EXPLANATION.-The numbers in the left-hand column and across the top are the addends. The sums will be found in the horizontal columns opposite the numbers.

Thus, 1 and 0 are 1; 1 and 1 are two; 1 and 2 are 3; 1 and 3 are 4, etc. The importance of thoroughly committing this table should not be overlooked. The neglect of it accounts, in large measure, for the waste of time and inaccuracy in adding columns of figures. The pupil should work out the sums for himself, thus investigating the results, and then thoroughly

master the table.

DRILL EXERCISES.

38. The following list contains all the combinations of two num bers from 1 to 9.

If the student has thoroughly committed the addition table, he can tell the sums at a glance. If he can not, he should practice adding bers daily until able to do so.

the

num

39. 1. Count or add by 2's from 0 to 20, thus: 0, 2, 4, 6, etc.

2. Add by 2's from 1 to 33, thus: 1, 3, 5, 7, 9, etc.

3. Add by 3's from 0 to 30.
4. Add by 3's from 2 to 41.
5. Add by 4's from 0 to 44.
6. Add by 4's from 3 to 67.
7. Add by 5's from 0 to 60.
8. Add by 5's from 2 to 82.
9. Add by 5's from 4 to 89.
10. Add by 6's from 0 to 66.
11. Add by 6's from 2 to 80.
12. Add by 6's from 4 to 100.
13. Add by 7's from 0 to 70.
14. Add by 7's from 2 to 93.
15. Add by 7's from 4 to 95.
16. Add by 8's from 0 to 88.
17. Add by 8's from 2 to 106.
18. Add by 8's from 5 to 101.
19. Add by 9's from 0 to 90.
20. Add by 9's from 2 to 92.
21. Add by 9's from 4 to 112.
22. Add by 10's from 0 to 100.
23. Add by 10's from 2 to 122.

From 1 to 40.

From 3 to 45.
From 1 to 52.
From 5 to 73.
From 1 to 76.
From 3 to 93.
From 6 to 91.
From 1 to 73.
From 3 to 99.
From 5 to 101.
From 1 to 85.
From 3 to 108.
From 5 to 103.
From 1 to 97.
From 3 to 115.
From 7 to 103.
From 1 to 100.
From 3 to 102.
From 5 to 122.

From 1 to 111.
From 3 to 113.

WRITTEN EXERCISES.

40. Copy and add from the bottom upwards, and then from the top downwards, each of the following:

TO ADD LARGER NUMBERS.

41. 1. What is the sum of 132, 324, and 203 ?

132 324

203 659

EXPLANATION.-For convenience, we write the numbers so that all the figures of the same order stand in the same column. We draw a line beneath, and begin with the units to add. 3 units and 4 units are 7 units, and 2 units are 9 units, which we write under the column of units; 0 tens and 2 tens are 2 tens, and 3 tens are 5 tens, which we write under the column of tens; 2 hundreds and 3 hundreds are 5 hundreds, and 1 hundred are 6 hundreds, which we write under the column of hundreds. Hence, the sum is 6 hundreds, 5 tens, and 9 units, or six hundred fifty-nine.

2. What is the sum of 636, 873, and 945 ?

636 873

945

2454

EXPLANATION.-We write the figures of the same order in the same column as before and begin with the units to add. 5 units and 3 units are 8 units, and 6 units are 14 units, which equal 1 ten and 4 units; we write the 4 units under the column of units, and add the 1 ten to the column of tens. 1 ten and 4 tens are 5 tens, and 7 tens are 12 tens, and 3 tens are 15 tens, which equal 1 hundred and 5 tens; we write the 5 tens under the column of tens, and add the 1 hundred to the column of hundreds. 1 hundred and 9 hundreds are 10 hundreds and 8 hundreds are 18 hundreds, and 6 hundreds are 24 hundreds, which equal 2 thousands and 4 hundreds. We write the 4 hundreds under the column of hundreds, and place the 2 thousands on the left in the order of thousands. Hence, the sum is 2 thousands, 4 hundreds, 5 tens, and 4 units, or two thousand four hundred fifty-four.

1. In practice we name only the results. Thus, we say : 5, 8, 14; write the 4 and carry the 1; 1, 5, 12, 15; write the 5 and carry the 1; 1, 10, 18, 24; write 24. Hence the sum is 2454.

2. When the sum of any column is an exact number of tens, as 10, 20, 30, a cipher is written under the column added, and the 1, 2 or 3, etc., is added to the next column.

3. The addition of each column should be proved correct by adding it in the opposite direction from that in which it was first added.

RULE. Write the numbers to be added so that the figures of the same order stand in the same column, and draw a line beneath. Begin at the right, add the numbers expressed by the figures of each column separately, and write the sum underneath, if less than ten.

When the sum is ten or more, write the right-hand figure underneath, and add (carry) the number expressed by the other figure or figures with the numbers of the next column.

Write the whole sum of the last column.

PROOF.-Add each column in the opposite direction from that in which it was added before, and if the work is correct the result of both additions will be the same.

33. Add 2645, 7621, 4327, 6423, 5467.

34. Add 4856, 7358, 234, 579, 724, 3712, 48.
35. Add 3916, 4237, 8326, 47, 326, 516, 792.