22. Add 37, 40, 63, 84, 67, 22, and 70. 23. Add 95, 46, 77, 66, 88, 63, 33, and 44. 24. Add 67, 76, 45, 54, 38, 83, 27, and 72. 25. Add 68, 86, 97, 79, 86, 68, 78, and 87. 26. Add 45, 60, 57, 86, 83, 76, 49, 58, and 84. 27. Add 56, 75, 83, 96, 69, 73, 37, 38, and 205. 28. Add 27, 72, 33, 38, 69, 96, 75, 57, and 336. 29. Add 235, 88, 77, 66, 55, 44, 33, 22, and 11. 30. Add 405, 56, 43, 47, 74, 36, 63, 75, and 66. 31. Add 46, 67, 72, 38, 99, 87, 65, 74, and 88. 32. Add 73, 86, 47, 56, 69, 65, 58, 33, 52, and 94.

DEFINITIONS AND PRINCIPLES.

23. Addition is the process of finding the sum of two or more numbers.

The Sum of two or more numbers is a number containing as many units as all of them, taken together. It is also called the Amount.

24. The Sign of Addition is a short vertical line bisecting an equal horizontal line, thus: +. It is called plus.

25. The Sign of Equality is two short horizontal parallel lines, thus: =. It is read equals or is equal to. Thus, 7815 is read 7 plus 8 equals 15.

26. Like Numbers are composed of units of the same kind. Thus, 4 balls and 8 balls, or 4 dimes and 8 dimes, or 4 and 8, are like numbers.

27. PRINCIPLES.-1. Only like numbers can be added. 2. Only like orders of figures can be added.

3. The sum is of the same kind or order as the numbers addet. 4. The sum is the same whatever be the order in which the numbers are added.

NOTE.--See appendix for method of proof by "casting out the 9's."

SECTION IV.

SUBTRACTION.

MENTAL PROBLEMS.

1. Count by 4's from 61 back to 1, 2. Count by 6's from 53 back to 5; 3. Count by 7's from 66 back to 3; 4. Count by 8's from 75 back to 3; 5. Count by 9's from 73 back to 1;

thus: 61, 57, 53, etc. from 74 back to 2. from 85 back to 1. from 94 back to 6. from 96 back to 6.

6. A grocer having a certain number of sacks of flour, bought 48 sacks, and sold 33 sacks, and then had 34 sacks on hand: how many sacks had he at first?

7. A man sold a horse for $95, which was $28 more than the horse cost him: what was the cost of the horse?

8. Two men start at once from the same point, and travel in the same direction, one traveling 52 miles a day, and the other but 39 miles: how far will they be apart at the close of the second day?

9. A man earns $85 a month, and pays $18 for house rent, and $35 for other expenses: how much does he save each month?

10. A gentleman being asked his age said, that if he should live 27 years longer, he should then be three score and ten what was his age?

11. From a piece of carpeting containing 68 yards, a merchant sold 27 yards to one man and 18 yards to another how many yards of the piece were left?

12. A man bought a carriage for $135, paid $21 for repairing it, and then sold it for $170: how much did he gain?

13. A boy earned 65 cents, and his father gave him 33 cents; he paid 45 cents for an arithmetic and 18 cents for a slate: how much money had he left?

14. There are 85 sheep in three fields; there are 36 sheep

C.Ar.-2

in the first field, and 28 sheep in the second: how many sheep in the third field?

15. John had 33 chestnuts, and Charles 25; John gave Charles 14 chestnuts, and Charles gave his sister as many as he then had more than John: how many chestnuts did the sister receive?

WRITTEN PROBLEMS.

16. A builder contracted to build a school-house for $25460, and the job cost him $21385: what were his profits?

17. The earth's mean distance from the sun (old value) is 95274000 miles, and that of Mars is 145168136: how much farther is Mars from the sun than the earth?

18. The population of Illinois in 1860 was 1711951, and in 1865 its population was 2141510: what was the increase in five years?

19. The population of Massachusetts in 1860 was 1231066, and in 1865 it was 1267031: what was the increase in five years?

20. The area of the Chinese Empire is 4695334 square miles, and the area of the United States is 3578392 square miles: how much greater is the Chinese Empire than the United States?

21. The area of Europe is 3781280 square miles: how much greater is Europe than the United States? The Chinese Empire than Europe?

22. In 1866, Ohio produced 99766822 bushels of corn, and Illinois 155844350 bushels: how many bushels did Illinois produce more than Ohio?

23. A man bought a farm for $5867, and built upon it a house at a cost of $1850, and then sold the farm for $7250: how much did he lose?

24. An estate of $13450 was divided between a widow and two children; the widow's share was $6340, the son's $1560 less than the widow's, and the rest fell to the daughter: what was the daughter's share?

25. A man deposited in a bank at one time $850, at an

other, $367, and at another, $670; he then drew out $480, and $375: how much money had he still in bank?

26. A man bought a farm for $6450, giving in exchange a house worth $4500, a note for $1150, and paying the difference in money: how much money did he pay?

27. A grain dealer bought 15640 bushels of wheat, and sold at one time 3465 bushels, at another, 4205, and at another, 1080: how many bushels remained?

28. A has 320 acres of land; B has 65 acres more than A; C has 124 acres less than both A and B; and D has as many acres as both A and C less the number of acres owned by B. How many acres have B, C, and D respectively? How many have all?

29. From the sum of 45003 and 13478, take their differ

ence.

DEFINITIONS AND PRINCIPLES.

28. Subtraction is the process of finding the difference between two numbers.

The Difference is the number found by taking one number from another. It is also called the Remainder.

The Minuend is the number diminished.

The Subtrahend is the number subtracted.

29. The Sign of Subtraction is a short horizontal line, made thus It is called minus. Thus 12 5 is read 12 minus 5.

30. PRINCIPLES.-1. The minuend, subtrahend, and difference are like numbers.

2. The minuend is the sum of the subtrahend and difference. 3. If the minuend and subtrahend be equally increased, the difference will not be changed.

4. The adding of 10 to a term of the minuend and 1 to the next higher term of the subtrahend, increases the minuend and subtrahend equally.

SECTION V.

MULTIPLICATION.

MENTAL PROBLEMS.

1. There are 24 hours in a day: how many hours in 7 days? In 9 days? 11 days?

2. There are 60 minutes in an hour: how many minutes in 8 hours? In 12 hours? 15 hours?

3. If a man earn $63 a month, and spend $48, how much will he save in 12 months?

4. If 12 men can do a piece of work in 15 days, how long will it take one man to do it?

5. If 35 bushels of oats will feed 8 horses 25 days, how long will they feed one horse?

6. Two men start from the same place and travel in opposite directions, one at the rate of 28 miles a day, and the other at the rate of 32 miles a day: how far will they be apart at the end of five days?

7. Two men are 450 miles apart: if they approach each other, one traveling 30 miles a day and the other 35 miles a day, how far apart will they be at the end of 6 days?

8. A cask has two pipes, one discharging into it 90 gallons of water an hour, and the other drawing from it 75 gallons an hour: how many gallons of water will there be in the cask at the end of 12 hours?

9. A had $24, B four times as much as A less $16, and C twice as much as A and B together plus $17: how much money had B and C?

10. A farmer sold to a grocer 15 pounds of butter, at 30 cents a pound, and bought 8 pounds of sugar, at 15 cents a pound, and 9 pounds of coffee, at 20 cents a pound: how much was still due him?