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63. The object of division is twofold.

First. To find how many times one number is contained in another of the same kind.

Second. To separate a given number into as many equal parts as there are units in another.

Thus, if 30 cents are divided equally among a number of boys, giving each boy 5 cents, how many boys are there?

Here the whole number and one of the equal parts are given, to find the number of equal parts.

Again, if 30 cents are divided equally among 6 boys, how many cents does each boy receive?

Here the whole number and the number of equal parts are given, to find the size or value of one of the equal parts.

64. The equal parts of a number or thing are named according to their size or number. Thus,

1. If a number or thing is separated into two equal parts, each part is called one half.

Written,

One half () of 16 is 16-28;

1

of 12 is 12÷÷2 = 6.

2. If a number is separated into three equal parts, each part is called one third. Written, Written, 1

One third (3) of 21 is 21÷3 = 7; } of 24 is 24÷÷3 = 8.

3. If a number is separated into four equal parts, each part is called one fourth. Written, 4

1

One fourth (1) of 28 is 28÷4 = 7; of 40 is 40÷4=10.

4. In like manner, we obtain the parts named fifths, sixths, sevenths, eighths, twentieths, etc.

5. How is one of 2 equal parts, or one half of a number found? One of 3 equal parts, or one third? One of 8 equal parts, or one eighth? One of 12 equal parts, or one twelfth ?

6. The equal parts into which a number or thing is divided are called fractional parts, or fractions.

PRINCIPLES.-To find how many times one number is contained in another:

I. The divisor and dividend must be like numbers, and the quotient an abstract number.

To find one of the equal parts of a number:

II. The dividend and quotient must be like numbers, and the divisor an abstract number.

III. The dividend is the product of the divisor and the quotient.

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65. 1. The product of two factors is 48 miles, and one of the factors is 8 miles; what is the other? If one factor is 6 miles, what is the other? Why?

2. If the dividend is 84 feet and the divisor 12 feet, what is the quotient? If the divisor is 7?

In 80?

In 800 ?

In 8000?

In 80?

In 800?

In 8000 ?

In 120?

In 1200?

In 12000?

In 160?

In 1600?

In 16000?

3. How many 2's in 8? 4. How many 4's in 8? 5. How many 6's in 12? 6. How many 8's in 16? 7. How many times 5 feet are 10 feet? 100 feet? 1000 ft.? 8. How many times $8 are $16? $160? $1600?

9. How many times 10 men are 60 men? 7 days are 42 days? 8 miles are 64 miles?

10. If an acre of land is divided into 3 equal parts, what is each part called? If into 4 equal parts? 5? 6? 7? 8? 9? 10? 12? 15? 20?

11. What is one third (1) of 24 cents? 12. What is one fourth (1) of 32 rods? 13. What is one eighth of 72 miles?

One twelfth (1) of 84 bushels?

Of 36 men?
One sixth (†) of $60?
One ninth of 54 acres?

14. How do we obtain 1, 1, 1, 1, etc., of any number?

The following dictation and oral exercises and examples are intended simply as models or forms, each to be largely increased, at the option of the teacher.

15. If 36 marbles are equally divided among 4 boys, what part of 36 marbles does each boy receive? If among 9 boys? 16. If $40 are paid to 8 men in equal parts, what part of $40 is paid to each man? If paid to 5 men?

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ANALYSIS.-One is of 4; 2 is 2 times, or of 4; 3 is of 4.

25. How do we obtain one fourth of a number? fourths? Three fourths? Two thirds?

Two

26. What part of $5 is $1? Is $2 ? Is $3? Is $4? Of 6 pounds is 1 pound? 2 pounds? 3 pounds? 5 pounds? 27. What part of 7 is 3? Of 8 is 5? Of 9 is 2? 28. Seven are how many times 3?

ANALYSIS.-Three is in 7, 2 times and 1 remainder, which is of 3. Hence, 7 is 2 times 3 and 1 of 3.

29. 13 are how many times 4? 30. 35 are how many times 8? 31. 42 are how many times 4? 32. 63 are how many times 5?

