52. The preceding examples illustrate all the principles in subtraction. Hence, to perform subtraction,

RULE. 1. Write the less number under the greater, units, under units, tens under tens, etc., and draw a line beneath.

2. Beginning at the right hand, take each figure of the subtrahend from the figure above it, and set the remainder under the line.

3. If any figure in the subtrahend is greater than the figure above it, add TEN to the upper figure and take the lower figure from the SUM; set down the remainder and, considering the next figure in the minuend ONE LESS, or the next figure in the subtrahend ONE GREATER, proceed as before.

53. PROOF. Add the subtrahend and the remainder to gether, and the sum should be the minuend.

NOTE 1. This proof rests upon the self-evident truth, that the whole of a thing is equal to the sum of all its parts; thus, the minuend is separated into the two parts, subtrahend and remainder; hence the sum of those parts must be the minuend.

Ex. 28.
(5) (9) (12)

Minuend, 6 0 2
Subtrahend, 4 3 8

27.

7777777

566 6669

Here we cannot take 8 from 2, nor can we borrow from the tens' place, as that place is occupied by 0; but we can borrow one of the 6 hundreds and separate the one hundred into 9 tens and 10 units; then, putting the 9 tens in the place of tens and adding the 10 units to the 2 units, we can subtract 8 from 12, 3 from 9, and 4 from 5.

Remainder, 1 6 4

NOTE 2. This process will probably be more readily understood by the young learner than the second method given in the rule, though the latter, being thought more convenient, is usually adopted.

between two numbers is 365 and the Ans. 511.

greater number is 876; what is the less?

45. What number added to 3876 will give 7469? 46. What number taken from 8742 leaves 3748?

53. What is there peculiar in Ex. 28? Explain the

process.

47. The sum of two numbers is 8629, and the less of the two numbers is 2689; what is the greater? Ans. 5940.

48. The sum of two numbers is 8426, and the greater is 7162; what is the less?

49. From fourteen million, eight hundred and sixty-two thousand, three hundred and twenty-five, take six million, six hundred and eighty-six thousand, two hundred and fourteen.

Ans. 8176111. 50. From seven hundred and thirty-three thousand, six hundred and fifty-four, take two hundred and twenty-seven thousand, five hundred and fifteen.

51. How many years from the discovery of America by Columbus in 1492 to the birth of Washington in 1732?

52. How many years have elapsed since the discovery of America in 1492?

53. By the census of 1860, the number of inhabitants in Massachusetts was 1231065, and the number in Vermont was 315116; how many more in Massachusetts than in Vermont?

54. The population of the United States was 23191876 in 1850, and 17063353 in 1840; what was the increase in ten years?

55. The area of New England is 64230 square miles and the area of Maine is 35000 square miles; what is the area of the other five New England States?

56. About 56619608 bushels of corn were raised in Ohio in 1850, and 73436690 bushels in 1853; what was the increase? 57. Bought a paper mill for $15475, and sold it for $17925; what did I gain?

58. How many are 876942 - 468279?

59. How many are 742006

- 387429 ?

60. How many are 820654 -260408 ?

61. Washington was born in 1732 and died in 1799; at what age did he die?

62. A merchant sold goods to the amount of $4276, and thereby gained $1142; what did the goods cost him?

63. A farm was sold for $3462, which was $876 more than it cost; what did it cost?

How

64. The distance from the earth to the sun is about 95000000 miles; the distance to the moon is about 240000 miles. much farther to the sun than to the moon?

65. Methuselah died at the age of 969 years, and Washington

at 67; what was the difference of their ages?

66. Mr. Hale, owing a debt of $3762, paid $2486; how much remained unpaid?

EXAMPLES IN ADDITION AND SUBTRACTION.

1. From the sum of 76 and 92 take 14.

Ans. 154.

Ans. 1046.

2. From the sum of the three numbers, 876, 493, and 916, take the sum of 842 and 397. 3. I owe 3 notes, whose sum is $600. One of these notes is for $150, another for $200; for what is the third one?

4. My real estate is valued at $4500 and my personal property at $2596. I owe to A $600, to B $1358, and to C $318; what am I worth? Ans. $4820.

5. Bought a barrel of flour for $9, four yards of cloth for $2, and 8 pounds of sugar for $1. In payment I gave a ten and a five dollar bill; what change shall the merchant return to me?

6. Mr. Fox, owning 3762 acres of land, gave 563 acres to his oldest son, and 672 acres to his youngest son; how many acres had he remaining?

7. The area of Maine is 35000 square miles; N. H., 8030; Vt., 8000; Mass., 7250; R. I., 1200; Ct., 4750. Which is the greater, Maine or the rest of N. E.? How much?

8. Gave my note for $3465. Paid $1300 at one time, and $575 at another; how much do I still owe? Ans. $1590.

9. Mr. T., opening an account at the Andover Bank, deposited $187 on Monday, $362 on Tuesday, $580 on Thursday, and $675 on Friday. On Tuesday he withdrew $67, on Wednesday $213, on Friday $350, and on Saturday $125; how much remained on deposit at the close of the week? Ans. $1049.

10. A traveler who was 875 miles from home, traveled toward home 144 miles on Monday, 127 miles on Tuesday, 156 miles on Wednesday, and 157 miles on Thursday; how far from home was he on Friday morning?

11. From the discovery of America by Columbus in 1492, to the settlement of Jamestown in 1607, was 115 years, from the settlement of Jamestown to the Declaration of Independence in 1776, was 169 years, and from the Declaration of Independence to the present time (1862) is 86 years. Methuselah died at the age of 969 years; how much longer did he live than from the discovery of America to the year 1862 ?

12. Four men, A, B, C, and D, commencing business together, furnished money as follows: A, $2475; B, $3475; C, $2850; and D, $4500. At the end of a year they closed business, having lost $3225; how much money had they to divide between them?

MULTIPLICATION.

54. MULTIPLICATION is a short method of adding equal numbers; that is, multiplication is a short method of finding the sum of the repetitions of a number.

Or, MULTIPLICATION is a short method of finding how many units there are in any number of times a given number.

The MULTIPLICAND is the number to be repeated.

The MULTIPLIER is the number which shows how many times the multiplicand is to be taken.

The PRODUCT is the sum of the repetitions, or the result of the multiplication.

The Multiplicand and Multiplier are called FACTORS.

Ex. 1. There are 7 days in 1 week; how many days in 4 weeks?

This example may be solved by addition; thus, 7+7+7+7 28; or more briefly, by multiplication; thus, 4 times 7 are 28, Ans.

54. What is Multiplication? Another definition? What is the Multiplicand} Multiplier? Product? What are the Multiplicand and Multiplier called?