194

106

18

130

56

168

80

192

104

72

184

96

8

120

32

144

70

182

94

6

118

86

198

110

22

134

46

158

84

196

108

20

132

100

12

124

36

148

60

172

98

10

122

34

146

No. 46

3

171

39

207

75

243

111

279

168

36

204

72

240

24

192

60

228

96

264

132

21

189

57

225

93

261

45

213

81

249

117

285

153

42

210

78

246

114

282

66

234

102

270

138

6

174

63

231

99

267

135

87

255

123

291

159

27

195

84

252

120

288

156

108

276

144

12

180

48

216

105

273

141

9

177

129

297

165

33

201

69

237

126

294

162

30

198

150

18

186

54

222

90

258

147

15

183

51

219

No. 47

4

228

52

276

100

324

148

372

224

48

272

96

320

32

256

80

304

128

352

176

28

252

76

300

124

348

60

284

108

332

156

380

204

56

280

104

328

152

376

88

312

136

360

184

8

232

84

308

132

356

180

116

340

164

388

212

36

260

112

336

160

384

208

144

368

192

16

240

64

288

140

364

188

12

236

172

396

220

44

268

92

316

168

392

216

40

264

200

24

248

72

292

120

344

196

20

244

68

296

No. 48

1. $3433540.07

2. $2509179.07

3. $3688667.60

4. $3251326.81

5. $3449296.55

6. $3353169.99

No. 49

1. $18.53

2. $25.66

3. $23.95

4. $14.78

5. $41.76

6. $38.38

7. $15.74

8. $42.95

9. $60.76

10. $71.19

11. $66.57

12. $59.85

13. $93.72

14. $80.90

15. $75.68

16. $61.52

No. 50

5

285

65

345

125

405

185

465

280

60

340

120

400

40

320

100

380

160

440

220

35

315

95

375

155

435

75

355

135

415

195

475

255

70

350

130

410

190

470

110

390

170

450

230

10

290

105

385

165

445

225

145

425

205

485

265

45

325

140

420

200

480

260

180

460

240

20

300

80

360

175

455

235

15

295

215

495

275

55

335

115

395

210

490

270

50

330

250

30

310

90

370

150

430

245

25

305

85

365

No. 51

(Same as No. 49)

No. 52

6

342

78

414

150

486

222

558

336

72

408

144

480

48

384

120

456

192

528

264

42

378

114

450

186

522

90

426

162

498

234

570

306

84

420

156

492

228

564

132

468

204

540

276

12

348

126

462

198

534

270

174

510

246

582

318

54

390

168

504

240

576

312

216

552

288

24

360

96

432

210

546

282

18

354

258

594

330

66

402

138

474

252

588

324

60

396

300

36

372

108

444

180

516

294

30

366

102

438

No. 53

7

399

91

483

175

567

259

651

392

84

476

168

560

56

448

140

532

224

616

308

49

441

133

525

217

609

105

497

189

581

273

665

357

98

490

182

574

266

658

154

546

238

630

322

14

406

147

539

231

623

315

203

595

287

679

371

63

455

196

588

280

672

364

252

644

336

28

420

112

504

245

637

329

21

413

301

693

385

77

469

161

553

294

686

378

70

462

350

42

434

126

518

210

602

343

35

427

119

51.1

No. 54

1. $6537136.94

2. $6295852.28

3. $6328194.91

4. $5945296.77

No. 55

1. $19.76

2. $18.86

3. $44.51

4. $26.39

5. $41.42

6. $6.20

7. $12.22

8. $19.63

9. $87.27

10. $84.51

11. $71.61

12. $55.60

13. $97.15

14. $73.69

15. $61.63

16. $68.20

No. 56

8

456

104

552

200

648

296

744

448

96

544

192

640

64

512

160

608

256

704

352

56

504

152

600

248

696

120

568

216

664

312

760

408

112

560

208

656

304

752

176

624

272

720

368

16

464

168

616

264

712

360

232

680

328

776

424

72

520

224

672

320

768

416

288

736

384

32

480

128

576

280

728

376

24

472

344

792

440

88

536

184

632

336

784

432

80

528

400

48

496

144

592

240

688

392

40

488

136

584

No. 57

(Same as No. I5)

No. 58

(Same as No. 55)

No. 59

1. 795

2. 682

3. 564

4. 814

5. 598

6. 924

7. 810

8. 946

9. 1032

10. 912

11. 901

12. 621

13. 665

14. 308

15. 962

16. 714

17. 1008

18. 364

19. 736

20. 782

21. 855

22. 864

23. 865

24. 988

25. 667

No. 60

9

513

117

621

225

729

333

837

504

108

612

216

720

72

572

180

684

288

792

396

63

567

171

675

279

783

135

639

243

747

351

855

459

126

630

234

738

342

846

198

702

306

810

414

18

522

189

693

297

801

405

261

765

369

873

477

81

585

252

756

360

864

468

324

828

432

36

540

144

648

315

819

423

27

531

387

891

495

99

603

207

711

378

882

486

90

594

450

54

558

162

666

270

774

441

45

549

153

657

No. 61

11

627

143

759

275

891

407

1023

616

132

748

264

880

88

704

220

836

352

968

484

77

693

209

825

341

957

165

781

297

913

429

1045

561

154

770

286

902

418

1034

242

858

374

990

506

22

608

231

847

363

979

495

319

935

451

1067

583

99

715

308

924

440

1056

572

396

1012

528

44

660

176

792

385

1001

517

33

649

473

1089

605

121

737

253

869

462

1078

594

110

726

550

66

682

198

814

330

946

539

55

671

187

803

No. 62

1. $11230083.55

2. $10797546.08

3. $8876665.99

4. $8230948.08

No. 63

1. $47.65

2. $6.21

3. $79.61

4. $34.74

5. $14.68

6. $27.74

7. $27.93

8. $21.85

9. $54.46

10. $13.83

11. $36.49

12. $4.46

13. $50.47

14. $8.53

15. $27.16

16. $39.87

No. 65

(Same as No. 63)

