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plying and subtracting as before, the remainder is 15. Bringing down the next figure, we have 153 to be divided by 435. But 435 is not contained in 153; we therefore place a cipher in the quotient, and bring down the next figure. Then 435 in 1534, 3 times. Place the 3 in the quotient, and proceed as before.

Note. -After the first quotient figure is obtained, for each figure of the dividend which is brought down, either a significant figure, or a cipher, must be put in the quotient. (Art. 117.)

120. From the preceding illustrations and principles we derive the following

GENERAL RULE FOR DIVISION.

I. When the divisor contains but one figure.

Write the divisor on the left of the dividend, with a curve line between them. Begin at the left hand, divide successively each figure of the dividend by the divisor, and place each quotient figure directly under the figure divided. (Arts. 116, 118. Obs. 1, 2.)

If there is a remainder after dividing any figure, prefic it to the next figure of the dividend and divide this number as before ; and if the divisor is not contained in any figure of the dividend, place a cipher in the quotient and prefix this figure to the next one of the dividend, as if it were a remainder. (Arts. 117, 118.)

II. When the divisor contains more than one figure.

Beginning on the left of the dividend, find how many times the divisor is contained in the fewest figures that will contain it, and place the quotient figure on the right of the dividend with a curve line between them. Then multiply the divisor by this figure and subtract the product from the figures divided ; to the right of the remainder bring down the next figure of the dividend and divide this number as before. Proceed in this manner till all the figures of the dividend are divided.

Quest.-120. How do you write the numbers for division? When the divisor contains but one figure, how proceed? Why place the divisor on the left of the dividend and the quotient under the figure divided ? When there is a remainder after dividing a figure, whai is to be done with it? When the divisor is not contained in any figure of the divi. dend. how proceed? Why? Why begin to divide at the left hand ? When the divisor contains more than one figure, how proceed?

Whenever there is a remainder after dividing the last figure, write it over the divisor and annex it to the quotient. (Art. 118.)

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Demonstration.—The principle on which the operations in Division depend, is that a part of the quotient is found, and the product of this part into the divisor is taken from the dividend, showing how much of the latter remains to be divided; then another part of the quotient is found, and its product into the divisor is taken from what remained before. Thus the operation proceeds till the whole of the dividend is divided, or till the remainder is less than t'e divisor. (Art. 113. Obs. 2.)

OBs. When the divisor is large, the pupil will find assistance in determining the quotient figure, by finding how many times the first figure of the divisor is contained in the first figure, or if necessary, the first two figures of the dividend. This will give pretty nearly the right figure. Some allowance must, however, be made for carrying from the product of the other figures of the divisor, to the product of the first into the quotient figure.

121. PROOF.Multiply the divisor by the quotient, to the product add the remainder, and if the sum is equal to the dividend, the work is right.

OBS. Since the quotient shows how many times the divisor is contained in the dividend, (Art. 111,) it follows, that if the divisor is repeated as many times as there are units in the quotient, it must produce the dividend. Ex. 8. Divide 256329 by 723. Operation.

Proof 723)256329(354711 Ans.

723 divisor. 2169

354 quotient. 3942

2892 3615

3615 3279

2169 2892

387 rem. 387 rem.

256329 dividend.

122. Second Method.–Subtract the remainder, if any, from the dividend, divide the dividend thus diminished, by the quotient; and if the result is equal to the given divisor, the work is right.

QUEST.-When there is a remainder after dividing the last figure of the dividend, what must be done with it? 121. How is division proved ? Obs. How does it appear that the product of the divisor and quotient will be equal to the dividend, if the work is right? Can division be proved by any other methods ?

123. Third Method.--First cast the 9s out of the divisor and quotient, and multiply the remainders together; to the product add the remainder, if any, after division ; cast the 9s out of this sum, and set down the excess; finally cast the 9s out of the dividend, and if the excess is the same as that obtained from the divisor and quotient, the work may be considered right.

Note. Since the divisor and quotient answer to the multiplier and multiplicand, and the dividend to the product, it is evident that the principle of casting out the 9s will apply to the proof of division, as well as that of multiplication. (Art. 90.)

124. Fourth Method.—Add the remainder and the respective products of the divisor into each quotient figure together, and if the sum is equal to the dividend, the work is right.

Note.—This mode of proof depends upon the principle that the whole of a quantity is equal to the sum of all its parts. (Ax. 11.)

