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DEFINITIONS AND PRINCIPLES. ART. 39.—Subtraction is the process of finding the difference between two numbers. 8 less three is 5.

ART. 40.-The Minuend is the number from which another number is to be subtracted. 8 is the minuend.

ART. 41.—The Subtrahend is the number to be subtracted. 3 is the subtrahend.

ART. 42.—The Difference is the number obtained by subtraction. 5 is the difference.

ART. 43.—The Sign of Subtraction (-) is read minus. When placed between two numbers, it shows that the number on the right is to be subtracted from the number on the left. Thus, 8 - 3 is read 8 minus 3, and means that 3 is to be subtracted from 8.

ART. 44.—Principles. 1. Only like numbers can be subtracted.

2. The sum of the subtrahend and difference equals the minuend.

3. If the minuend and subtrahend are equally increased or diminished, the difference is unchanged.

Thus : y - 4 3. If 2 be added to both the minuend and the subtrahend, we have 9 – 6 3; or, if 2 be subtracted from both the minuend and the subtrahend, we have 5 2=3.

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ORAL EXERCISES. ART. 45.-1. Newton had 13 apples and gave away 9: how many had he left?

Solution. He had left the difference between 13 apples and 9 apples, or 4 apples.

2. Thomas had 14 cents and spent 9: how many had he left ?

3. In a school numbering 35 pupils, 15 were absent : how many were present ?

4. A watch which was bought for $23 was sold for $18: what was the loss ?

5. John solved 21 examples, and Harry solved 11 : how many more did John solve than Harry ?

6. If a man earns $27 a week and his expenses are $15, how much does he save ?

7. Fritz earns $55 a month ; he spends $15 for board and $12 for other expenses : how much does he save ?

8. Harry bought a hat for $2 and a book for $1 : how much change was due him from a five-dollar bill ?

9. From a piece of cloth containing 65 yards, a merchant sold 15 yards to one customer and 20 yards to another : how many yards were left ?

10. Anna's father gave her 10 cents, her mother gave her 12 cents, and her grandmother gave her 15 cents. She spent 37 cents: how much was left ?

11. Amos sold a horse for $120, which was $15 more than he gave for it: what was the cost of the horse ?

12. Mr. Glenn bought a carriage for $75, paid $10 for repairing it, and then sold it for $88: how much did he gain?

13. The difference is 10 and the subtrahend 19: what is the minuend ? The difference is 12 and the subtrahend 45, what is the minuend ?

14. The minuend is 36; subtrahend 14: what is the difference ? The minuend is 49; difference 34: what is the subtrahend ?

15. The difference is 40; minuend 60: what is the subtrahend? The minuend is 35; subtrahend 18: what is the difference? The difference is 21 ; minuend 52: what is the subtrahend ?

WRITTEN EXERCISES.

ART. 46.-1. From 8756 take 4894.

Process. Analysis.-We write the subtrahend under the 8756

minuend, placing units of the same order in the 4894 same column, and begin at the right to subtract. 3862

Four units from 6 units leave units, which we

write in the units' place. Nine tens cannot be subtracted from 5 tens, so we add 10 tens to 5 tens in the minuend making 15 tens. Nine tens from 15 tens leave 6 tens, which we write in the tens' place. Since we increased the minuend by 10 tens, we must increase the subtrahend by 1 hundred, equal to 10 tens. Nine hundreds cannot be taken from 7 hundreds, so we add 10 hundreds to 7 hundreds in the minuend, making 17 hundreds. Nine hundreds from 17 hundreds leave 8 hundreds, which we write in the hundreds' place. Since we increased the minuend by 10 hundreds, we must increase the subtrahend by 1 thousand, equal to 10 hundreds. Five thousands from 8 thousands leave 3 thousand, which we write in the thousands' place.

Therefore, 4894 subtracted from 8756 leaves 3862.
Proof.-4894 + 3862 = 8756.

ART. 47.-Rule for Subtraction.— Write the subtrahend under the minuend, placing units of the same order in the same column, and begin at the right to subtract.

Subtract each order in the subtrahend from the order above it, placing the remainder beneath. If any order in the subtrahend is greater than the order above it, add 10 to the latter and subtract. In such case increase the next subtrahend order by 1 before subtracting.

Proof. Add the subtrahend to the remainder, and if the sum equals the minuend the work is correct.

Solve and prove the following: 2.

3. From 6405621402

7204325021 Take 3233313323

2323032313

4 5024301382 2233223323

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23. A contractor received $18,250 for building a house that cost him $16,119 : how much was his profit ?

24. A gained $1,785.75 in one year, and B gained $2,348.90 in one year : how much more did B gain than A ?

25. In a storm at sea, a ship was obliged to throw overboard $23,250 worth of goods. If the value of the cargo was $89,784, how much was saved ?

26. A gentleman bequeathed $38,475 to his two sons,

To one he gave $19,360 : how much did he give to the other ?

27. If a man's income is $2,875 and his expenses $1,976, how much does he save ?

28. The population of the United States in 1790 was 3,929,214, and in 1800 it was 5,308,483 : how much was the increase in 10 years ?

29. In 1880, there were in Missouri 1,557,631 persons not less than 10 years old, of whom 138,818 could not read : how many were able to read ?

30. The number of people in the United States who were not less than 10 years of age in 1880, was 28,761,607; of these 4,923,451 could not read : how many were able to read ?

31. The minuend is $4,750.80, and the difference $3,240.66 : what is the subtrahend ?

32. The subtrahend is $66,580.05, and the difference $29,333.08 : what is the minuend?

33. The difference is $17,482.04, and the subtrahend $23,541.09 : what is the minuend ?

34. From ten thousand six hundred fifty-four take eight thousand three hundred ninety-six.

35. From ten thousand ten take five thousand five.

36. From four million four thousand fourteen take two million six hundred thousand eighteen.

37. From seventeen thousand four hundred twenty-five dollars take eleven thousand fifty dollars.

38. From one hundred twenty-five cents take eighty cents.

39. From one thousand one dollars take eleven dollars.

40. From one million one thousand dollars take one million nine dollars.

41. Illustrate by an original problem the subtraction of one number from another.

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