23. Divide 944,580 dollars equally among 12 men, and what will be the share of each ? Ans. 78,715 dollars.

per

24. Divide 154,503 acres of land equally among 9 Ans. 17,167 acres.

sons.

25. A plantation in Cuba was sold for 7,011,608 dollars, and the amount was divided among 8 persons. What was paid to each person?

Ans. 876,451 dollars. 26. A prize, valued at 178,656 dollars, is to be equally divided among 12 men; what is the share of each?

Ans. 14,888 dollars. 27. Among 7 men, 67,123 bushels of wheat are to be distributed; how many bushels does each man receive?

Ans 9,589 bushels.

28. If 9 square feet make 1 square yard, how many yards in 895,347 square feet? Ans. 99,483 yards. 29. A township of 876,136 acres is to be divided among 8 persons; how many acres will be the portion of each? Ans. 109,517 acres. 30. Bought a farm for 5670 dollars, and sold it for 7896 dollars, and I divide the net gain among 6 persons; what does each receive? Ans. 371 dollars. 31. If 6 shillings make a dollar, how many dollars in 7890 shillings? Ans. 1315.

ART. 51. The method of operation by Long Division, or when the divisor exceeds 12.

Ex. 1. A gentleman divided 896 dollars equally among his 7 children; how much did each receive? Ans. 128 dollars.

QUESTIONS. Art. 51. What is long division? What is the difference between long division and short division? How do you arrange the numbers for long division? What do you first do after arranging the numbers for long division? Where do you place the figures of the quotient?

multiply the divisor, 7, by it, placing the product, 7, under the 8, from which we subtract it, and to the right of the remainder, 1, bring down 9, the next figure of the dividend, making 19. We now inquire how many times 7 is contained in 19, and place the number, 2, at the right of the quotient figure before obtained. We then multiply the divisor by it, and place the product under the 19, and subtract as before, and to the remainder, 5, we bring down 6, the next and last figure of the dividend, making 56. We proceed, as before, to find the next quotient figure, and, after subtracting the product of the divisor multiplied by it, from 56, find there is no remainder left. Hence we learn that each one of the 7 children must receive 128 dollars.

NOTE. - The preceding example, and the four that follow, are usually performed by short division, but are here introduced to illustrate more clearly the method of operation by long division.

Ex. 6. A gentleman divided 4712 dollars equally among his 19 sons; what was the share of each?

OPERATION.

Dividend.

Divisor 19) 47 12 (248 Quotient.

38

91

76

152

152

Ans. 248 dollars.

Having arranged the divisor and dividend as before, we first inquire how many times 19, the divisor, is contained in 47, the two left-hand figures of the dividend; and finding it to be 2 times, we write the 2 in the quotient, multiply the divisor by it, and subtract the product from the 47; and to the right of the re

mainder, 9, bring down 1, the next figure of the dividend, making 91. We next inquire how many times 19 is contained in 91, place the number, 4, in the quotient, then multiply and subtract as before, and to the remainder, 15, bring down 2, the last figure of the dividend, and, proceeding as before, after finding the quotient figure, no remainder is left. Hence the share of each of the 19 sons is 248 dollars.

QUESTIONS. After the quotient figure is found, what is the next thing you do? Where do you place the product? What do you next do? What is the next step? How do you then proceed? Is long division the same in principle as short division?

ART. 52. From the preceding illustrations, the pupil will perceive the propriety of the following general

RULE.-1. Write down the divisor and dividend as in short division, and draw a curved line at the right hand of the dividend.

2. Then inquire how many times the divisor is contained in an equal number of figures on the left hand of the dividend, or in one more, if an equal number will not contain the divisor, and place the result in the quotient at the right hand of the dividend.

3. Multiply the divisor by the quotient figure, writing the product under the figures of the dividend that were taken. Subtract this product from the figures of the dividend above it, and to the right of the remainder bring down the next figure of the dividend.

4. Find how many times the divisor is contained in the number thus formed; place the figure denoting it at the right hand of the former quotient figure; multiply the divisor by it, and subtract the product from the number divided, and to the remainder bring down the next figure of the dividend, as before. Thus proceed until all the figures of the dividend are divided; and if there is a remainder, write it as directed in the preceding rule.

NOTE 1.-The proper remainder is in all cases less than the divisor. If, in the course of the operation, it is at any time found to be larger than the divisor, it will show that there is an error in the work, and that the quotient figure should be increased.

NOTE 2. — If, at any time, the divisor, multiplied by the quotient figure, produces a product larger than the part of the dividend used, it shows that the quotient figure is too large, and must be diminished.

NOTE 3.- It will often happen, that, when a figure is brought down, the number will not contain the divisor, and in that case a cipher must be placed in the quotient, and another figure of the dividend brought down, and so on until the number is large enough to contain the divisor.

ART. 53. Second Method of Proof. Add together the remainder, if any, and all the products that have been produced by multiplying the divisor by the several quotient figures, and the result will be like the dividend, if the work is right.

ART. 54. Third Method. Subtract the remainder, if any, from the dividend, and divide the difference by the quotient. The result will be like the original divisor, if the work is right.

NOTE. The first method of proof (Art. 50) is usually most convenient, and is most commonly employed.

QUESTIONS. Art. 52. What is the general rule for long division? How may you know when the quotient figure is too small? How may you know when it is too large? What do you do when the part of the dividend used will not contain the divisor?-Art. 53. What is the second method of proof for division? Art. 54. What is the third method? Can long division be proved by the first method of proof (Art. 50)?

EXAMPLES FOR PRACTICE.

Ex. 7. It is required to find how many times 48 is contained in 28618.

Ans. 596.

* This sign of addition denotes the several products to be added.

2295060

8428688

34. Divide 678957000107 by 10789561. 62927

35. Divide 990070171009 by 9007Q0601. 1099 200210510 36. Divide three hundred twenty-one thousand three hundred dollars equally among six hundred seventy-five men.

Ans. 476 dollars. 37. Four hundred seventy-one men purchase a township containing one hundred eighty-six thousand forty-five acres; what is the share of each? Ans. 395 acres.

38. A railroad, which cost five hundred eighteen thousand seventy-seven dollars, is divided into six hundred seventy-nine shares; what is the value of each share? Ans. 763 dollars. 39. Divide forty-two thousand four hundred thirty-five bushels of wheat equally among one hundred twenty-three men. Ans. 345 bushels each.

40. A prize, valued at one hundred eighty-four thousand seven hundred seventy-five dollars, is to be divided equaly, among four hundred seventy-five men; what is the share of each? Ans. 389 dollars.

41. A certain company purchased a valuable township for nine millions six hundred ninety-one thousand eight hundred