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We first inquire how many times Divisor 6 ) 8574 Dividend. 6, the divisor, is contained in 8, the
first figure of the dividend, which 1429 Quotient.
is thousands, and find it to be 1
time, and 2 thousands remaining: We write the 1 directly under the 8, its dividend, for the thousands figure of the quotient. To 5, the next figure of the dividend, which is hundreds, we regard as prefixed the 2 thousands remaining, which equal 20 hundreds, thus forming 25 hundreds, in which we find the divisor 6 to be contained 4 times, and 1 hundred remaining. We write the 4 for the hundreds' figure in the quotient, and the 1 hundred remaining, equal 10 tens, we regard as prefixed to 7, the next figure of the dividend, which is tens, forming 17 tens, in which the divisor 6 is contained 2 times, and 5 tens remaining.
We write the 2 for the tens' figure in the quotient, and the 5 tens remaining, equal 50 units, we regard as prefixed to 4, the last figure of the dividend, which is units, forming 54 units, in which the divisor 6 is contained 9 times. Writing the 9 for the units' figure of the quotient, we have 1429 as the entire quotient.
RULE. Write the divisor at the left hand of the dividend, with a curved line between them, and draw a horizontal line under the dividend.
Then, beginning at the left, find how many times the divisor is contained in the fewest figures of the dividend that will contain it, and write the quotient under its dividend.
If there be a remainder, regard it as prefixed to the next figure of the dividend, and divide as before.
Should any dividend be less than the divisor, write a cipher in the quotient, and annex another figure, if any remains, for a new dividend.
Note 1.- When there is a remainder after dividing the last figure of the dividend, write it with the divisor underneath, with a line between them, at the right of the quotient.
NOTE 2. — Prefix means to place before ; annex, to place after.
50. First Method of Proof. — Multiply the divisor and quotient together, and to the product add the remainder, if any, and, if the work is right, the result obtained will equal the dividend.
48. How are the numbers arranged for short division? At which hand do you begin to divide ? Why not begin at the right, where you begin to multiply? Where do you write the quotient? If there is a remainder after dividing a figure, what is done with it? — 49. The rule for short division ? Repeat the notes ?
Note. - This method of proof depends upon the fact, that division is the reverse of multiplication. The dividend answers to the product, the divisor to one of the factors, and the quotient to the other.
50. How is short division proved? Of what is division the reverse ? To what do the three terms in division answer in multiplication? What, then, is the reason for this proof of division ?
23. Divide 944,580 dollars equally among 12 men, and what will be the share of each ?
Ans. 78,715 dollars. 24 Divide 154,503 acres of land equally among 9 persons.
Ans. 17,167 acres. 25. A plantation in Cuba was sold for 7,011,608 dollars, and the amount was divided among 8 persons. What was paid to each person?
Ans. 876,451 dollars. 26. A prize valued at 178,656 dollars is to be equally divided among 12 men; what will be the share of each ?
Ans. 14,888 dollars. 27. Among 7 men, 67,123 bushels of wheat are to be distributed; how many bushels will each man receive ?
Ans. 9,589 bushels. 28. If 9 square feet make 1 square yard, how many yards in 895,347 square feet?
Ans. 99,483 yards. 29. A township of 876,136 acres is to be divided among 8 persons ; how many acres will be the portion of each ?
Ans. 109,517 acres. 30. Bought a farm for 5670 dollars, and sold it for 7896 dollars, and I divide the net gain among 6 persons ; what does each receive ?
Ans. 371 dollars. 31. If 6 shillings make a dollar, how many dollars in 7890 shillings?
Ans. 1315 dollars. 51. Long Division, or, in general, when the divisor exceeds 12.
Ex. 1. A gentleman divided 896 dollars equally among his 7 children ; how much did each receive ? Ans. 128 dollars.
