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III. SUBTRACTION.

MENTAL Exercises.

ART. 25. WHEN it is required to find the difference be tween two numbers, the process is called Subtraction. The operation is the reverse of addition.

984

Ex. 1. John has 7 oranges, and his sister but 4; how many more has John than his sister?

ILLUSTRATION. To solve this question, we first inquire what number added to 4 will make 7. From addition we learn that 4 and 3 are 7; consequently, if 4 oranges be taken from 7 oranges, 3 will remain. Hence John has 3 oranges more than his sister.

QUESTIONS. Art. 25. What does subtraction teach? Of what is it the

reverse?

The following table will be of service to facilitate the progress of the learner in the solution of questions in subtraction.

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9 from 9 leaves
9 from 10 leaves
9 from 11 leaves
9 from 12 leaves
9 from 13 leaves
9 from 14 leaves
9 from 15 leaves
9 from 16 leaves
9 from 17 leaves
9 from 18 leaves
9 from 19 leaves 10
9 from 20 leaves 11
9 from 21 leaves 12

2

3

4

5

6

7

8

9

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5

6

7

8

9

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7 from 7 leaves
7 from 8 leaves

7 from 9 leaves

7 from 10 leaves

1

2

3

4

5

6

7

8

0

1

2

3

4

7 from 11 leaves

7 from 12 leaves

7 from 13 leaves

7 from 14 leaves
7 from 15 leaves
7 from 16 leaves
7 from 17 leaves 10
7 from 18 leaves 11
7 from 19 leaves 12

5

6

7

8

9

0

1

0

10 from 10 leaves
110 from 11 leaves
10 from 12 leaves
10 from 13 leaves
10 from 14 leaves
10 from 15 leaves

0 11 from 11 leaves
111 from 12 leaves
211 from 13 leaves
3 11 from 14 leaves
411 from 15 leaves

12 from 14 leaves 2 12 from 15 leaves 3 412 from 16 leaves 4

11 from 16 leaves

12 from 17 leaves 5

10 from 16 leaves

11 from 17 leaves

12 from 18 leaves 6

10 from 17 leaves

11 from 18 leaves

12 from 19 leaves 7

10 from 18 leaves

11 from 19 leaves

12 from 20 leaves 8
10 from 19 leaves
12 from 21 leaves 9
10 from 20 leaves 10
12 from 22 leaves 10
10 from 21 leaves 11
12 from 23 leaves 11
10 from 22 leaves 1211 from 23 leaves 1212 from 24 leaves 12

11 from 20 leaves
11 from 21 leaves 10
11 from 22 leaves 11

2

3

4 from 4 leaves 0 4 from 5 leaves 1

4 from

4 from

4 from

5

6

7

8

9

6 leaves 2

7 leaves 3

8 leaves 4

4 from 9 leaves 5

4 from 10 leaves 6 4 from 11 leaves 7

4 from 12 leaves 8 4 from 13 leaves 9 4 from 14 leaves 10 4 from 15 leaves 11 4 from 16 leaves 12

8 from 8 leaves 0 8 from 9 leaves 1 8 from 10 leaves 2 8 from 11 leaves 3 8 from 12 leaves 4 8 from 13 leaves 5 8 from 14 leaves 6 8 from 15 leaves 7

8 from 16 leaves - 8

8 from 17 leaves 9 8 from 18 leaves 10 8 from 19 leaves 11 8 from 20 leaves 12

12 from 12 leaves 0 12 from 13 leaves 1

2. Thomas had five oranges, and gave two of them to John; how many had he left?

3. Peter had six marbles, and gave two of them to Samuel; how many had he left?

4. Lydia had four cakes; having lost one, how many had she left?

5. Daniel, having eight cents, gives three to Mary; how many has he left?

6. Benjamin had ten nuts; he gave four to Jane, and three to Emily; how many had he left?

7. Moses gives eleven oranges to John, and eight to Enoch ; how many more has John than Enoch?

8. Agreed to labor for a man twelve days; how many remain, after I have been with him five days?

9. I owed Thomas nine dollars, and, having paid him seven, how many remain due?

10. From ten dollars, I paid four dollars to one man, and three dollars to another; how much have I left?

11. Timothy had eleven marbles, and lost seven; how many had he left?

12. John is thirteen years old, and his brother Thomas is seven; how much older is John than Thomas ?

13. From fifteen dollars, I paid five; how many had I left? 14. Sold a barrel of flour for eight dollars, and a bushel of wheat for two dollars; what was the difference in the prices?

15. Paid seven dollars for a pair of boots, and two dollars for shoes; how much did the boots cost more than the shoes? 16. How many are 4 less 2? 4 less 1? 4 less 4? 17. How many are 4 less 3? 5 less 1? 5 less 5? 18. How many are 5 less 2? 5 less 3? 5 less 4? 19. How many are 6 less 1? 6 less 2? 6 less 4? 20. How many are 7 less 2? 7 less 3? 7 less 4? 21. How many are 8 less 6? 8 less 5? 8 less 2? 4? 8 less 1?