No. 66

1. 1827

2. 1705

3. 1170

4. 1376

5. 2511

6. 2624

7. 3772

8. 1200

9. 1537

10. 1235

11. 1408

12. 1428

13. 1407

14. 1408

15. 2016

16. 2418

17. 3772

18. 1164

19. 2015

20. 2592

No. 67

1. $846.98

2. $836.87

3. $666.99

4. $829.97

5. $634.22

6. $827.43

7. $857.76

8. $527.72

9. $418.44

10. $906.92

11. $447.71

12. $586.87

13. $407.46

14. $510.63

15. $533.62

16. $663.85

No. 68

(Same as No. 17)

No. 69

(Same as No. 67)

No. 71

1. $276.69

2. $855.51

3. $682.90

4. $520.36

5. $773.79

6. $891.54

7. $326.93

8. $245.59

9. $371.93

10. $471.54

11. $386.88

12. $330.44

13. $878.62

14. $696.89

15. $770.20

16. $674.87

No. 72

(Same as No. 22)

No. 73

1. 755717535

2. 756410013

3. 824293224

4. 824985702

5. 3674994324

6. 1167178458

7. 1236433047

8. 6091457406

9. 1690209807

10. 1752668607

11. 1511041308

12. 3675686802

13. 1306128921

14. 1031412036

15. 1442533509

No. 74

1. 1536

2. 4606

3. 2646

4. 1495

5. 5313

6. 3230

7. 7347

8. 4814

9. 4284

10. 1295

11. 6624

12. 1624

13. 1886

14. 3618

15. 5494

16. 3861

17. 3344

18. 8608

19. 1612

20. 2655

No. 75

(Same as No. 71)

No. 76

(Same as No. 26)

No. 77

12

684

156

828

300

972

444

1116

672

144

816

288

960

96

768

240

912

384

1056

528

84

756

228

900

372

1044

180

852

324

996

468

1140

612

168

840

312

984

456

1128

264

936

408

1080

552

24

696

252

924

396

1068

540

348

1020

492

1164

636

108

780

336

1008

480

1152

624

432

1104

576

48

720

192

864

420

1092

564

36

708

516

1188

660

132

804

276

948

504

1176

648

120

792

600

72

744

216

888

360

1032

588

60

732

204

876

No. 78

(Same as No. 34)

No. 79

1. $451.84

2. $189.86

3. $343.97

4. $352.59

5. $188.21

6. $145.71

7. $291.97

8. $664.63

9. $136.68

10. $86.14

11. $440.45

12. $221.48

13. $196.63

14. $146.23

15. $586.21

16. $568.49

No. 80

1. 17081

2. 13361

3. 25543

4. 22632

5. 37893

6. 34323

7. 52643

8. 45201

9. 68302

10. 62693

11. 19602

12. 12312

13. 77922

14. 33033

15. 25662

16. 12831

17. 16086

18. 20274

19. 22263

20. 47583

21. 44896

No. 81

1. 123782280

2. 123895704

3. 135014592

4. 135128016

5. 601943392

6. 191177264

7. 202520776

8. 997746448

9. 276846856

10. 287077256

11. 247500064

12. 602056816

13. 213936568

14. 168939488

15. 236278872

No. 82

(Same as No. 38)

No. 83

1. $451.84

2. $189.86

3. $343.97

4. $352.59

5. $188.21

6. $145.71

7. $291.97

8. $664.63

9. $136.68

10. $86.14

11. $440.45

12. $221.48

13. $196.63

14. $146.23

15. $586.21

16. $568.49

No. 84

1. 19584

2. 23793

3. 28288

4. 24466

5. 17344

6. 21483

7. 24208

8. 21346

9. 25164

10. 12691

11. 17138

12. 21918

13. 30702

14. 36206

15. 33355

16. 17199

17. 27846

18. 31003

19. 29120

20. 33948

21. 16238

No. 86

1. $95513.02

2. $102635.78

3. $98506.46

4. $117398.69

5. $95153.78

6. $99073.91

No. 89

1. 170810

2. 133610

3. 255430

4. 226320

5. 378930

6. 343230

7. 526430

8. 452010

9. 683020

10. 626930

11. 196020

12. 123120

13. 779220

14. 330330

15. 256620

16. 128310

17. 160860

18. 202740

19. 222630

20. 465830

21. 448960

No. 90

13

741

169

897

325

1053

481

1209

728

156

884

312

1040

104

832

260

988

418

1144

572

91

819

247

975

403

1131

195

923

351

1079

507

1235

663

182

910

338

1066

494

1222

286

1014

442

1170

598

26

754

273

1001

429

1157

585

377

1105

533

1261

689

117

845

364

1092

520

1248

676

468

1196

624

52

780

208

936

455

1183

611

39

767

559

1287

715

143

871

299

1027

546

1274

702

130

858

650

78

806

234

962

390

1118

637

65

793

221

949

No. 91

(Same as No. 48)

