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DIVISION TABLE.

NOTE.The expression used by the pupil in reciting the table may be, 2 in 2 one time, 2 in 4 two times, 4 in 12 three times, &c

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84÷12, or

=

how many? how many? how many? how many?

549, or 5 = how 328, or 32-how many? 33+11, or

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how many? | 108÷12, or 108-how many?

NOTE. The pupil should be thoroughly exercised in the foregoing table.

T32. The principles of division will be made more plain to the pupil by turning his attention to the same diagram to which it was directed while illustrating the principles of muntiplication, since division is the reverse of multiplication.

DIAGRAM OF STARS.

In multiplication, we call the whole number of stars a symbol of the product. In division, a symbol of the dividend.

In multiplication, stars in a row, and number of rows, are symbols of the multiplicand and mutiplier, which are factors of the product.

In division, stars in a row, and rows of stars, are symbols of the divisor and quotient, which are factors of

the dividend.

T33. When the object in division is to find how many times one number, or quantity, is contained in another number, or quantity, the divisor must be of the same kind as the dividend, (stars in a row,) and the quotient will be a number telling how many times (rows of stars.)

On the other hand-when the object is to divide a number or quantity, into a given number of equal parts, the quotien will be of the same name or kind as the dividend, (stars in a row.) If we divide 35 apples into 5 parts, the quotient, 7 apples, will be one part, (stars in a row,) and the divisor, 5, will be a number, that is, the number of parts, (rows of stars.)

T34. It has been remarked that division is a short way of performing many subtractions. How often can 3 be subtracted from 963? Ans. 321 times. To set down 963 and subtract 3 from it 321 times would be a long and tedious process; but by division we may decompose the number 963 thus; 963-900+60+3, and say 3 is contained in 9(00,) 3(00) times, in 6(0,) 2(0) times, and in 3, 1 time=321 times, which brings us to the same result in a much shorter way.

35. 1. How many yards of cloth, at 3 dollars a yard, can be bought for 936 dollars?

SOLUTION. As many yards as 3 dollars are contained times in 936

Questions. 32. What, in the diagram of stars, may be taken as a symbol of the dividend in division? Stars in a row and rows of stars are taken as symbols of what, in multiplication? of what in división? If the dividend be 108, the divisor 12, and the quotient 9, how would you make a diagram to correspond?

33. When the object is to find how many times one number or quantity is contained in another, the divisor will be of what name or kind? the quotient will be what? When the object is to divide a number or quantity into a given number of parts, the quotient will be of what name or kind? the divisor will be what?

34. How can you make it appear from the diagram that d'vision is a shorter way of performing many su otractions? Find on the blackboard the quotient of 963 divided by 3 How would this example be performed by subtraction?

dollars, or as many as 3 can be subtracted times from 936; 936 dollars is the dividend, (number of stars,) 3 dollars (stars in a row) the divisor.

PREPARATION.

Write the divisor on the left of the diviDividend, dend, separate them by a curved line, and

Divisor, 3) 936

OPERATION.

3)936

draw a line underneath.

We may decompose the dividend thus, 936= 900+30+6, and divide each part separately. Beginning at the left hand, we say, 3 in 9, 3 times. This quotient 3 is 3 hundred, because the 9 which we divided is hundreds; therefore we write it under the 9 in the place of hundreds.

Quotient, 312

Proceeding to the next figure, we say, 3 in 3, 1 time, which, being 1 ten, we write it in tens' place. Lastly, 3 in 6, 2 times, which, being units, we write the 2 in units' place, and the work is done. The quotient (number of rows) is 3 hundred, (300,) 1 ten, (10,) and 2 units, or 312 yards, Ans.

NOTE.-The quotient figure will always be of the same order of units as the figure divided to obtain it.

2. 28462= how many? 840÷4= how many? 500 ÷ 5 = how many?

3. If you give 856 dollars to 4 men, how many dollars will you give to each?

OPERATION.

Divisor, Dividend,
4 men,) 856 dollars.

SOLUTION. Write down the numbers as before. Divide the first figure, 8, (hundreds,) in the dividend as before. Proceeding to the next figure, 5, (tens,) 4 is contained 1 (ten) time in 4 of the tens, or 40, and there is 1 ten left, which, added to the 6 units, will make 16, and 4 in 16 units, 4 (units) times. Ans. 214 dollars.

Quotient, 214 dollars.

Here again we see that the 856 is taken in three parts, 800, 40, and 16, and each part is divided separately. When this decomposing into parts can be done in the mind, as in these examples, the process is called Short Division. It can always be done when the divisor does not exceed 12.

