47. Benjamin Franklin was born in 1706, and died in 1790. How old was he when he died?

48. Cotton was first planted in the United States in 1769. How many years since ?

49. Glass windows were first used in England in 1180. How many years since ?

50. Newspapers were first published in 1630. How many years since ?

51. Quills were used for writing in 636. How many years since ?

52. The first permanent settlement in Virginia was made in 1607. How many years since?

DIVISION.

SECTION VI.

30. DIVISION is the process of finding how many times one number is contained in another.

31. The number divided is called the dividend. The number divided by is called the divisor. The result is called the quotient. When any thing remains after dividing, it is called the remainder, and is always of the same kind as the dividend.

OBS. The term quotient is from the Latin word quoties, signifying how many times.

32. The sign of division is a horizontal line between two dots,, and is read divided by. Thus 12÷3 4 is read, 12 divided by 3 is equal to 4.

What is division? What is the dividend? What is the divisor? What is the quotient? What is the remainder? What is the sign of division?

33. When the divisor does not exceed 12,

RULE. Write the divisor at the left of the dividend. Find how many times the divisor is contained in the first left hand figure or figures, and write underneath the result. If there be no remainder, divide the next figure or figures in the same manner. If there be a remainder, suppose it to be prefixed to the next figure of the dividend, and divide as before.

OBS. When the divisor is not contained in any figure of the dividend, excepting the first, a cipher must be written in the quotient. 1. Divide 8756 by 6.

Divisor, 6) 8756, dividend.

Quotient, 1459, and 2 remainder.

Write the 1 un

6 is contained in 8 once, and 2 over. derneath, in the quotient, and prefix the 2 to the next figure, 7, making 27. 6 in 27 4 times, and 3 over. Write the 4 in the quotient, and prefix the 3 to the 5, making 35. 6 in 35 5 times, and 5 over. Write the 5 in the quotient, and prefix the 5 to the next figure, 6, making 56. 6 in 56 9 times, and 2 over. Write the 9 in the quotient, and the 2 at the right for the remainder.

EXAMPLES.

2. Divide 845678 by 4. 3. Divide 967834 by 2. 4. Divide 603406 by 3. 5. Divide 734842 by 8.

6. Divide 356742 by 9. 7. Divide 498756 by 12. 8. Divide 643275 by 7. 9. Divide 734562 by 5.

34. When the divisor exceeds 12,

RULE. Write the divisor at the left of the dividend. Find how many times the divisor is contained in the smallest number of figures that will contain it one or more times, and write the result in the quo

Recite the rule for division when the divisor does not exceed 12. When the divisor exceeds 12, what is the rule?

Multiply the divi

tient at the right of the dividend. sor by this quotient figure, and subtract the product from the figures divided, and to the remainder annex the next figure of the dividend, and divide this number as before, and continue dividing in the same manner till all the figures are divided.

PROOF. Multiply the divisor by the quotient, and to the product add the remainder, and if the sum be equal to the dividend, it is supposed to be right.

OBS. 1. The dividend, divisor, and quotient must be separated by a line between them.

OBS. 2. If the remainder, after having one figure annexed, will not contain the divisor, write a cipher in the quotient, and annex another figure to the dividend.

OBS. 3. If the product of the divisor by the quotient figure be larger than the dividend, the quotient figure is too large.

OBS. 4. If the remainder, before a figure of the dividend has been annexed, be greater than the divisor, or equal to it, the last figure of the quotient is too small.

OBS. 5. Annexing a figure is placing it at the right of another figure; prefixing a figure is placing it at the left of another.

10. Divide 8756424 by 324.

324) 8756424 ( 27026

648

2276

2268

842

648

1944

1944

The smallest number of figures that will contain the divisor is three, 875; in which the divisor is contained 2 times and 227 remainder, to which annex 6, the next figure, making 2276, which contains the divisor 7 times and 8 remainder, to which annex 4, the next figure of the dividend, making 84, which is less than the divisor; write a cipher in the quotient, and annex 2, the next figure of the dividend, making 842,

which contains the divisor 2 times and 194 remainder ; to which annex 4, the next figure, making 1944, which ontains the divisor 6 times, without a remainder.

35. When the divisor consists of two or more figures, products may be first formed of the divisor and the nine digits, which will enable the pupil to determine at once how many times the divisor is contained in any partial dividend.

OBS. The figures first selected to be divided, and those consisting of the remainders, with the several figures of the dividend annexed, are called the partial dividends.

29. Divide 6543742 by 295. 30. Divide 4526437 by 425.

By comparing the several products of the divisor and the partial dividends together, the pupil will discover how many times the divisor is contained in any partial dividend. Thus it will be seen that 36 is contained in 96, twice; in 246, 6 times; in 308, 8 times, &c.

36. When the divisor is a composite number,

RULE. Divide the dividend by one of the factors of the divisor, and the quotient thus obtained by the other.

To find the true remainder when there are two factors,

RULE. Multiply the first divisor by the last remainder, and to the product add the first remainder, which will be the true remainder.

When there are more than two factors,

RULE. Multiply the product of the first and second divisor by the last remainder, and the first divisor by the second remainder; to the sum of their products add the first remainder, which will be the true remainder.

37. This rule depends upon the principle stated in art. 31, that the remainder, after division, is of the same kind as the dividend.

When the divisor is a composite number, what may be done? How do you find the remainder when there are two factors? More than two factors?