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duce of each tree is 7 barrels of fruit, worth 3 dollars per barrel. What was the income of the orchard?

Ans. 2205 dollars.

DIVISION OF SIMPLE NUMBERS.

22. DIVISION teaches the method of finding how many times one number is contained in another.

'The number to be divided is called the dividend.

The number by which we divide is called the divisor. The number of times which the dividend contains the divisor is called the quotient.

Besides these three parts there is sometimes a remainder, which is of the same name as the dividend, since it is a part of it.

The sign usually employed to indicate division is. Thus, 123, denotes that 12 is to be divided by 3. By using this sign we may form the following

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23. Division may also be represented by placing the divisor under the dividend, with a short horizontal line be tween them; thus, denotes that 10 is to be divided by 2. In the same way we have

212-2; 44-13-3; 7-17-5; 4-53-7.

This method is employed, when in division there is a remainder, to express accurately the value of the quotient.

What does division teach? What is the number to be divided called? What is the number by which we divide called? What is the number of mes which the dividend contains the divisor called? There is sometimes another part, what is it? of what name is the remainder? What is the symbol of division? By what other method is division denoted ?

When the divisor consists of only one figure, we proceed as follows:

Divide 973 by 7.

Having placed the divisor at the left of the dividend, keeping them separate by means of a curved line, we draw a straight horizontal line underneath.

OPERATION.

7)973

139 quotient

We then say, 7 is contained in 9, 1 time and 2 re mainder; we write the 1 underneath. As the 9 occupies the hundreds' place, the 2 remainder must be 2 hundreds. The next figure, 7, to be divided, is tens, to which we add the 2 hundreds, or 20 tens, making 27 tens; which result is obtained by prefixing the 2 to the 7. Next, we see how many times 7 is contained in 27, which is 3 times and 6 remainder; we place the 3 for the next figure of the quotient, and conceive the 6 to be prefixed to the next figure of the dividend, making 63; which is the same as adding 6 tens or 60 units to the 3 units. Finally, we find

7 is contained in 63, 9 times.

Thus 7 is contained 139 times in 973. Hence, 139 repeated 7 times must equal 973.

24. Suppose we wish to know how many times 8 is contained in 32. We might proceed as follows: since 32 's greater than 8, we know that 8 is contained in it, at least once; therefore, subtracting 8 from 32, we find 24 for

remainder. Again, we know that 8 is contained at least once in 24; therefore, subtracting 8 from 24, we have 16, from which, subtracting 8, we have left 8; finally, from 8 subtracting 8, we have no remainder. Hence, we perceive that 8 has been subtracted 4 times from 32, that is, 8 is contained just four times in 32. It is obvious that by continued subtractions any operation in division may be performed.

For this reason division is said to be a concise way of performing several subtractions.

CASE I.

25. Short Division is the method of operation wher divisor consists of only one figure.

From the preceding operation we infer the following

RULE.

1. Place the divisor at the left of the dividend, keeping them separate by a curved line, and draw a straight line underneath the dividend.

II. Seek how many times the divisor is contained in the left-hand figure or figures of the dividend, and place the result directly beneath, for the first figure of the quotient.

III. If there is no remainder, divide the next figure of the dividend for the next figure of the quotient. But when there is a remainder, conceive it to be prefixed to the next succeeding figure of the dividend before making the next division. If a figure of the dividend, which is required to be divided, is less than the divisor, we must write 0 in the quotient, and consider that figure as a remainder.

Division is said to be a concise way of performing what? What is Short Division? Repeat the rule.

EXAMPLES.

1. Divide 2345675 by 8.

OPERATION.

Divisor 8)2345675 dividend.

Quotient 293209 with 3 remainder.

26. When there is a remainder, we may place it over the divisor, with a short horizontal line between them, thus

indicating that this remainder is still to be divided by the

divisor, agreeably to ART. 23.

2. Divide 12456789 by 4. 3. Divide 78900346 by 7.

Ans. 31141974.

4 Divide 131305678 by 6. 5. Divide 357020348 by 3.

Ans. 11271478. Ans. 218842794. Ans. 1190067824.

CASE II.

27. Long Division is the method of operation when the divisor consists of more than one figure.

EXAMPLES.

1. Divide 4703598 by 354. It requires 3 figures, 470, of the dividend to contain the divisor 354. This is contained once in 470; we place the 1 at the right of the dividend for the first figure of the quotient, keeping it separate from the dividend by a curved line. Multiplying the divisor by this quotient figure, and subtracting the product

OPERATION.

DIVISOR. DIVIDEND. QUOTIENT

354) 4703598 (13287
354 first product.
1163

1062 second product.

1015

708 third product.

3079

2832 fourth product.

2478

2478 fifth product.

from 470, we have 116 for a remainder, to which wo annex the next figure, 3, of the dividend, thus forming the number 1163. We now seek how many times the divisor is contained in 1163, which is 3 times. We place the 3 for a second figure of the quotient. Multiplying the divisor

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