125. Fifth Method.--First cast the 11s out of the divisor and quotient, and multiply the remainders together; to the product add the remainder, if any, after division, and casting the 11s out of this sum, set down the excess; finally, cast the 11s out of the dividend, and if the excess is the same as that obtained from the divisor and quotient, the work is right. (Art. 92. Note 2.)

EXAMPLES FOR PRACTICE.

127. Ex. 1. A farmer raised 2970 bushels of wheat on 66 acres of land : how many bushels did he raise per acre ?

2. A garrison consumed 8925 barrels of flour in 105 days : how much was that per day ?

3. The President of the United States receives a salary of 25000 dollars a year: how much is that per day ?

4. A drover paid 2685 dollars for 895 head of cattle : how much did he pay per head ?

5. If a man's expenses are 3560 dollars a year, how much are they per week ?

6. If the annual expenses of the government are 27 millions of dollars, how much will they be per day?

07. How long will it take a ship to sail from New York to Liverpool, allowing the distance to be 3000 miles, and the ship 10 sail 144 miles per day ?

8. Sailing at the same rate, how long would it take the same ship to sail round the globe, a distance of 25000 miles ?

10. 47839-42.
11. 75043:52.
12. 93840-63.
13. 421645; 74.
14. 325000; 85.
15. 400000-96.
16. 999999;47.
17. 352417--29.
18. 47981: 251.
19. 423405; 485,
20. 16512;344.
21. 304916--6274.
22. 12689-145.
23. 145260=1345.
24. 147735; 3283.

25. 1203033: 327.
26. 1912500-425.
27. 5184673-102.
28. 301140:-478.
29. 8893810:37846.
30. 9302688; 14356.
31. 9749320-365.
32. 3228242=5734.
33. 75843639426:8593.
34. 65358547823:- 2789.
35. 102030405060=123456.
36. 908070605040-654321.
37. 1000000000000000-111.
38. 1000000000000000; 1111.
39. 1000000000000000:11111.

CONTRACTIONS IN DIVISION. 128. The operations in division, as well as those in multiplication, may often be shortened by a careful attention to the application of the preceding principles.

CASE 1.- When the divisor is a composite number.
Ex. 1. A man divided 837 dollars equally among 27

persons, who belonged to 3 families, each family containing 9 persons : how many dollars did each

person

receive ? Analysis.-Since 27 persons received 837 dollars, each one must have received as many dollars, as 27 is contained times in 837. But as 27 (the number of persons), is a composite number whose factors are 3 (the number of families), and 9 (the number of persons in each family), it is obvious we may first find how many dollars each family received, and then how many each person received. Operation.

If 3 families received 837 3)837 whole sum divided. dollars, 1 family must have

9)279 portion of each Fam. received as many dollars, as Ans. 31

3 is contained times in 837 ; and 3 in 837,279 times. That is, each family received 279 dollars.

person.

Again, if 9 persons, (the number in each family,) received 279 dollars, 1 person must have received as many dollars, as 9 is contained times in 279; and 9 in 279, 31 times. Ans. 31 dollars.

PROOF.—31X27=837, the same as the dividend. Hence, 129. To divide by a composite number.

I. Divide the dividend by one of the factors of the divisor, then divide the quotient thus obtained by another factor; and so on till all the factors are employed. The last quotient will be the answer.

II. To find the true remainder.

If the divisor is resolved into but two factors, multiply the last remainder by the first divisor, to the product add the first remainder, if any, and the result will be the true remainder.

When more than two factors are employed, multiply each remainder by all the preceding divisors, to the sum of their products, add the first remainder, and the result will be the true remainder.

Obs. 1. The true remainder may also be found by multiplying the quotient by the divisor, and subtracting the product from the dividend.

2. This contraction is exactly the reverse of that in multiplication. (Art. 97.) The result will evidently be the same, in whatever order the factors are taken.

2. A man bought a quantity of clover seed amounting to 507 pints, which he wished to divide into parcels containing 64 pints each : how many parcels can he make ? Note.-Since 64=2X8X4, we divide by the factors respectively. Operation.

2)507
8)253—1 rem.

1 pt.
4)31–5 rem. Now 5X2 = 10 pts.

7-3 rem. and 3 X 8 X2 = 48 pts. Ans. 7 parcels, and 59 pts. over. 59 pts. True Rem.

Demonstration.—1. Dividing 507 the number of pints, by 2, gives 253 for the quotient, or distributes the seed into 253 equal parcels, leaving 1 pint over. Now the units of this quotient are evidently of a different value from those of the given dividend; for since there are but half as many parcels as at first, it

GUEST.–129. How proceed when the divisor is a composite number? How find the true remainder?

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