No. 93

1. 195840

2. 237930

3. 282880

4. 244660

5. 173440

6. 214830

7. 242080

8. 213460

9. 251640

10. 126910

11. 171380

12. 219180

13. 307020

14. 362060

15. 333550

16. 171990

17. 278460

18. 310030

19. 291200

20. 339480

21. 162380

No. 94

1. 13502509

2. 13514882

3. 14727760

4. 14740133

5. 65661630

6. 20854138

7. 2209151S

8. 10883691

9. 30199211

10. 30315171

11. 26997983

12. 65674003

13. 23336785

14. 18438381

15. 25773945

No. 96

(Same as No. 54

No. 97

1. 11211

2. 24642

3. 40051

4. 57902

5. 77691

6. 92412

7. 29432

8. 21311

9. 35742

10. 52151

11. 71002

12. 91791

13. 25521

14. 48155

15. 24442

16. 49184

17. 76146

18. 44844

19. 37296

20. 97902

21. 39693

No. 99

1. $11230083.55

2. $10797546.08

3. $8876665.99

4. $8230948.08

No. 101

1. 36156

2. 59290

3. 80618

4. 22869

5. 36696

6. 52624

7. 71918

8. 93555

9. 97856

10. 103972

11. 108988

12. 84058

13. 103474

14. 108580

15. 79165

16. 57318

17. 65778

18. 77744

19. 91086

20. 35547

21. 80690

No. 103

1. 365

2. 268

3. 371

4. 433

5. 257

6. 327

7. 209

8. 270

9. 287

10. 410

11. 257

12. 404

13. 231

14. 217

15. 311

16. 303

17. 254

18. 237

19. 308

20. 343

21. 350

22. 360

23. 308

24. 271

25. 341

No. 105

1. 116081

2. 142272

3. 165481

4. 107512

5. 132181

6. 159372

7. 156996

8. 191522

9. 181692

10. 217894

11. 110564

12. 110940

13. 121598

14. 120273

15. 134316

16. 120990

17. 113970

18. 145262

19. 122811

20. 139635

21. 144284

No. 106

14

798

182

966

350

1134

518

1302

784

168

952

336

1120

112

896

280

1064

448

1232

616

98

882

266

1050

434

1218

210

994

378

1162

546

1330

714

196

980

364

1148

532

1316

308

1092

476

1260

644

28

812

294

1078

462

1246

630

406

1190

574

1358

742

126

910

392

1176

560

1344

728

504

1288

672

56

840

224

1008

490

1274

658

42

826

602

1386

770

154

938

322

1106

588

1372

756

140

924

700

84

868

252

1036

420

1204

686

70

854

238

1022

No. 107

(Same as No. 17)

No. 109

1. 136004

2. 229024

3. 268746

4. 128064

5. 160446

6. 236496

7. 195853

8. 223096

9. 368063

10. 145673

11. 187146

12. 305283

13. 355096

14. 291014

15. 348928

16. 145728

17. 336414

18. 395324

19. 430265

20. 247275

21. 575276

No. 110

1. 146267910

2. 146401938

3. 159540624

4. 159674652

5. 711289224

6. 225905508

7. 239309622

8. 1178991756

9. 327137382

10. 339226182

11. 292459608

12. 711423252

13. 252799146

14. 199628136

15. 279200034

No. 111

(Same as No. 26)

No. 113

1. 164232

2. 227238

3. 301464

4. 377910

5. 456576

6. 497502

7. 658752

8. 172104

9. 243320

10. 279396

11. 354252

12. 427652

13. 484432

14. 588078

15. 671944

16. 175392

17. 173514

18. 257237

19. 341968

20. 429525

21. 519302

No. 116

(Same as No, 34)

No. 118

(Same as No. 38)

No. 119

15

855

195

1035

375

1215

555

1395

840

180

1020

360

1200

120

960

300

1140

480

1320

660

105

945

285

1125

465

1305

225

1065

405

1245

585

1425

765

210

1050

390

1230

570

1410

330

1170

510

1350

690

30

870

315

1155

495

1335

675

435

1275

615

1455

795

135

975

420

1260

600

1440

780

540

1380

720

60

900

240

1080

525

1365

705

45

885

645

1485

825

165

1005

345

1185

630

1470

810

150

990

750

90

930

270

1110

450

1290

735

75

915

255

1095

No. 120

(Same as No. 41)

No. 122

(Same as No. 48)

No. 123

1. 157510725

2. 157655055

3. 171803640

4. 171947970

5. 765962140

6. 243269630

7. 257704045

8. 1269714410

9. 352282645

10. 365300645

11. 314939380

12. 766106470

13. 272230435

14. 214972460

15. 300660615

No. 124

(Same as No. 54)

No. 126

(Same as No. 62)

No. 128

(Same as No. 38)

