6 from 29. 6 from 69. 6 from
6 from 99.
6 from 89.

6. Take 6 from 19. 49. 6 from 59. 6 from 79.

7. Take 7 from 16. 7 from 26. 7 from 46. 7 from 7 from 76. 7 from 66. 7 from 86.

36. 7 from 56. 7 from 96.

8. 2 from II.

from 51. 2 from

2 from 21.

2 from 31.

2 from 41. 2

81.

2 from 91.

61. 2 from 71. 2 from

9. 4 from 22. 4 from 32. 4 from 62. 4 from 52. 4 from 42. 4 from 82. 4 from 72. 4 from 92.

10. 5 from 13. 5 from 23. 5 from 43. 5 from 53. 5 from 33. 5 from 63. 5 from 83. 5 from 73. 5 from 93.

II. 6 from 23. 6 from 43. 6 from 33. 6 from 53. 6 6 from 83. 6 from 93. 6 from 73.

12. 7 from 12. 7 from 22. 7 from 52. 7 from 72. 7 from 62. 7 from 42. 7 from 82. 7 from 92. 13. 8 from 94. 8 from 84. 8 from 74. 8 from 64. 8 from 54. 8 from 44. 8 from 34.

9

14. 9 from 85. 9 from 75. 9 from 65. 9 from 55. from 45. 9 from 35. 9 from 25.

15. Count backward by nines from 90 to o, as above. 16. Count backward by tens from 100 to 0.

MENTAL EXERCISES.

1. The age of a father is 50 years, and that of his son 20 years: how much older is the father than the son? ANALYSIS.--50 is 5 tens, and 20 is 2 tens; now 2 tens from 5 tens leave 3 tens, or 30. Therefore, etc.

When a figure in the Lower Number is Larger than the one

above it.

1. Find the difference between 723 dols. and 476 dols. ?

OPERATION.

723 dols. 476 dols. Ans. 247 dols.

IST METHOD.-Set down the numbers and begin at the right hand as before. Since 6 units cannot be taken from 3 units, we borrow I of the 2 tens and add it to the 3, making 13 units. Now 6 from 13 leaves 7, which we set under the figure subtracted. As we borrowed 1 of the 2 tens there is but I left; and 7 tens cannot be taken from 1 ten. We therefore borrow I of the 7 hundred and add it to the I ten, making II tens; and 7 from II leaves 4, which we set in tens' place. As we borrowed 1 of the 7 hundred, there are but 6 hundred left; and 4 from 6 leaves 2, which we set in hundreds' place. Borrowing Illustrated by Unit Marks.

ANALYSIS.-Let the 7 hundreds of the minuend be representen

by 7 marks, the 2 tens by 2 marks, and the 3 units by 35 marks Let the subtrahend be represented in like manner.

Since we cannot take 6 units from 3 units, we borrow one of the 2 tens, which reduced to units, we add to the 3 units, making 13 units; and 6 from 13 leaves 7. Next, 7 tens cannot be taken from I ten (I ten being erased and transferred to the units), we therefore borrow one of the hundreds, and add it to the I ten, making II tens; then 7 from 11 leaves 4. Finally, 4 hundreds from 6 hundreds (1 hundred being erased and transferred to the tons), leave 2 hundred. The result is 247 dollars.

2D METHOD.-As 6 units cannot be taken from 3 units, we add Io to the 3, making 13; and 6 from 13 leaves 7, which we set under the figure subtracted. To balance the 10 added to the 3, instead of considering the next upper figure 1 less than it is, we add I ten to the 7 tens, the next figure in the lower number, making 8 tens. But 8 tens cannot be taken from 2 tens; we again add 10 to the 2, making 12 tens, and 8 from 12 leavcs 4, which we set in tens' place. Finally, to balance the 10 added to 2, we add 1 to the next figure in the lower number, making 5 hundred, and 5 from 7 leaves 2, which we set under the figure subtracted. The result is 247 dols., the same as before.

10. What is adding 10 to the upper figure called?

Borrowing ten.

11. Why does not borrowing 10 affect the difference between

the two numbers?

The First Method simply transfers a unit from a higher to the next lower order of the minuend; therefore its value is not altered.

By the Second Method the two numbers are •qually increased; and when two numbers are equally increased, their difference is not altered.

NOTE. This method is the less liable to mistakes, and is more generally practiced by business men.

12. How proceed by the second method, when the figure in the lower number is larger than the one above it ?

Add 10 to the upper figure, then subtract, and add 1 to the next figure in the lower number.

15. The preceding principles may be summed up in the following

I. Place the less number under the greater, units under units, tens under tens, etc.

II. Begin at the right, and subtract each figure in the lower number from the one above it, setting the remainder under the figure subtracted.

III. If a figure in the lower number is larger than the one above it, add 10 to the upper figure; then subtract, and add to the next figure in the lower number.

I

PROOF.-Add the remainder to the subtrahend; if the sum is equal to the minuend, the work is right.

NOTE. This proof depends upon the Axiom that the whole is equal to the sum of all its parts.

9. From 6435 quarts, take 4268 quarts.

10. From 265045 barrels, take 120328 barrels.

11. A farmer having 2568 bushels of corn, sold 1830 bushels: how many bushels had he left?

12. A's income is 2345 dollars, B's 3068 dollars: what is the difference between their incomes?

3. A man paid 1730 dollars for his horses, and 2135 dollars for his carriage: what was the difference in their cost?

14. What is the difference between nineteen hundred and nine, and nine hundred and nineteen?

15. What is the difference between two thousand and four, and one thousand and fourteen?

16. Find the difference between eight hundred and eight, and eight thousand and eighty.

17. 7800461 - 4560231.

18. 8000030 - 6234521.

20. 9603245 2896750. 22. 1900000

-

899996.

19. 7930451 — 4000459. 21. 6235672-4000563. 23. If a farmer has 738 sheep, how many more must

he buy to make up 1320 ?

24. A man bought goods for 1943 dollars, and sold the same for 2365 dollars: what did he gain:

25. A man born in 1783, died in 1866: how old was he?

26. A person bought a drove of cattle for 5263 dollars, and sold them for 4675 dollars: how much did he lose? 27. The difference between the ages of two persons is 15 years, and the older is 79 years: how old is the younger?

28. 1463 and what number make 3185?

29. A has 765 dollars, B 1695 dollars, and C's money was equal to the difference between A's and B's: how much money had C?

30. The Pilgrim Fathers landed at Plymouth Rock in 1620, and the independence of the colonies was declared in 1776: how many years between these two events?