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.

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 6-24 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.

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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.

OPERATION.

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, 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.

Minuend 468
Subtrahend 245
Remainder 223

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.

2.

EXAMPLES FOR PRACTICE.

2.

3.

3.

OPERATION. OPERATION AND PROOF. OPERATION. OPERATION AND PROOF.

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

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

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?

10. A gentleman gave his son 3692 dollars, and his daugh ter 1212 dollars less than his son; how much did his daughter receive?

ART. 30. Method of operation when any figure in the subtrahend is greater than the figure above it in the minuend. Ex. 1. If I have 624 dollars, and lose 342 of them, how many remain ? Ans. 282.

OPERATION.

Minuend 624
Subtrahend 342

In performing this example, we first take the 2 units from the 4, and find the difference to be 2, which we write directly under the figure subtracted. We then proceed to take Remainder 282 the 4 tens from the 2 tens above it; but we here find a difficulty, since the 4 is greater than 2, and cannot be subtracted from it. We therefore add 10 to the 2, which makes 12, and then subtract 4 from 12 and 8 remains, which we write directly below. Then, to compensate for the 10 thus added to the 2 in the minuend, we add one to the 3 in the next higher place in the subtrahend, which makes 4, and subtract 4 from 6, and 2 remains. The remainder, therefore, is 282.

The reason of this operation depends upon the self-evident truth, That, if any two numbers are equallly increased, their difference remains the same. In this example 10 tens, equal to 1 hundred, were added to the 2 tens in the upper number, and 1 was added to the 3 hundreds in the lower number. Now, since the 3 stands in the hundreds' place, the 1 added was in fact 1 hundred. Hence, the two numbers being equally increased, the difference is the same.

NOTE. This addition of 10 to the minuend is sometimes called borrowing 10, and the addition of 1 to the subtrahend is called carrying 1.

ART. 31. From the preceding examples and illustrations in subtraction, we deduce the following general

RULE.

1. Place the less number under the greater, units under units, tens under tens, &c., and draw a line under them.

2. Then commencing with the units, subtract each figure of the subtrahend from the figure above it in the minuend, and write the difference below.

QUESTIONS. Art. 30. How do you proceed when a figure of the subtrahend is larger than the one above it in the minuend? How do you compensate for the 10 which is added to the minuend? What is the reason for this addition to the minuend and subtrahend? How does it appear that the 1 added to the subtrahend equals the 10 added to the minuend? What is the addition of 10 to the minuend sometimes called? The addition of 1 to the subtrahend? Art. 31. What is the general rule for subtraction?

3. If any figure of the subtrahend is larger than the figure above it in the minuend, add 10 to that figure of the minuend, and from their sum subtract the lower figure; then carry 1 to the next figure of the subtrahend, and subtract as before, till all the figures of the subtrahend are subtracted, and the result will be the difference or remainder.

ART. 32. Second Method of Proof. — Subtract the remainder or difference from the minuend, and the result will be like the subtrahend if the work is right.

This method of proof depends on the principle, That the smaller of any two numbers is equal to the remainder obtained by subtracting their difference from the greater.

EXAMPLES FOR PRACTICE.

2.

2.

3.

3.

OPERATION.

Minuend 376

OPERATION AND PROOF. OPERATION. OPERATION AND PROOF.

QUESTIONS. Art. 32. What is the second method of proving subtraction? What is the reason for this method of proof, or on what principle does it depend?

19. From 8054010657811 take 76909748598.

20. From 7100071641115 take 10071178.

21. From 501505010678 take 794090589.

22. Take 99999999 from 100000000.
23. Take 44444444 from 500000000.
24. Take 1234567890 from 9987654321.

25. From 800700567 take 1010101.

26. Take twenty-five thousand twenty-five from twenty-five millions.

27. Take nine thousand ninety-nine from ninety-nine thousand.

28. From one hundred one millions ten thousand one hundred one take ten millions one hundred one thousand and ten.

29. From one million take nine.

30. From three thousand take thirty-three.

31. From one hundred millions take five thousand.

32. From 1,728 dollars, I paid 961 dollars; how many re-, main?

33. Our national independence was declared in 1776; how many years from that period to the close of the last war with Great Britain in 1815?

34. The last transit of Venus was in 1769, and the next will be in 1874; how many years will intervene ?

35. In 1830, the number of inhabitants in Bradford was 1,856, and in 1840 it was 2,222; what was the increase?

36. How many more inhabitants were there in New York city than in Boston, in 1840, there being, by the census of that year, 312,710 inhabitants in the former, and 93,383 in the · latter city?

37. In 1821 there were imported into the United States 21,273,659 pounds of coffee, and in 1839, 106,696,992 pounds; what was the increase?

38. By the census of 1840, 11,853,507 bushels of wheat were raised in New York, and 13,029,756 bushels in Pennsylvania; how many bushels in the latter State more than in the former?