No. 131

16

912

208

1104

400

1296

92

1488

896

192

1088

384

1280

128

1024

320

1216

512

1408

704

112

1008

304

1200

496

1392

240

1136

432

1328

624

1520

816

224

1120

416

1312

608

1504

352

1248

544

1440

736

32

928

336

1232

528

1424

720

464

1360

656

1552

848

144

1040

448

1344

640

1536

832

576

1472

768

64

960

256

1152

560

1456

752

48

944

688

1584

880

176

1072

368

1264

672

1568

864

160

1056

800

96

992

288

1184

480

1376

784

80

976

272

1168

No. 132

1. 168753540

2. 168908172

3. 184066656

4. 184221288

5. 820635056

6. 260633752

7. 276098468

8. 1360237064

9. 377427908

10. 391375108

11. 337419152

12. 820789688

13. 291661724

14. 230316784

15. 322121196

No. 140

17

969

221

1173

425

1377

629

1581

952

204

1156

408

1360

136

1088

340

1292

544

1496

748

119

1071

323

1275

527

1479

255

1207

459

1411

663

1615

867

238

1190

442

1394

646

1598

374

1326

578

1530

782

34

996

357

1309

561

1513

765

493

1445

697

1649

901

153

1105

476

1428

680

1632

884

912

1564

816

68

1020

272

1224

595

1547

799

51

1003

731

1683

935

187

1139

391

1343

714

1666

918

170

1122

850

102

1054

306

1258

510

1462

833

85

1037

289

1241

No. 141

1. 179996355

2. 180161289

3. 196329672

4. 196494606

5. 875307972

6. 277997874

7. 294492891

8. 1450859718

9. 402573171

10. 417449571

11. 359898924

12. 875472906

13. 311093013

14. 245661108

15. 343581777

No. 148

18

1026

234

1242

450

1458

666

1674

1008

216

1224

432

1440

144

1152

360

1368

576

1584

792

126

1134

342

1350

558

1566

270

1278

486

1494

702

1710

918

252

1260

468

1476

684

1692

396

1404

612

1620

828

36

1044

378

1386

594

1602

810

522

1530

738

1746

954

162

1170

504

1512

720

1728

936

648

1656

864

72

1080

288

1296

630

1638

846

54

1062

774

1782

990

198

1206

414

1422

756

1764

972

180

1188

900

108

1116

324

1332

540

1548

882

90

1098

306

1314

No. 149

1. 191239170

2. 191414406

3. 208592688

4. 208767924

5. 929980808

6. 295361996

7. 312887314

8. 1541482372

9. 427718434

10. 443524034

11. 382378696

12. 930156124

13. 330524302

14. 261005432

15. 365042358

No. 156

19

1083

247

1311

475

1539

703

1767

1064

228

1292

456

1520

152

1216

380

1444

608

1672

836

133

1197

361

1425

589

1653

285

1349

513

1577

741

1805

969

266

1330

494

1558

722

1786

418

1482

646

1710

874

38

1102

399

1463

627

1691

855

551

1615

779

1843

1007

171

1235

532

1596

760

1824

988

684

1748

912

76

1140

304

1368

665

1729

893

57

1121

817

1881

1045

209

1273

437

1501

798

1862

1026

190

1254

950

114

1178

342

1406

570

1634

931

95

1159

323

1387

No. 159

1. 202481985

2. 202667523

3. 220855704

4. 221041242

5. 984653804

6. 312726118

7. 331281737

8. 1632105026

9. 452863697

10. 469598497

11. 404858468

12. 984839342

13. 349955591

14. 276349756

15. 386502939

No. 165

20

1140

260

1380

500

1620

740

1860

1120

240

1360

480

1600

160

1280

400

1520

640

1760

880

140

1260

380

1500

620

1740

300

1420

540

1660

780

1900

1020

280

1400

520

1640

760

1880

440

1560

680

1800

920

40

1160

420

1540

660

1780

900

580

1700

820

1940

1060

180

1300

560

1680

800

1920

1040

720

1840

960

80

1200

320

1440

700

1820

940

60

1180

860

1980

1100

220

1340

460

1580

840

1960

1080

200

1320

1000

120

1240

360

1480

600

1720

980

100

1220

340

1460

No. 166

1. 213724800

2. 213920640

3. 233118720

4. 233314560

5. 1039326720

6. 330090240

7. 349676160

8. 1722727680

9. 478008960

10. 495672960

11. 427338240

12. 1039522560

13. 369386880

14. 291694080

15. 407963520

No. 172

21

1197

273

1449

525

1701

777

1953

1176

252

1428

504

1680

168

1344

420

1596

672

1848

924

147

1323

399

1575

651

1827

315

1491

567

1743

819

1995

1071

294

1470

546

1722

798

1974

462

1638

714

1890

966

42

1218

441

1617

693

1869

945

609

1785

861

2037

1113

189

1365

588

1744

840

2016

1092

756

1932

1008

84

1260

336

1512

735

1911

987

63

1239

903

2079

1155

231

1407

483

1659

882

2058

1134

210

1386

1050

126

1302

378

1554

630

1806

1029

105

1281

357

1533

No. 173

1. 224967615

2. 225173757

3. 245381736

4. 245587878

5. 1093999636

6. 347454362

7. 368070583

8. 1813350334

9. 503154223

10. 521747423

11. 449818012

12. 1094205778

13. 388818169

14. 307038404

15. 429424101

No. 179

22

1254

286

1518

550

1782

814

2046

1232

264

1496

528

1760

176

1408

440

1672

704

1936

968

154

1386

418

1650

682

1914

330

1562

604

1826

858

2090

1122

308

1540

572

1804

836

2068

484

1716

748

1980

1012

44

1276

462

1694

726

1958

990

638

1870

902

2134

1166

198

1430

616

1848

880

2112

1144

792

2024

1056

88

1320

352

1584

770

2002

1034

66

1298

946

2178

1210

242

1474

506

1738

924

2156

1188

220

1452

1100

132

1364

396

1628

660

1892

1078

110

1342

374

1606

No. 180

1. 236210430

2. 236426874

3. 257644752

4. 257861196

5. 1148672552

6. 364818484

7. 386465006

8. 1903972988

9. 528299486

10. 547821886

11. 472297784

12. 1148888996

13. 408249458

14. 322382728

15. 450884682

No. 186

23

1311

299

1587

575

1863

851

2139

1288

276

1564

552

1840

184

1472

460

1748

736

2024

1012

161

1449

437

1725

713

2001

345

1623

621

1909

897

2185

1173

322

1610

598

1886

874

2162

506

1794

782

2070

1058

46

1334

483

1771

759

2047

1035

667

1955

943

2231

1219

207

1495

