ADDITION.

49. Addition is the process of combining several numbers into one equiva lent number.

50. The Sum or Amount is the result obtained by the addition of two or more numbers.

51. The Sign of Addition is +, and is called Plus, which signifies more. When placed between two numbers or combinations of numbers, it indicates their addition; as, 5 + 2 is read 5 plus 2, and shows that 5 and 2 are to be added.

52. The Sign of Equality is. When placed between two numbers or combinations of numbers, it indicates that there is no difference in their value; thus, 5 + 2 = 7, is read 5 plus 2 equals 7, and indicates that the value of 7 equals the value of the sum of the numbers at the left of the sign of equality.

53. Carrying the Tens is the process of reserving the tens and adding them with the next column.

54. Principles.-1. Only like numbers and like unit orders can be added one to another.

2. The sum or amount contains as many units as all the numbers added.

3. The sum or amount is the same in whatever order the numbers be added.

55. Addition is the Reverse of Subtraction and may be proved by it; as, 5 + 2 = 7. Now if 7 be diminished by 5, the result will be 2, while if 7 be diminished by 2, the result will be 5.

56. Numbers are written for addition either in vertical or horizontal order.

57. General Rules.-1. If the sum of two numbers and one of the numbers be given, the unknown number may be found by taking the given number from the sum.

2. If the sum of several numbers and all of the numbers but one be given, the unknown number may be found by subtracting the sum of those given from the sum of all the numbers.

NOTES TO TEACHER.-1. Classes should have frequent and extended drill in rapid mental addition.

2. The following table is given simply to facilitate class drill, preparatory to work in rapid addition.

1721 38 | 25 + 25 = 50

59. There are three methods of addition in common use, viz.; the Elementary method, the Result method, and the Group method.

REMARKS.-1. These methods of addition are recommended to be taught in their order to pupils in elementary work; the first, as soon as mastered, should be abandoned for the second, and the second in its turn, when mastered, abandoned for the third.

2. Daily drill in the third method is urgently advised with all pupils during the entire period of their study of Arithmetic. Too much importance can scarcely be attached to this suggestion.

60. The Elementary Method of Addition.

EXAMPLE.-Add 32, 71, 25, 48, 90, 12, and 63.

OPERATION.

32

71

25

48

90

12

63

341

EXPLANATION. -Having arranged the numbers so that units of like orders stand directly under each other, begin with the last figure in the right-hand, or units' column, and add upward as follows: 3 and 2 are 5, 5 and 8 are 13, 13 and 5 are 18, 18 and 1 are 19, 19 and 2 are 21. Having thus obtained the sum, place the 1 beneath the line, in units' column, and treat the 2 as a part of the second, or tens' column, which add upward as before; thus, 2 and 6 are 8, 8 and 1 are 9, 9 and 9 are 18, 18 and 4 are 22, 22 and 2 are 24, 24 and 7 are 31, 31 and 3 are 34. Having obtained the sum, write it in full at the left of the figure 1 before written, and the result is 341, the numerical expression of the sum of the numbers added.

TO PROVE.-Add the columns downward; if the two results agree, the work is presumed to be correct.

61. The Result Method of Addition.

EXAMPLE.

OPERATION.

32

71

25

48

90

12

63

Add 32, 71, 25, 48, 90, 12, and 63.

EXPLANATION.-Beginning as before, with the lower figure in units' column, name the result only of each successive addition, thus: 3, 5, 13, 18, 19, 21; then, as before, write the 1 beneath the line in units' column and carrying the 2 to tens' column as a part of it, add upward, thus: 2, 8, 9, 18, 22, 24, 31, 34; as before, write 34 at the left, and the result is 341, the same as before.

TO PROVE.-Add the columns downward.

341

62. The Group Method of Addition.

EXAMPLE.-1. Add 32, 71, 25, 48, 90, 12, and 63.

90 10

12

63

341

EXPLANATION. - Treat the same numbers thus: add upward; 3, 13, 21; grouping 2 and 8 for 10 to add to 3, making 13, and 5, 1, and 2 for 8, to add to 13, making 21. Having written the 1 beneath the line, in units' place, carry the 2 or 2 tens, to its column, and again add; 2, 8, 18, 24, 34; grouping 2 and 6 for 8, 9 and 1 for 10, 4 and 2 for 6, and 7 and 3 for 10; then write the result in full as before.

TO PROVE. Review the first column by adding downward; 8, 18, 21; grouping 2, 1, and 5 for 8, 8 and 2 for 10, to add for 18, and to this add the remaining figure 3, for 21, the same result as before. Then review the second column by adding downward; 10, 16, 26, 34; grouping 3 and 7 for 10, 2 and 4 for 6, 9 and 1 for 10, and 6 and 2, for 8, with the same result.

EXAMPLE.-2. Add 3417, 2140, 439, 7164, 1538, 5046, 6116, 8735, 971, 4880, 1263, 9270, 192, and 634.

OPERATION.