4. Answer the following questions after the same manner, viz., 650 ÷ 5= how many? 84906; or, what the same thing, 8490 how many? 21840:

=

how many?

expresses how many?

5. What is the quotient of 14371 divided by 7?

OPERATION.

7)14371

Quotient, 2053

There are two other things to be learned in this operation. First, the divisor, 7, is not contained in 1, the first figure of the dividend, then take two figures, or so many as shall contain the

divisor, and say, 7 in 14, 2 times; we write 2 in the quotient, in thousands' place, because we divided 14 thousands.

Then, again, proceeding to the next figure, 3 in the dividend will not contain the divisor, 7; to obviate this difficulty, we place a cipher in the quotient, joining the 3 to the 7 tens, calling it 37 tens, and so proceed. Ans. 2053.

Hence, for Short Division, this general

RULE.

I. Write the divisor at the left hand of the dividend, sep. arate them by a line, and draw a line under the dividend, to separate it from the quotient.

II. Find how many times the divisor is contained in the first left hand figure or figures of the dividend, and place the result directly under the last figure of the dividend taken, for the first figure of the quotient.

III. If there be no remainder, divide the next figure in the dividend in the same way; but, if there be a remainder, join it to the next figure of the dividend as so many tens, and then find how many times the divisor is contained in this amount, and set down the result as before.

IV. Proceed in this manner till all the figures in the dividend are divided.

EXAMPLES FOR PRACTICE.

6. A man has 256 hours' work to do; how many days will it take him, if he work 8 hours each day?

Ans. 32 days.
Ans. 215477.

7. 2370247 how many ? 8. In 1 gallon are 4 quarts; how many gallons in 2784 quarts? Ans. 696 gallons. 9. Seven men undertake to build a barn, for which they are to receive 602 dollars; into how many equal parts must the money be divided? How much will 1 part be?

parts ?

5 parts? (See ¶ 30.)

Ans. to the last, 430 dollars.

3

Questions. T35. When the dividend is large, how must it be taken? how divided? How is it done when the divisor does not exceed 12? What is the preparation? Where do you begin the division? If you divide units, what will the quotient be? if tens, what? hundreds, what? If at any time you have a remainder, what do you do with it? In Ex. 5 there are two things to be learned; what is the first thing? the second thing? What then is to be done? How do you obviate this 'difficulty? What does the cipher you write in the quotient show? What is short division? When employed? Repeat the rule.

10. Divide 24108 by 12.

Quotient, 2009.

¶ 36. 1. A man gave 86 apples to 5 boys; how many

apples did each boy receive?

Dividend,

Divisor, 5)86

Quotient, 17 1 Remainder.

SOLUTION. - Here, dividing the number of the apples (86) by the number of boys, (5,) we find that each boy's share would be 17 apples; but there is 1 apple left, and this apple, which is called the remainder, is a portion of the dividend yet undivided. Wherefore this 1 apple must be divided equally among the 5 boys. But when a thing is divided into 5 equal parts, one of the parts is called, ( 30.) So each boy will have of an apple more, or 17 apples in all. Ans. 17 apples.

NOTE 1.-The 17 (apples) expressing whole apples, are called Integers, that is, whole numbers.

Integers are numbers expressing whole things; thus, 86 oranges, 4 dollars, 5 days, 75, 268, &c., are integers, or whole numbers. NOTE 2. The (1 fifth) of an apple given to each boy, expressing part of a divided apple, is called a Fraction, or broken num ber.

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Fractions are the parts into which a unit or whole thing may be divided. Thus, (1 half) of an apple, (2 thirds) of an orange, (4 sevenths) of a week, are fractions.

NOTE 3. — A number composed of a whole number and a fraction, is called a Mixed Number; thus, the number 17 (apples) in the above example, is a mixed number, being composed of the integers 17 and the fraction

If we examine the fraction, we shall see, that it consists of the`remainder (1) for its numerator, and the divisor (5) for its denominator Therefore,

If there be a remainder, set it down at the right hand of the quotient for the numerator of a fraction, under which write the divisor for its denominator.

2. Eight men drew a prize of 453 dollars in a lottery, how many dollars did each receive?

Dividend, Divisor, 8) 453

Quotient,

56

Here, after carrying the division as far as possible by whole numbers, we have a remainder of 5 dollars, which, written as above directed, gives for the answer 56 dollars and (5 eighths) of another dollar, to each man. What are integers? fractions? a mixed number? If there be a remainder after division, it is a portion of what? What do you do with it? If you have a quotient of 2311, what was the remainder?

Questions.

T36.

What was the divisor?

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