644

1932

920

2208

1196

828

2116

1104

92

1380

368

1656

805

2093

1081

69

1357

989

2277

1265

253

1541

529

1817

966

2254

1242

230

1518

1150

138

1426

414

1702

690

1978

1127

115

1403

391

1679

No. 187

1. 247453245

2. 247679991

3. 269907768

4. 270134514

5. 1203345468

6. 382182606

7. 404859429

8. 1994595642

9. 553444749

10. 573896349

11. 494777556

12. 1203572214

13. 427680747

14. 337727052

15. 472345263

No. 193

24

1368

312

1656

600

1944

888

2232

1344

288

1632

576

1920

192

1536

480

1824

768

2112

1056

168

1512

456

1800

744

2088

360

1704

648

1992

936

2280

1224

336

1680

624

1968

912

2256

528

1872

816

2160

1104

48

1392

504

1848

792

2136

1080

696

2040

984

2328

1272

216

1560

672

2016

960

2304

1248

864

2208

1152

96

1440

384

1728

840

2184

1128

72

1416

1032

2376

1320

264

1608

552

1896

1008

2352

1296

240

1584

1200

144

1488

432

1776

720

2064

1176

120

1464

408

1752

No. 194

1. 258696060

2. 258933108

3. 282170784

4. 282407832

5. 1258018384

6. 399546728

7. 423253852

8. 2085218296

9. 578590012

10. 599970812

11. 517257328

12. 1258255432

13. 447112036

14. 353071376

15. 493805844

No. 200

25

1425

325

1725

625

2025

925

2325

1400

300

1700

600

2000

200

1600

500

1900

800

2200

1100

175

1575

475

1875

775

2175

375

1775

675

2075

975

2375

1275

350

1750

650

2050

950

2350

550

1950

850

2250

1150

50

1450

525

1925

825

2225

1125

725

2125

1025

2425

1325

225

1625

700

2100

1000

2400

1300

900

2300

1200

100

1500

400

1800

875

2275

1175

75

1475

1075

2475

1375

275

1675

575

1975

1050

2450

1350

250

1650

1250

150

1550

450

1850

750

2150

1225

125

1525

425

1825

No. 201

1. 269938875

2. 270186225

3. 294433800

4. 294681150

5. 1312691300

6. 416910850

7. 441648275

8. 2175840950

9. 603735275

10. 626045275

11. 539737100

12. 1312938650

13. 466543325

14. 368415700

15. 515266425

No. 204

(Annex 0 to Answers to No. 45)

No. 208

(Annex 0 to Answers to No. 46)

No. 212

(Annex 0 to Answers to No. 47)

No. 215

(Annex 0 to Answers to No. 60)

No. 219

(Annex 0 to Answers to No. 52)

No. 222

(Annex 0 to Answers to No. 68)

No. 226

(Annex 0 to Answers to No. 56)

No. 228

(Annex 0 to Answers to No. 60)

No. 229

1. 242

2. 464

3. 686

4. 902

5. 1124

6. 1246

7. 1462

8. 1684

9. 1906

10. 322

11. 444

12. 666

13. 882

14. 1104

15. 1326

16. 1442

17. 1664

18. 1886

19. 302

20. 524

No. 232

(Annex 0 to Answers to No. 61)

No. 233

1. 393

2. 726

3. 1059

4. 1392

5. 1713

6. 1896

7. 2229

8. 2562

9. 2883

10. 516

11. 699

12. 1032

13. 1353

14. 1686

15. 2019

16. 2202

17. 2523

18. 2856

19. 489

20. 822

No. 236

(Annex 0 to Answers to No. 77)

No. 237

1. 564

2. 1008

3. 1452

4. 1896

5. 2340

6. 2564

7. 3008

8. 3452

9. 3892

10. 740

11. 964

12. 1408

13. 1852

14. 2296

15. 2740

16. 2964

17. 3408

18. 3852

19. 696

20. 1140

No. 239

(Annex 0 to Answers to No. 90)

No. 240

1. 755

2. 1310

3. 1865

4. 2420

5. 2975

6. 3280

7. 3805

8. 4360

9. 4915

10. 970

11. 1275

12. 1830

13. 2355

14. 2910

15. 3465

16. 3770

17. 4325

18. 4880

19. 905

20. 1460

No. 242

(Annex 0 to Answers to No. 106)

No. 243

1. 846

2. 1512

3. 2178

4. 2844

5. 3510

6. 4176

7. 4482

8. 5106

9. 5772

10. 1038

11. 1704

12. 2370

13. 2676

14. 3342

15. 3966

16. 4632

17. 5298

18. 5964

19. 870

20. 1536

No. 244

(Annex 0 to Answers to No. 119)

No. 245

1. 917

2. 1694

3. 2471

4. 3248

5. 4025

6. 4802

7. 5579

8. 5866

9. 6587

10. 1064

11. 1841

12. 2618

13. 3395

14. 4172

15. 4459

16. 5236

17. 5957

18. 6734

19. 1211

20. 1988

No. 246

(Annex 0 to Answers to No. 181)

No. 247

1. 1128

2. 2016

3. 2904

4. 3792

5. 4680

6. 5568

7. 5976

8. 6864

9. 7752

10. 1368

11. 2256

12. 3144

13. 355?

14. 4440

15. 5328

16. 6216

17. 7104

18. 7992

19. 5928

20. 5216

No. 248

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 249

(Annex 0 to Answers to No. 140)

No. 250

1.

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3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

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30.

31.

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35.

36.

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38.

39.

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No. 251

1. 1368

2. 2367

3. 3366

4. 4365

5. 5364

6. 5823

7. 6822

8. 7821

9. 8757

10. 1656

11. 2655

12. 3114

13. 4113

14. 5112

15. 6111

16. 7056

17. 8055

18. 8514

19. 1413

20. 2412

No. 252

1. 121

2. 232

3. 343

4. 451

5. 562

6. 623

7. 731

8. 842

9. 953

10. 161

11. 222

12. 333

13. 441

14. 552

15. 663

16. 721

17. 832

18. 943

19. 151

20. 262

No. 253

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 254

(Annex 0 to Answers to No. 148)

No. 255

1. 131

2. 242

3. 353

4. 464

5. 571

6. 632

7. 743

8. 854

9. 961

10. 172

11. 233

12. 344

13. 451

14. 562

15. 673

16. 734

17. 841

18. 952

19. 163

20. 274

No. 256

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 257

(Annex 0 to Answers to No. 166)

No. 258

1. 141

2. 252

3. 363

4. 474

5. 585

6. 641

7. 752

8. 863

9. 974

10. 185

11. 241

12. 352

13. 463

14. 574

15. 685

16. 741

17. 852

18. 963

19. 174

20. 285

No. 259

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 260

(Annex 0 to Answers to No. 166)

No. 261

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 262

1. 151

2. 262

3. 373

4. 484

5. 595

6. 656

7. 761

8. 872

9. 983

10. 194

11. 255

12. 366

13. 471

14. 582

15. 693

16. 754

17. 865

18. 976

19. 181

20. 292

No. 263

1.