Having set down the divisor Dividend.
and dividend as in short diviDivisor 7) 8 9 6 (1 2 8 Quotient. the right of the dividend, to
sion, we draw a curved line at 7
mark the place for the quo19
tient. We then inquire how
many times 7, the divisor, is 14
contained in 8, the first figure 5 6
of the dividend, which is hun
dreds ; and, finding it to be 1 5 6
hundred times, we write the 1 for the hundreds' figure in the
51. What is long division ? The difference between long division and short division? How do you arrange the numbers for long division ? What do you first do after arranging the numbers for long division? Where de you place the figures of the quotient?
quotient, and multiply the divisor, 7, by it, writing the product, 7, under the 8, from which we subtract it, and to the remainder, 1, annex the 9 of the dividend, making 19 tens. We now inquire how many times 7 is contained in the 19, and write the 2 obtained for the tens' figure of the quotient. We then multiply the divisor by it, and place the product under the 19, and subtract as before ; and to the remainder, 5, we annex the 6 of the dividend, making 56 units. We proceed next to find the units' figure, and, after subtracting the product of the divisor multiplied by it from 56, find there is no remainder. Hence each one of the 7 children must receive 128 dollars.
Note. — The preceding example and the four that follow are usually performed by short division, but are here introduced to illustrate more clearly the method of operation by long division.
Ex. 6. A gentleman divided 4712 dollars equally among his 19 sons; what was the share of each ? Ans. 248 dollars.
We first inquire how Dividend.
many times 19, the divisor, Divisor 19 ) 4712 (2 48 Quotient.
is contained in 47, the two 38
left-hand figures of the div
idend; and, finding it to be 91
2 times, we write the 2 in 7 6
the quotient, multiply the
divisor by it, and subtract 1 5 2
the product from the 47; 152
and to the remainder, 9,
annex 1, the next figure of the dividend, making 91. We next inquire how many times 19 is contained in 91, place the number, 4, in the quotient, then multiply and subtract as before, and to the remainder, 15, annex 2, the last figure of the dividend, and, proceeding in like manner as before, after finding the quotient figure, there is no remainder. Hence the share of each of the 19 sons is 248 dollars. This illustration, except in omissions, is essentially like the preceding one.
51. After the quotient figure is found, what is the next thing you do? Where do you place the product? What do you next do? What is the next step? How do you then proceed? Is long division the same in prin. ciple as short division?
52. RULE. Write the divisor and dividend as in short division, and draw a curved line at the right hand of the dividend.
Then inquire how many times the divisor is contained in the fewest figures on the left hand of the dividend that will contain it, and write the result at the right hand of the dividend for the first quotient figure.
Multiply the divisor by the quotient figure, and subtract the product from the figures of the dividend used, and to the remainder annex the next figure of the dividend.
Find how many times the divisor is contained in the number thus formed ; write the figure denoting it at the right hand of the former quotient figure.
Thus proceed until all the figures of the dividend are divided.
Note 1.- If, when a figure is brought down, the number formed will not contain the divisor, a cipher must be placed in the quotient, and another figure of the dividend brought down, and so on until the number is large enough to contain the divisor.
NOTE 2. If there is a remainder after dividing all the figures of the dividend, it must be written as directed in the preceding rule. (Art. 49, Note 1.)
NOTE 3. — The proper remainder is in all cases less than the divisor. If, in the course of the operation, it is at any time found to be as large as, or larger than, the divisor, it will show that there is an error in the work, and that the quotient figure should be increased.
Note 4. — If, at any time, the divisor, multiplied by the quotient figure, produces a product larger than the part of the dividend used, it shows that the quotient figure is too large, and must be diminished.
53. Second Method of Proof. — Add together the remainder, if any, and all the products that have been produced by multiplying the divisor by the several quotient figures, and the result will be like the dividend, if the work is right.
54. Third Method. Subtract the remainder, if any, from the dividend, and divide the difference by the quotient. The result will be like the original divisor, if the work is right.
NOTE. - The first method of proof (Art. 50) is usually most convenient.
52. The rule for long division? How may you know when the quotient figure is too small? How may you know when it is too large ? 53. What is the second method of proof? - 54. What is the third method? Can long division be proved by the first method of proof (Azt. 50) ?