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bring down 4, the next figure in the dividend, and divide the 84 by 15, which gives 5 the next figure in the quotient... We then bring down 1, the remaining figure of the dividend to the right hand of 9 the remainder, and find that 15 can be taken 6 times from the 91. We then multiply the divisor by 6, and subtract the product from 91, which leaves 1 for a remainder. 21. Divide 6556 by 32.

Operation. 3 2) 6 5 5 6 (2 O 4 Explanation. After 6 4 bringing down the 5 to the right of the 1, 15, the 1 5 6 number formed, is not so 1 2 8 large as the divisor, con- sequently, we cannot sub2 8 tract the divisor from it;

- we therefore, place a cipher in the quotient, and to the right of the 5 bring down 6, the next figure of the dividend. 22. Divide eight million, four hundred ninety thousand, nine hundred and sixty-five, by thirty-four thousand, six hundred and fifty-seven. Quotient 245. 23. How many times can nine thousand, eight hundred and seventy-six, be subtracted from two hundred twenty-seven thousand, one hundred and forty-eight? Ans. 23. 24. A gentleman has a field of corn containing nine thousand one hundred and two hills; each row has two hundred and twenty-two hills; how many

rows in the field 7 Ans, 41. 25. What is the quotient of 484848, divided by 1036 7 Ans. 468.

26. A bookseller purchased 591 arithmeticks for $437,34, what does he give a piece? Ans. 74 cents.

Mote 3. Dollars and cents, when both are given, are divided in the same way as simple numbers, when the divisor is a simple number; but the two right hand figures of the quotient, must be separated, which will be cents,

27. How many arithmeticks at 74 cents a piece can be bought for $437,342 Ans. 591.

28. If a person spend $101,40 in one year, what does he spend per week, there being 52 weeks in a year? Ans. $1,95.

Illustration of Rule 2. The quotient will always contain one more place of figures than there are in the dividend to the right of the product of the first quotient figure. Although we multiply the divisor by each figure in the quotient, as though it were in units’ place, by placing the product as far towards the left of the dividend as the quotient figure is from units’ place in the quotient, we really subtract the divisor as many times from the dividend as there are units in the quotient figure.

Let it be required to divide 5775 by 25.

Operation. 2 5) 5 7 7 5 (2 3 There will be 3 figures in the 5 0. quotient the 2 therefore is real

ly 200, and by placing the pro

- 7 7 duct of 25 multiplied by 2 un, 7 5. der the 57, or really 5700, it —— becomes 5000, which is the

2 5, product of 25 multiplied by 2 5 200, consequently we subtract - 25 the divisor, 200 times from

- a part of the dividend. As 75 belong to the right hand of 7 the remainder, there remains 775 yet to be divided. The product of 25 multiplied by 3, is 75, but as the 3 in the quotient is really 3 tens, the product must be increased by 10, and this we do, by placing 5, the right hand figure under 7 which was brought from tens' place in the dividend ; therefore we now subtract 25, 30 times from another part of the dividend. We have now subtracted the whole of the dividend except 25, and as that is just equal to the divisor, the divisor can be taken from it but once ; therefore when division is correctly performed, we place

an unit in the quotient every time we subtract the divisor from the dividend.

29. Bought 7860 yards of cloth for $5266,20; what did it cost per yard 2 Ans. 67 cents.

30. If in the preceding example the whole cost of the cloth had been $6602,40 what would have been the cost per yard 7 Ans. 84 cents.

31. A general has an army of 97”440 men, which he wishes to divide into 14 divisions; how many men in each division ? Ans. 6'960.

Proof.

Multiply the divisor by the quotient and add the remainder, if any, to the product; the product or the sum, will equal the dividend, if the work be right.

Illustration. By dividing, we subtract the divisor from the dividend as many times as we put an unit in the quotient, consequently were we to add the divisor together as many times as we subtracted it from the dividend, the sum must equal the dividend, if there be no remainder ; but as multiplying the divisor by the quotient is the same as adding it together as many times as the quotient contains an unit, the product must equal the dividend. When a remainder occurs, it is what is left over the number which we subtracted from the dividend, and by adding it to the product of the divisor into the quotient, the sum must equal the dividend.

A short horizontal line between two points is used as a sign of division, and is frequently expressed by the word by. It shows that the number before it is to be divided by the number after it.

Thus 12-i-6=2; and is read 12 divided by 6 is equal to 2; or 12 by 6 equals 2.

Let the scholar perform the following questions as dirécted by the sign and prove them. 6

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Remark. When a remainder occurs in division, it is usually written at the right hand of the quotient with the divisor below it, separated by a line drawn between them. The remainder and divisor thus written, is called a fraction. 38. How many times can 12 be subtracted from 625 2

Operation. 1 2) 6 2 5 ( 5 2 Explanation. 12 can be sub6 0 tracted from 625, 52 times and - 1 remains. As there is a re

2 5 mainder, the 52 is not the ex2 4 act quotient, we therefore write -- the remainder 1 at the right

1 hand of the 52 and the divisor

below it. We read this quotient, thus, fifty-two and one twelfth.

39. What number must be multiplied by 47 to make a product of 4°089'423 * Ans. 87°009. 40. A gentleman said he owned an equal part with several others of a factory valued at $244° 125, and that his share was worth $6'975; how many owners to the factory ! Ans. 35. 41. A factory worth $376'569 is owned equally by 47 men; what is the share of each Ans. 38'012,3,.. 42. A merchant has 7800 bushels of salt, which he wishes to put into bins of 121 bushels each; how many will be required 2 Ans. 64%or.

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When the divisor contains but one figure, or when a cipher or ciphers stand at the right hand of the

divisor, the operations in division may be contracted or shortened.

Contraction 1.

When the divisor consists of but one figure, the multiplying and subtracting are performed in the mind, and the quotient set directly under the dividend. This method is usually called Short Division.

43. What is the quotient of 940, divided by 4?

Operation 1. Operation 2.
4 ) 9 40 4 ) 9 4 0 ( 2 3 5
- 8
2 3 5 -
1 4
1 2
2 O
2 0

Illustration of contraction 1. Twice 4 are 8, we can take 4 from 9, 2 times, as in the 1st operation, and 1 remains. We next suppose the 1 written at the left hand of the 4, which makes 14 ; then 4 can be taken from 14, 3 times, and 2 remains. Lastly, suppose the 0 written at the right of the remaining 2, which makes 20; then 4 can be taken from 20, 5 times, and nothing remains. The remainders are called so many tens, because they belong to the left of the next figure. By comparing the first with the second operation, it will be seen that the numbers which we divide in the mind in the first, are the same as those formed by bringing down the next figures of the dividend in the second operation.

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