2.

3.

4.

5.

6.

7.

8.

9.

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11.

12.

13.

14.

15.

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17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

No. 264

(Annex 0 to Answers to No. 172)

No. 265

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 266

1. 141

2. 252

3. 363

4. 474

5. 585

6. 696

7. 747

8. 851

9. 962

10. 173

11. 284

12. 395

13. 446

14. 557

15. 661

16. 772

17. 883

18. 994

19. 145

20. 256

No. 267

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 268

(Annex 0 to Answers to No. 179)

No. 269

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 270

1. 131

2. 242

3. 353

4. 464

5. 575

6. 686

7. 797

8. 838

9. 941

10. 152

11. 263

12. 374

13. 485

14. 596

15. 637

16. 748

17. 851

18. 962

19. 173

20. 284

No. 271

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 272

{Annex 0 to Answers to No. 186)

No. 273

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 274

1. 141

2. 252

3. 363

4. 474

5. 585

6. 696

7. 747

8. 858

9. 969

10. 171

11. 282

12. 393

13. 444

14. 555

15. 666

16. 777

17. 888

18. 999

19. 741

20. 652

No. 275

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 276

(Annex 0 to Answers to No. 193)

No. 277

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 278

1. 152

2. 263

3. 374

4. 485

5. 596

6. 647

7. 758

8. 869

9. 973

10. 184

11. 295

12. 346

13. 437

14. 568

15. 679

16. 784

17. 895

18. 946

19. 157

20. 268

No. 279

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 280

{Annex 0 to Answers to No. 200)

No. 281

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 282

1. 2r86

2. 2rl29

3. 2rl08

4. 2r347

5. 2r456

6. 2r589

7. 2r312

8. 2rl02

9. 2r208

10. 2rll7

11. 3rl3

12. 3r50

13. 3rl05

14. 3rl82

15. 3r285

16. 4rl26

17. 4r200

18. 4r252

19. 4r282

20. 4r280

No. 283

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 284

1. 1066

2. 1377

3. 1708

4. 2059

5. 2511

6. 2912

7. 1023

8. 1394

9. 1326

10. 1647

11. 1988

12. 2349

13. 2821

14. 992

15. 1353

16. 1734

17. 1586

18. 1917

19. 2268

20. 2639

No. 285

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 286

1. 2rl

2. 2r29

3. 2r376

4. 2r551

5. 2r374

6. 3r378

7. 3r518

8. 3r680

9. 3r864

10. 3rl7

11. 4r266

12. 4r225

13. 4rl72

14. 4r93

15. 4rl62

16. 5r90

17. 5rl30

18. 5rl48

19. 5rl44

20. 5rll9

No. 287

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 288

1. 1470

2. 1872

3. 2294

4. 2736

5. 3198

6. 3772

7. 1344

8. 1806

9. 1820

10. 2232

11. 2664

12. 3116

13. 3588

14. 1312

15. 1764

16. 2236

17. 2108

18. 2520

19. 2952

20. 3404

No. 289

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 290

1. 2r37

2. 2r771

3. 2rl50

4. 2r85

5. 2r99

6. 3r46

7. 3rl02

8. 3rl70

9. 3r280

10. 3r402

11. 4rl92

12. 4r235

13. 4r276

14. 4r285

15. 4r272

16. 5r67

17. 5r693

18. 5r564

19. 5r632

20. 5r97

No. 291

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 292

1. 1892

2. 2385

3. 2898

4. 3431

5. 3984

6. 4557

7. 1683

8. 2236

9. 2332

10. 2835

11. 3358

12. 3901

13. 4464

14. 1617

15. 2193

16. 2756

17. 2772

18. 3510

19. 3818

20. 4371

No. 293

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 294

1. 3r51

2. 3r69

3. 3r95

4. 3r32

5. 3r54

6. 4r226

7. 4r85

8. 4r864

9. 4rll9

10. 4r208

11. 5rl46

12. 5r288

13. 5r321

14. 5r465

15. 5rl08

16. 6rl25

17. 6r200

18. 6r77

19. 6rlll

20. 6r310

No. 295

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 296

1. 2332

2. 2916

3. 3520

4. 4144

5. 4788

6. 5452

7. 2006

8. 2684

9. 2862

10. 3456

11. 4070

12. 4704

14. 1972

15. 2596

16. 3599

17. 3392

18. 3996

19. 4620

20. 5264

No. 297

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 298

1. 5r219

2. 5r642

3. 5r312

4. 5r97

5. 5rl06

6. 6r310

7. 6rl50

8. 6rl00

9. 6r609

10. 6rll5

11. 7r65

12. 7rl35

13. 7r235

14. 7rl85

15. 7r64

16. 8r72

17. 8rl25

18. 8rl80

19. 8r360

20. 8r421

No. 299

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 300

1. 2790

2. 3465

3. 4160

4. 4875

5. 5610

6. 6365

7. 2380

8. 3105

9. 3410

10. 4095

11. 4800

12. 5525

13. 6270

14. 2345

15. 3060

16. 3795

17. 4030

18. 4725

19. 5440

20. 6175

No. 301

1.

2.

3.

4.

5.

6.

7.

8.

9.

