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

In division there are three principal terms: the Dividend, the Divisor, and the Quotient, or answer.

The dividend is the number to be divided.

The divisor is the number by which we divide.

The quotient is the number of times the divisor is contained in the dividend.

NOTE. Quotient is derived from the Latin word quoties, which signifies how often, or how many times.

When the dividend does not contain the divisor an exact number of times, the excess is called a remainder, and may be regarded as a fourth term in the division. The remainder, being part of the dividend, will always be of the same denomination or kind as the dividend, and must always be less than the divisor.

ART. 47. SIGNS. The sign of division is a short horizontal line, with a dot above it and another below; thus, . It shows that the number before it is to be divided by the number after it. The expression 6 ÷ 2 = 3 is read, 6 divided by 2 is equal to 3.

Division is also indicated by writing the dividend above a short horizontal line and the divisor below; thus, §. The expression 3 is read, 6 divided by 2 is equal to 3.

=

There is a third method of indicating division, by a curved line placed between the divisor and dividend. Thus, the expression 6) 12 shows that 12 is to be divided by 6.

EXERCISES FOR THE SLATE.

ART. 48. The method of operation by Short Division, or when the divisor does not exceed 12.

Ex. 1. Divide 8574 dollars equally among 6 men.

Ans. 1429 dollars.

What are the three principal What is the divisor? What is

QUESTIONS. Art. 46. What is division? terms in division? What is the dividend? the quotient? What the remainder? What will be the denomination of the remainder? How does it compare with the divisor?- Art. 47. What is the first sign of division, and what does it show? What is the second, and what does it show? What is the third, and what does it show?-Art. 48. What is short division?

OPERATION.

We first inquire how many times

Divisor 6) 8574 Dividend. 6, the divisor, is contained in 8, the first figure of the dividend,

1429 Quotient. which is thousands, and find it to be 1 time, and 2 remaining.

We

write the 1 directly under the 8, its dividend, for the thousands' figure of the quotient. To 5, the next figure of the dividend, which is hundreds, we regard as prefixed the 2 thousands remaining, which equal 20 hundreds, and thus form the number 25 hundreds, in which we find the divisor 6 to be contained 4 hundred times, and 1 hundred remaining. We write the 4 for the hundreds' figure in the quotient, and the 1 hundred remaining, equal 10 tens, we regard as prefixed to 7, the next figure of the dividend, which is tens, forming 17 tens, in which the divisor 6 is contained 2 tens' times, and 5 tens remaining. We write the 2 for the figure in the quotient, and the 5 tens remaining, equal 50 units, we regard as prefixed to 4, the last figure of the dividend, which is units, forming 54 units, in which the divisor 6 is contained 9 units' times. Writing the 9 for the units' figure of the quotient, we have 1429 as the entire quotient, or the number of times which the dividend contains the divisor 6.

ART. 49. From the foregoing illustration we deduce the following RULE.

Write the divisor at the left hand of the dividend, with a curved line between them, and draw a horizontal line under the dividend.

Then, beginning at the left, divide each figure of the dividend by the divisor, and write each quotient under its dividend.

If there be a remainder, regard it as prefixed to the next figure of the dividend, and divide as before.

Should any figure to be divided be less than the divisor, write a cipher in the quotient, and regard it as prefixed to the next figure of the dividend.

NOTE.-1. When there is a remainder after dividing the last figure of the dividend, write the divisor under it, with a line between them, and annex the same to the quotient.

NOTE.-2. Prefix means to place before; annex, to place after.

ART. 50. First Method of Proof. - Multiply the divisor by the quotient, and to the product add the remainder, if any, and, if the work is right, the sum thus obtained will be equal to the dividend.

At

QUESTIONS. How are the numbers arranged for short division? which hand do you begin to divide? Why not begin at the right, where you begin to multiply? Where do you write the quotient? If there is a remainder after dividing a figure, what is done with it? Art. 49. What is the rule for short division? Repeat the notes.

NOTE. It will be seen, from this method of proof, that division is the reverse of multiplication. The dividend answers to the product, the divisor to one of the factors, and the quotient to the other.

QUESTIONS. Art. 50. How is short division proved? Of what is division the reverse? To what do the three terms in division answer in multiplication? What, then, is the reason for this proof of division?

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

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

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

OPERATION.

Dividend.

Ans. 128 dollars.

Having set down the divisor and dividend as in short divi

Divisor 7) 8 9 6 (128 Quotient. sion, we draw a curved line at

7

19

14

56

56

the right of the dividend, to mark the place for the quotient. We then inquire how many times 7, the divisor, is contained in the 8 hundreds of the dividend; and, finding it to be one hundred times, we write the 1 in the quotient,

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?

and multiply the divisor, 7, by it, writing the product, 7 hundreds, under the 8 hundreds, from which we subtract it, and to the remainder, 1, annex the 9 tens of the dividend, making 19 tens. We now inquire how many times 7 is contained in 19 tens, and write 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 annex 6 units, the next and last figure of the dividend, making 56 units. 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.

Ans. 248 dollars.

We first inquire how many times 19, the divisor,

Divisor 19) 4712 (248 Quotient. is contained in 47, the two

38

91

76

152

152

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 remainder, 9, annex 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, annex 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. This illustration, except in omissions, is essentially like the preceding one.

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?