22. How many are 9 less 2? 9 less 3?

7?

23. How many are 10 less 8? less 3? 10 less 1?

24. How many are 11 less 9? less 3? 11 less 4 ?

25. How many are 12 less 10? less 4? 12 less 7?

27. How many are less 6? 14 less 7?

28. How many are less 7? 15 less 9?

26. How many are 13 less 11? 13 less 9 ? 13 less 5?

29. How many are less 9? 16 less 11? 30. How many are less 7? 17 less 8?

9 less 4?

10 less 7?

31. How many are less 8? 18 less 10?

32. How many are less 7?

19 less 9?

11 less 7?

12 less 8?

6 less 5? 7 less 6?

8 less

11 less 5? 11

12 less 6? 12

13 less 10? 13 less 7?

16 less 4? 16 less 7? 16

14 less 11? 14 less 9? 14 less 8? 14
14 less 3?
15 less 2?
15 less 13?
16 less 3?
16 less 15?
17 less 1?
17 less 12?
18 less 2? 18 less 4?
18 less 12?

17 less 3?

17 less 5? 17

18 less 7? 18

15 less 4 ?

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15 less 5? 15

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33. How many are 20 less 5? 20 less 8? 20 less 9? 20 less 12? 20 less 15? 20 less 19'

34. Bought a horse for 60 dollars, and sold him for 90 dollars; how much did I gain?

ILLUSTRATION. We may divide the two prices of the horse into tens, and subtract the greater from the less. Thus 60 equals 6 tens, and 90 equals 9 tens; 6 tens from 9 tens leave 3 tens or 30. Therefore I gained 30 dollars.

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35. Sold a wagon for 70 dollars, which cost me 100 dollars; how much did I lose?

36. John travels 30 miles a day, and Samuel 90 miles; what is the difference?

37. I have 100 dollars, and after I shall have given 20 to Benjamin, and paid a debt of 30 dollars to J. Smith, how many dollars have I left?

38. John Smith, Jr. had 170 dollars; he gave his oldest daughter, Angeline, 40 dollars, his youngest daughter, Mary, 50 dollars, his oldest son, James, 30, and his youngest son, William, 20 dollars; he also paid 20 dollars for his taxes; how many dollars had he remaining?

ART. 26. The pupil, having solved the preceding questions, will perceive, that

SUBTRACTION is the taking of a less number from a greater to find the difference.

The greater number is called the Minuend, and the less number, the Subtrahend.* The answer, or number found by the operation, is called the Difference or Remainder.

ART. 27. SIGNS. - Subtraction is denoted by a short horizontal line, thus, signifying minus or less. It indicates that the number following is to be taken from the one that precedes it. The expression 6-2 = 4 is read, 6 minus, or less, 2 is equal to 4.

The words minuend and subtrahend are derived from two Latin words, the former from minuendum, which signifies to be diminished or made less, and the latter from subtrahendum, which means to be subtracted or taken away.

QUESTIONS. Art. 26. What is subtraction? What is the greater number called? What is the less number called? What the answer? - Art. 27. What is the sign of subtraction? What does it signify and indicate?

EXERCISES FOR THE SLATE.

ART. 28. Method of operation, when the numbers are large, and each figure in the subtrahend is less than the figure above it in the minuend.

Ex. 1. Let it be required to take 245 from 468, and to find their difference. Ans. 223.

In this operation, we place the less number under the greater, units under units, tens under tens, &c., and draw a line below them. We then begin at the right hand, and say, 5 from 8 leaves 3, and write the 3 directly below. We then say,

Remainder 223

4 from 6 leaves 2, and write the 2 below the line, as before, and proceed with the next figure and say, 2 from 4 leaves 2, which we also write below. We thus find the difference to be 223.

OPERATION.

Minuend 468
Subtrahend 245

ART. 29. First Method of Proof. — Add the remainder and the subtrahend together, and their sum will be equal to the minuend, if the work is right.

This method of proof depends on the obvious principle, That the greater of any two numbers is equal to the less added to the difference between them.

EXAMPLES FOR PRACTICE.

2.

3.

3.

OPERATION AND PROOF. OPERATION. OPERATION AND PROOF.

547

986

235

763

223

2.

OPERATION.

Minuend 547
Subtrahend 235

Remainder 312

4.

From 684
Take 462

312

Min. 5 47

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986

763

223

Min. 986

7. 948 746

8. A farmer paid 539 dollars for a span of fine horses, and sold them for 425 dollars; how much did he lose?

Ans. 114 dollars.

9. A farmer raised 896 bushels of wheat, and sold 675 bushels of it; how much did he reserve for his own use? Ans. 221 bushels.

QUESTIONS.-Art 28. How are numbers arranged for subtraction? Where do you begin to subtract? Why? Where do you write the difference? Art. 29. What is the first method of proving subtraction? What is the reason of this proof, or on what principle does it depend?

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