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No. 302

1. 6rl0

2. 6r29

3. 6r38

4. 6rl65

5. 6r651

6. 7r501

7. 7r307

8. 7r799

9. 7r646

10. 7r20

11. 8rl89

12. 8r612

13. 8r325

14. 8r486

15. 8rl7

16. 9rl25

17. 9rl35

18. 9r74

19. 9r85

20. 9r59

No. 303

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 304

1. 3266

2. 4032

3. 4818

4. 5624

5. 6450

6. 7296

7. 2772

8. 3588

9. 3976

10. 4752

11. 5548

12. 6364

13. 7200

14. 2736

15. 3542

16. 4368

17. 4686

18. 5472

19. 6278

20. 7104

No. 305

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 306

1. 6r706

2. 6r95

3. 6r37

4. 6r38

5. 6r40

6. 7rl8

7. 7rll8

8. 7r211

9. 7r346

10. 7r252

11. 8r28

12. 8r39

13. 8r404

14. 8r355

15. 8r626

16. 9r64

17. 9r301

18. 9r400

19. 9r500

20. 9r65

No. 307

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 308

1. 3713

2. 4617

3. 5494

4. 6391

5. 7308

6. 8245

7. 3182

8. 4089

9. 4503

10. 5427

11. 6314

12. 7221

13. 8148

14. 3145

15. 4042

16. 4959

17. 5293

18. 6237

19. 7134

20. 8051

No. 309

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 310

1. 7rl29

2. 7r642

3. 7r711

4. 7r32

5. 7r232

6. 8r77

7. 8r444

8. 8r312

9. 8rl47

10. 8r25

11. 9r27

12. 9r297

13. 9r358

14. 9r555

15. 9r609

16. 9r775

17. 9r862

18. 9r927

19. 9rl50

20. 9r215

No. 311

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 312

1. 4224

2. 5162

3. 6188

4. 7176

5. 8184

6. 9212

7. 3610

8. 4608

9. 5104

10. 6052

11. 7098

12. 8096

13. 9114

14. 3572

15. 4560

16. 5568

17. 5984

18. 6942

19. 8008

20. 9016

No. 313

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 314

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 315

1. 4655

2. 5664

3. 6693

4. 7742

5. 8811

6. 9405

7. 3744

8. 4753

9. 5782

10. 6831

11. 7505

12. 8544

13. 9603

14. 3822

15. 4851

16. 5605

17. 6624

18. 7663

19. 8722

20. 9801

No. 316

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 317

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 318

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 319

1. 41

2. 51

3. 61

4. 71

5. 81

6. 91

7. 31

8. 41

9. 51

10. 61

11. 71

12. 81

13. 91

14. 31

15. 41

16. 51

17. 61

18. 71

19. 81

20. 91

No. 320

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 321

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 322

1. 42

2. 52

3. 62

4. 72

5. 82

6. 92

7. 32

8. 42

9. 52

10. 62

11. 72

12. 82

13. 92

14. 32

15. 42

16. 52

17. 62

18. 72

19. 82

20. 92

No. 323

1.

2.

3.

4.

5.

6.

No. 324

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 325

1. 43

2. 53

3. 63

4. 73

5. 83

6. 93

7. 33

8. 43

9. 53

10. 63

11. 73

12. 83

13. 93

14. 33

15. 43

16. 53

17. 63

18. 73

19. 83

20. 93

No. 327

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 328

1. 44

2. 54

3. 64

4. 74

5. 84

6. 94

7. 34

8. 44

9. 54

10. 64

11. 74

12. 84

13. 94

14. 34

15. 44

16. 54

17. 64

18. 74

19. 84

20. 94

No. 330

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 331

1. 45

2. 55

3. 65

4. 75

5. 85

6. 95

7. 35

8. 45

9. 55

10. 65

11. 75

12. 85

13. 95

14. 35

15. 45

16. 55

17. 65

18. 75

19. 85

20. 95

No. 332

1. 46

2. 56

3. 66

4. 76

5. 86

6. 96

7. 36

8. 46

9. 56

10. 66

11. 76

12. 86

13. 96

14. 36

15. 46

16. 56

17. 66

18. 76

19. 86

20. 96

No. 333

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

No. 334

1. 47

2. 57

3. 67

4. 77

5. 87

6. 97

7. 37

8. 47

9. 57

10. 67

11. 77

12. 87

13. 97

14. 37

15. 47

16. 57

17. 67

18. 77

19. 87

20. 97

No. 335

1.

2.

3.

4.

5.

6.

No. 336

1. 48

2. 58

3. 68

4. 78

5. 88

6. 98

7. 38

8. 48

9. 58

10. 68

11. 78

12. 88

13. 98

14. 38

15. 48

16. 58

17. 68

18. 78

19. 88

20. 98

No. 337

1. 49

2. 59

3. 69

4. 79

5. 89

6. 99

7. 39

8. 49

9. 59

10. 69

11. 79

12. 89

13. 99

14. 39

15. 49

16. 59

17. 69

18. 79

19. 89

20. 99

No. 338

1.

2.

3.

4.

5.

6.

7.

8.

9. .20

10. .40

11. .60

12. .80

No. 339

1. 2886

2. 5994

3. 9268

4. 12818

5. 17081

6. 19584

7. 23793

8. 28288

9. 24466

10. 4104

No. 340

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

No. 341

1. 4235

2. 8352

3. 12691

4. 17138

5. 21918

6. 25543

7. 30702

8. 36206

9. 33355

10. 5796

No. 342

1. $17887

2. $9818

3. 9865

4. 25775

5. 39540

6. 23332

7. 17313

8. 31383

9. $14822.40

10. 243062

No. 343

1. 5764

2. 10890

3. 16238

4. 21808

5. 27408

6. 30968

7. 37893

8. 44408

9. 42284

10. 7740

No. 344

1. .0625

2. .1875

3. .3125

4. .4375

5. .5625

6. .6875

7. .8125

8. .9375

9.

10.

11.

12.

13.

14.

