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DIVISION.

DIVISION TABLE-Continued. 7 in 63 9 times. 9 in 63 7 times. 11 in 55 5 times. 7 70 10 times. 9 « 72 8 times. 11 6 66 6 times.

77 11 times. 96 81 9 times. 11 “ 77 7 times. 7 " 84 12 times. 9" 90 10 times. 11 66 88 8 times.

8 once. 9" 99 11 times. / 11 6 99 9 times. 8 " 16 twice, 9 6 108 12 times. | 11 6 110 10 times.

24 3 times. 10 6 10 once. 11 6 121 11 times.

32 4 times. | 10 6 20 twice. 11 “ 132 12 times. 8 " 40 5 times. | 10 6 30 3 times. 12 “ 12 once. 8 " 48 6 times. 1066 40 4 times. 12 24 twice. 8 " 56 7 times. / 10 6 50 5 times. 12 “ 36 3 times. 8 64 8 times. 10 66 60 6 times. | 12 66 48 4 times.

72 9 times. / 10 6 70 7 times. 12 “ 60 5 times.

80 10 times. 10 « 80 8 times. 12 " 72 6 times. 8 " 88 11 times. | 10 6 90 9 times. | 12 « 84 7 times. 8

96 12 times. 10 6 100 10 times. 12 6 96 8 times. 9 9 once. 10 “ 110 11 times. | 12 - 108 9 times. 9 18 twice. 10 “ 120 12 times. ( 12 " 120 10 times. 9 27 3 times. 11 6 11 once. 12 6 132 11 times. 96 36 4 times. 11 “ 22 twice. 12 “ 144 12 times. 9 45 5 times. 11 “ 33 3 times. 9 54 6 times. / 11 “ 44 4 times. How many tens in a hundred ? How many tens in a thousand ?

EXAMPLES.--For Mental Exercise. 1. How many times is 4 contained in 24? 2. A man gave 12 apples to 3 boys-how many apples did he give to each boy?

3. How many oranges could you buy for 18 cents, at 3 cents an orange?

4. Jane divided 12 peaches among 3 girls—how many did she give each girl ?

5. If a man can travel 4 miles an hour, how many hours would it take him to travel 24 miles ?

6. In an orchard there are 36 trees standing in rows, and there are 3 trees in a row-how many rows are there?

7. If you had 16 cents, how many pies could you buy at 3 cents apiece; and how much money left?

8. A man laid out 69 dollars for sheep, paying 3 dollars a head for them-how many did he buy?

9. How many times is 9 contained in 108 ? 10. How many times is 4 contained in 88 ? NOTE. When any one thing is divided into two equal parts, one of those parts is called a hall; if into three equal parts, one of those parts is called a third; if into four equal parts, one part is called a quarter, or a fourth; if into five, one part is called a fifth ; and so on.

Explanations. You have learned that Addition is joining or collecting numbers together; and that Subtraction is separating numbers, or taking them one from the other. You have also learned that . Multiplication is the bringing or collecting together of similar or equal sums. As Subtraction is the reverse of Addition, so is Division the reverse of Multiplication. Division is the separating larger numbers into smaller ones, the smaller being equal one to another; as 12 divided into 4 equal parts, gives us 3 for each part; and also shows us that 3 can be subtracted from 12 four times. In Multiplication there are two numbers given to find the third, that is, the multiplicand and multiplier given to find the

product: so in Division we have two numbers given to find the third. The larger number given to be divided is called the dividend, and answers to the product in Multiplication; and the less number, the number given to divide by, is called the divisor, and answers to one of the two, multiplicand or multiplier, in Multiplication. The result, or answer sought, is called the quotient, and answers to one of the two-the multiplier or multiplicand. Suppose the multiplicand in a Multiplication sum to be 12, the multiplier 8, and the product 96. Now, if we take the 96 for a dividend, and the 8 for a divisor, we will find 8 is contained in 96 twelve times, giving us the multiplicand; and if we take 12 the multiplicand, for a divisor, we will find 12 is contained in 96 eight times, giving us the multiplier. This is the best way to prove Multiplication. If we take 24 for a dividend in a Division sum, and 6 for a divisor, we will find that 6 is contained in 24 four times, which is our quotient. In this example we had the dividend and divisor given to find the

quotient. When we have the dividend and quotient given to find the divisor, we divide the dividend by the quotient, which gives us the divisor, and to multiply the divisor and quotient, one by the other, gives the dividend.

N. B. To divide any number by 2, the quotient will be one-half of that number; to divide by 3, gives one-third ; to divide by 4, gives one quarter of that number, &c.

DIVISION.

26

DIVISION teaches to divide a larger númber by a less, into equal parts; or it shows how often a less number may be taken from a larger:

There are four parts to be noted in Division.
1. The dividend, the number (giver) ta be divided,
2. The divisor, the number (given) to divide by.

3. The quotient, or answer to the question, which shows how often the divisor is contained in the dividend.

4. The remainder (if any) which is always less than the divisor, and of the same name with the dividend.

When the divisor does not exceed 12, the operation may be performed in one line: this is called Short Division.

RULE.

Write down the dividend, and draw a curve line at the left hand side, and a straight line under the dividend, and write the divisor at the left hand of it.