Ans. 3 times 4 and 1 of 4. Ans. 4 times 8 and 3 of 8. 5? 6? 7? 8? 9? 10? 6? 7? 8? 9? 10? 12?

For additional exercises, use Drill Tables Nos. 1 and 2.

For example, in Drill Table No. 1, divide 10 by each number in a column, naming only the quotient and remainder. In the same way, divide 11, 12, etc., to 20.

Taking two columns, divide the larger number by the smaller, naming the quotient and remainder. Taking three columns, divide the product of the first and second numbers by the third.

Many similar exercises for drill and dictation may be arranged from these tables by the apt teacher.

66. The pupil should early learn by practice to divide any number not greater than 100 by any number not greater than 12, and give the result at sight. The following methods will afford a pleasant exercise, and help in so desirable an acquisition for business purposes:

1. First, to divide any number, as 35, by the numbers less than 13.

Write on the slate or board the following forms of expression:

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The answers should be given orally, and not written; and the remainders may be read, either as so many “remainder,” or in fractional form; thus, 35 is 8 times 4 and 3 remainder; 5 times 6 and 5 remainder, etc.; or thus, 35 is 8 times 4 and of 4; 5 times 6 and of 6, etc.

In like manner, practice upon all dividends, from 12 to 100 inclusive.

2. Another and better method, since there will be less writing, is to use the same device, or the Chart, used to memorize the Multiplication Table.

1. Draw on the slate or board a small square, and around the inside write the numbers from 1 to 12 inclusive, without regard to order.

2. In the centre, write for a dividend any number greater than 12 and less than 100, as 27.

Commencing with any number in the margin, as 2, use the consecutive numbers as divisors, reading to the right around the square; thus, 2 in 27, 13 times and 1 remain

2

4 5

7

6

10

27

11

8

1 9 3 12

In the same

der; 4 in 27, 6 times and 3 remainder, etc.; or thus, 27 is 13 times 2 and one half (1) of 2; 6 times 4 and three fourths (4) of 4, etc. manner, commence and read to the left.

3. Then the teacher may point in rapid succession to the different figures in the margin, and the pupil promptly give the result orquotient.

4. Then erase the number in the center, and insert another for a new dividend, which will give a new set of quotients, and so on, until all the numbers from 13 to 100 have been used as dividends.

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67. 1. How many yards of cloth, at $4 a yard, can be bought for $36? *

ANALYSIS.-As many yards as $4 are contained times in $36, which are 9 times. Hence, 9 yards can be bought for $36.

2. At $6 a ton, how many tons of coal can be bought for $24? For $30? For $54? For $72?

3. If 7 barrels of flour cost $63, what will 1 barrel cost? ANALYSIS.-Since 7 barrels cost $63, 1 barrel will cost 1 seventh of $63, or $9.

4. If 6 barrels of flour cost $72, what will 1 barrel cost? 5. If a man travel 48 miles in 4 hours, how far does he travel in 1 hour?

6. What will be the cost of 1 ton of coal, if 8 tons cost $64? 7. If a farm of 120 acres is divided into 12 equal lots, how many acres does each lot contain?

8. At $9 a week, in what time will a man earn $36? $54? $72? $81? $108?

9. If 4 barrels of flour cost $36, what will 7 barrels cost?

ANALYSIS.-One barrel will cost 1 fourth of $36, or $9. and 7 barrels will cost 7 times $9. or $63.

10. What cost 12 yards of cloth, if 6 yards cost $24? 11. If 8 yards of silk cost $32, what will 12 yards cost? 12. What will 15 sheep cost, if 5 sheep cost $35?

13. How many cords of wood, at $4 a cord, will pay for 6 barrels of flour, at $8 a barrel?

ANALYSIS.-Six barrels of flour will cost 6 times $8, or $48; and $4, the price of 1 cord of wood, are contained in $48, 12 times.

14. How many days' labor, at $4 a day, will pay for 3 tons of coal, at $6 a ton, and 2 tons of hay, at $15 a ton?

The teacher should make plain the two forms of division, as shown in examples 1 and 3, and require the proper solution for each example.

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