No. 345

1. 7473

2. 13608

3. 19965

4. 26544

5. 33345

6. 37178

7. 44368

8. 52643

9. 51622

10. 9990

No. 346

1. $99.84

2. 96256

3. $117.76

4. 98304

5. 1728

6. $675.84

7. $8120.60

8. $30402.55

No. 347

1. 9362

2. 16506

3. 23872

4. 31460

5. 39270

6. 43952

7. 51748

8. 60168

9. 60946

10. 12222

No. 348

1. .03125

2. .09375

3. .15625

4. .21875

5. .28125

6. .34375

7. .40625

8. .46875

9. .53125

10. .59375

11. .65625

12. .71875

13. .78125

14. .84375

15. .90625

16. .96875

17. .04167

18. .20833

19. .29167

20. .45833

21. .54167

22. .70833

23. .79167

24. .95833

No. 349

1. 10011

2. 18144

3. 26499

4. 35076

5. 43875

6. 52896

7. 57519

8. 66378

9. 68302

10. 12456

No. 350

1. $424575

2. $84770

3. $733779.50

4. $26863.20

5. $830062.74

6. $526.32

7. $981088

8. $9603

9. $1007010

No. 351

1. 10349

2. 19602

3. 28946

4. 38512

5. 48300

6. 58310

7. 68542

8. 72906

9. 74339

10. 12312

No. 353

1. 12408

2. 22428

3. 33033

4. 43608

5. 54405

6. 65424

7. 70965

8. 82368

9. 85272

10. 15219

No. 354

1. $525

2. $756

3. $384

4. $810

5. $5400

6. $900

7. $13000

8. $14700

9. $7200

10. $1600

11. $630

12. $12600

13. $1200

14. $1200

15. $1200

No. 355

1. 14440

2. 25248

3. 36278

4. 47530

5. 59004

6. 61465

7. 72768

8. 84293

9. 95354

10. 19206

No. 357

1. 11211

2. 24642

3. 40051

4. 57902

5. 77691

6. 92412

7. 116081

8. 142272

9. 170321

10. 29032

No. 358

1. $247715.70

2. $243540

3. $60226335

4. $1087638.75

5. $5209451.52

6. $131602.24

7. $40102686.72

8. $8710669

No. 359

1. 24442

2. 49184

3. 76146

4. 104632

5. 136004

6. 156996

7. 191522

8. 229024

9. 268746

10. 47012

No. 361

1. 39693

2. 75746

3. 114019

4. 154512

5. 195853

6. 223096

7. 269709

8. 318542

9. 368063

10. 67596

No. 362

1. 138138

2. 115596

3. 74556

4. 186960

5. 89301

6. 235872

7. 119782

8. 73248

9. 193256

No. 363

1. 56964

2. 104328

3. 153912

4. 205716

5. 259740

6. 291014

7. 348928

8. 409062

9. 471416

10. 91390

No. 364

1. 210

2. 342

3. 255

4. 240

5. 195

6. 247

7. 272

8. 224

9. 361

No. 365

1. 76255

2. 134930

3. 195825

4. 258940

5. 324275

6. 364080

7. 429965

8. 501400

9. 575055

10. 115430

No. 366

1. $56496

2. $799018

3. $5663152

4. $410091.55

5. $453952.95

6. $36033.25

7. $530895.75

8. $1043606.30

No. 367

1. 85446

2. 155232

3. 227238

4. 301464

5. 377910

6. 456576

7. 497502

8. 575276

9. 659932

10. 120408

No. 368

1. $139510.50

2. $147804.75

3. $158233.30

4. $131011.65

5. $452339.40

6. $754503.75

7. $151524.65

8. $238939.80

No. 369

1. 92617

2. 173514

3. 256631

4. 341968

5. 429525

6. 519302

7. 611299

8. 651126

9. 740567

10. 121144

No. 370

1. 5476

2. 8649

3. 6724

4. 4096

5. 1444

6. 12544

7. 15376

8. 21316

9. 28224

10. 38809

11. 1236544

12. 1471369

13. 1726596

14. 2298256

15. 2954961

No. 371

1. 113928

2. 206136

3. 300564

4. 397212

5. 496080

6. 597168

7. 648396

8. 753324

9. 860472

10. 153558

No. 372

1. 7616

2. 12561

3. 15824

4. 22425

5. 40716

6. 42749

7. 421056

8. 224196

9. 198989

No. 373

1. 138168

2. 241697

3. 347446

4. 455415

5. 565604

6. 620473

7. 734502

8. 850751

9. 962297

10. 183816

No. 374

1. 8556

2. 4030

3. 7308

4. 8924

5. 45795

6. 100152

7. 173888

8. 264171

9. 837221

No. 375

1. 2025

2. 3025

3. 4225

4. 5625

5. 7225

6. 9025

7. 13225

8. 18225

9. 24025

10. 30625

11. 38025

12. 99225

13. 112225

14. 126025

15. 140625

No. 376

1. 621

2. 2009

3. 1224

4. 11021

5. 13216

6. 24024

7. 30616

8. 27209

9. 38016

No. 377

1. 275625

2. 390625

3. 680625

4. 1050625

5. 1500625

6. 1755625

7. 2640625

8. 2975625

9. 3330625

10. 3705625

No. 378

1. 4896

2. 6391

3. 8084

4. 12019

5. 16851

6. 22484

7. 25536

8. 32351

9. 36036

No. 379

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

No. 380

1. 276

2. 800

3.

4. 950

5. 2552

6. 5952

7. 1422

8. 2100

9. 3363

No. 381

1. 23.2

2. 45

3. 36

4. 3.5

5. 5.12

6. 13.05

7. 10.18

8. 61.2

9. 77.6

No. 382

1. 2744

2. 19683

3. 35937

4. 97336

5. 205379

6. 238328

7. 274625

8. 357911

9. 389017

10. 592704

11. 636056

12. 681472

13. 857375

14. 912673

15. 970299