EXPLANATION.—Beginning with the lower unit figure add upward; 10, 15, 35, 55, grouping 4, 2, 3, and 1 for 10, which added to 5 gives 15; grouping 6, 6, and 8 for 20 to add to 15 obtaining 35; and grouping 4, 9, and 7 for 20 to add to 35 20 for 55 the result. Write the units' figure 5 in its place, and carrying the tens' figure 5 to its column proceed thus: 8, 24, 38, 48, 56, 66, 70, grouping the 5 carried and 3 for 8; 9 and 7 for 16 to add to 8 for 24; 6 and 8 for 14 to add to 504620 24 to make 38; 7 and 3 for 10, making 48; 1, 4, and 3 for 8, making 50; 6, 3, and 1 for 10, making 66, to which we add the 4 for 70, the result. Write the cipher of the 70 at the left of the unit figure already written beneath the line and carrying the 7 to the third, or hundreds' column group as before; 16, 26, 36, 48, 58, grouping upward thus: 7, 6, 1, 2 = 16; 2, 8=10; 9, 1= 10; 7, 5 = 12; 1, 4, 1, 4 = 10. Write the 8 in hundreds' column, and carrying the 5 to thousands' column, group 15, 27, 39, 49, 51. 5, 9, 1 = 15; 4, 8 = 12; 6, 5, 1 = 12; 7,3 10, and adding 2 write the result, 51, at the left of the figures before written, thus obtaining 51805 the numerical expression of the sum of the numbers added.

3417

2140

439

7164

1538

6116

8735

971 4880 1263 9270 192 634 51805

10

Prove by adding downward, grouping as illustrated above.

REMARK.-Practice in grouping will lead to great proficiency, and after the pupil becomes somewhat skilled, he should be encouraged to skip about somewhat along the column, in order to select those numbers which can be most conveniently grouped. Ordinarily thorough drill in the addition table will greatly assist in grouping, and multiples of the nine digits can be added with ease. Except with very bright pupils, groups greater than 25 are not to be recommended.

HORIZONTAL ADDITION.

63. Numbers when written in horizontal order, as in invoices and other business forms, may be added without being re-written in vertical columns.

REMARKS.—1. In adding numbers written horizontally, more care is requisite that the units added shall be of like order, and greater certainty of correctness can be had by adding first from left to right, and then from right to left.

2. The group method may be employed with equal advantage where numbers are written horizontally.

MENTAL EXERCISES.

64. Add from left to right, and review from right to left.

1. 5, 3, 6, 1, 8, 2, 7, 9, 4.
2. 21, 56, 12, 93, 47, 60, 17.
3. 66, 29, 5, 14, 71, 19, 2, 11.
4. 149, 865, 73, 40, 5, 13, 502.
5. 365, 10, 88, 46, 200, 175, 95.

6.

15, 23, 36, 18, 25, 53, 92. 7. 11, 85, 315, 125, 111, 206.

8.

8, 42, 87, 20, 112, 108, 94, 128.

9. 61, 400, 1, 126, 25, 440.

10.

25, 50, 511, 3, 209, 8, 804.

WRITTEN EXERCISES.

65. Copy, and add from left to right; review from right to left, preserving results.

1. 510, 297, 69, 841, 638, 203, 40, 7, 700, 28, 9.

2. 1260, 2700, 408, 9206, 51, 7240, 27, 1620.
3. 8809, 1492, 1000, 20, 1, 504, 6620, 7596, 10.
4. 50000, 20000, 8900, 21050, 47800, 14090.
5. 76030, 20500, 38037, 69000, 81, 107, 2, 19975.
6. 346211, 218040, 173508, 973200, 701001, 555555.
7. 604000, 181523, 51, 19406, 200, 309, 5, 2, 8000.
8. 2463911, 7054133, 4444044, 1371005, 6090400.
8500500, 1035660, 5000000, 2987400, 7020319.

9.

10. 416, 49, 2, 7967400, 81, 307, 21021, 190200, 40, 3.

REMARK.-Horizontal addition is rarely practiced with numbers containing more than four or five figures. It may sometimes be employed to advantage in adding dollars and cents; in such cases it is best to omit the dollar sign; as, for $5.25 write 5.25.

66. Copy and add horizontally; review and preserve results.

1. 5.25, 8.17, 11.40, 1.82, 16.02, 90.70.

2. 146.24, 9.11, 210.10, 46.98, 5.50, 108.12, 4.75.

3.

4.

5.

6.

7.

8.

29.30, 403, 51, 73, 1.14, 90, 300, 1.25.
1.13, 9.25, 14, 27.16, 5.01, 8, 25, 1.75.
87.50, 125, 36.21, 9.90, 14.75, 16, 25.25.

10. 117.82, 7.71, 19.03, 15, 49.55, 87.08.
11. 5.40, 88, 35, 90, 112.50, 45.95, 111.50.

12. 100, 79.22, 50.08, 2.25, 7.75, 10, 3, 8.24. 13. 216.24, 92, 15,.06, 138.50, 2.38, 9.25.

REMARK.—The teacher may give other examples of the same kind; he will find extensive drill in such work of great value to all grades of pupils, in developing accuracy and rapidity.

EXAMPLES FOR PRACTICE.

67. 1. A grocer's sales were, for Monday, $241; Tuesday, $306; Wednesday, $523; Thursday, $438; Friday, $497; on Saturday his sales amounted to $27 more than the sales of the first three days of the week. What were his total sales during the week?