1. Find how many times the divisor is contained in the first figure or figures of the dividend, and set it under the dividend, carry the remainder, if any, to the next figure as so many tens.

2. Find' how many times the divisor is contained in this dividend, and set it down as before, and so proceed as before, till the whole is divided.

PROOF.

1. Multiply the quotient by the divisor, adding in the remainder, if any, and the product will be equal to the dividend.

2. Subtract the remainder, if any, from the dividend, and divide by the quotient, and you will have the divisor.

EXAMPLES.

1. How many times is 3 contained in 786 ?
Divisor 3)786 dividend. Proof. 262 quotient.

3 divisor.
262 quotient.

786 dividend.

We first write down the dividend, making a curve line at the eft hand side, and a straight line under the dividend. We then say, 3 in 7, twice, and 1 over, we write down the 2 directly under the 7, and the 1 that was over we conceive to be joined to the next figure 8, making 18; we then say 3 is contained in 18 six times, which we write down, then 3 in 6 goes twice, which we write down, and the work is done.

Remark.--The reason for calling every one that is over ten, when you divide, is evident: for 1 in the left hand row of figures

ways equal to 10 in the next place to the right hand.

SHORT DIVISION.

2. A man gave 84 apples to 2 boys-how many apples would each boy receive ?

Ans. 42 apples. 3. A lady divided 18 peaches among some little girls, giving 3 to each girl-how many girls were there?

Ans. 6 girls, 4. How many times is 4 contained in 36?

Ans. 9 times. 5. How many lead-pencils could you buy for 48 cents, if they were sold at 3 cents apiece ?

Ans. 16 pencils. 6. How many times 5 in 35 ?

Ans. 7 times. 7. A farmer got 144 dollars for some sheep, that he sold at 4 dollars apiece-how many were there? Ans. 36 sheep.

8. A trader wishes to pack 192 hats in boxes, putting 12 hats in a box-how many boxes are wanted ? Ans. 16 boxes.

9. How many dozen of eggs can you buy for 207 cents, when they are sold at 9 cents a dozen ?

Ans. 23 dozen. 10. A drover bought 12 oxen for 643 dollars how many dollars did he pay for each?

Ans. 54 dollars. 11. How often can 8 be subtracted from 96?

Ans. 12 times. 12. At 10 dollars a barrel, how many barrels of pork can be bought for 68470 ?

Ans. 6847 barrels. 13. A man having 415 dollars, laid them all out in hats, at 5 dollars a hat-how many hats did he buy?

Ans. 83 hats. 14. If a man spend 1008 dollars in 12 aonths, how many dollars does he spend in each month?

Ans. 84 dollars. 15. A farmer sold a quantity of cheese for 2996 cents, at 7 cents a pound-how many pounds did he sel?

Ans. 428 pounds. 16. There were 6 merchants who sold a quantity of goods for 11514 dollars—how many dollars were each man's share ?

Ans. 1919 dollars. 17. How much is the quotient of 28092 divided by 12?

Ans. 2341. 18. Twelve men, by contract, are to receive 1500 dollars for a job of work-how many dollars will be each man's part, if they be divided equally among them?

Ans. 125 dollars. 19. What is the one-third of 7593?

Ans. 2531. 20. What is the one-twelfth of 9864?

Ans. 822. 21. How many times 8 in 97637, and how many over?

Ans. 12204, and 5 over.. 22. Divide 307050 by 8; then prove it to be right, by multiplying the quotient by the divisor.

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LONG DIVISION. When the divisor exceeds 12, the operation is called Long Division.

RULE.

1. Write down the dividend, and draw a curve line at the right and left sides of the dividend, and write the divisor at the left hand side (of the dividend.)

2. Take the same number of the first left hand figures of the dividend that there are in the divisor, if they be equal to, or larger than, the divisor; but if they be less than the divisor, take one more; find the number of times the divisor is contained in them, and write it at the right hand of the dividend.

3. Multiply the divisor by the quotient figure, and write the product under that part of the dividend taken.

4. Subtract this product from that part of the dividend under which it stands, and bring down the next figure of the dividend, and write it at the right hand of the remainder; then find a quotient figure, multiply and subtract as before directed.

5. If it be necessary to bring down more figures than one to the remainder, in order to make it larger than the divisor, a cipher must be written in the quotient for every figure so brought down.

NOTE, To Teachers. The learner should be required to answer the fol. lowing

Questions. 1. What is Division ? Recite the explanations given in page 25. 2. How many parts are there in Division ? 3. What are the names given to the four parts in Division ? 4. What are the usual given numbers in Division called ? 5. The dividend and divisor given to find what? How?

6. Having the dividend and quotient given, how is the divisor found?

7. If you have the divisor and quotient given, how can you find the dividend ?

8. What is Short Division ?
9. What is Long Division ? 10. Rule ?
11. At which hand of the dividend must the divisor be written?

12. Why do you begin at the left hand of the dividend to divide? (Ans. Because numbers decrease, &c.)

13. When the divisor exceeds 12, where must the quotient be written?

14. How many figures of the dividend must first be taken?
15. How do you find the one-half of any number? The third,
rth, &c. &c.

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