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Having written the numbers upon the Operation. slate, as in the margin, we proceed thus : 2 is contained in 6, 3 times. Now as the
2) 6 4 8 6 denotes hundreds, the 3 must also be hundreds. We therefore write it in hun
Quot. 3 2 4 dreds' place; that is, under the figure which we are dividing. 2 in 4, 2 times. Since the 4 is tens, the 2 must also be tens, and we write it in tens' place. 2 in 8, 4 times. The 8 is units; hence the 4 must be units, and we write it in units' place. The answer is 324 barrels. 2. Divide 63 by 7.
Ans. 9. 3. Divide 56 by 8.
4. Divide 42 by 7. 5. Divide 54 by 9.
6. Divide 72 by 8. 7: How many hats, at 2 dollars apiece, can be bought for 468 dollars ? Ans. 234 hats.
8. How many sheep, at 3 dollars a head, can be bought for 369 dollars ?
9. A man wished to divide 248 acres of land equally between his two sons: how many acres will each receive?
10. How many times is 4 contained in 488 ? 68. Hence, when the divisor contains but one figure,
Write the divisor on the left hand of the dividend with a curve line between them; then, beginning at the left hand, divide each figure of the dividend by the divisor, and set each quotient figure directly under the figure from which it
11. A farmer bought 96 dollars worth of dry goods, and agreed to pay in wood at 3 dollars a cord: how many cords will it lakie to pay liis bill? Ans. 32 cords.
12. In 963 feet, how many yards are there, allowing 3 feet to a yard ?
13. Divide 63936 by 3. 14. Divide 48848 by 4. 15. Divide 55555 by 5. 16. Divide 2486286 by 2. 69. When the divisor is not contained in the first
QUEST.-68. How do you write the numbers for division? Where legin to divide? Where place each quotient figure! 69. When the divisor is not contained in the first figure of the dividend, what must be done ?
figure of the dividend, we must find how many times it is contained in the first two figures.
17. How many hats, at 3 dollars apiece, can be bought for 249 dollars ? Operation Since the divisor (3) is not contained in 3)249
(2) the first figure of the dividend, we say,
3 is in 24, 8 times, and write the 8 under Quot. 83
the 4. 3 in 9, 3 times. Ans. 83 hats. 18. Divide 124 by 4. 19. Divide 366 by 6. 20. Divide 255 by 5. 21. Divide 1248 by 4. 22. Divide 24693 by 3. 23. Divide 4266 by 6. 24. Divide 35555 by 5. 25. Divide 5677 by 7. 26. Divide 64888 by 8. 27. Divide 8199 by 9.
70. After dividing any figure of the dividend, if there is a remainder, prefix it mentally to the next figure of the dividend, and then divide this number as before.
Obs. 1. To prefix means to place before or at the left hand.
28. A man bought 741 acres of land, which he divided equally among his 3 sons : how many acres did each receive? Operation.
When we divide 7 by 3, there is 1
remainder. This we prefix mentally 3)741
to the next figure of the dividend. Ans. 247 acres.
We then say, 3 in 14, 4 times, and 2 over. Prefixing the remainder 2 to the next figure, as before, we say, 3 in 21, 7 times.
29. If a man travel at the rate of 5 miles an hour, how long will it take him to travel 345 miles ? Ans. 69 hours.
30. If 192 pounds of flour were equally divided among 4 persons, how many pounds would each receive ?
31. Divide 45690 by 6. 32. Divide 52584 by 8. 33. Divide 81670 by 5. 34. Divide 28296 by 9.
35. When flour is 6 dollars a barrel, how much can be bought for 642 dollars ?
QUEST.-70. If there is a remainder after dividing a figure of the dividend, what must be done with it? Obs. What does the word prefix mean? When the divisor is not contained in a figure of the dividend, what must be done!
Obs. 2. In this example the divisor is not con- Operation. tained once in the tens' figure of the dividend; we 6)642 must therefore write a cipher in the quotient, and
Ans. 107 barrels. prefix the 4 to the next figure of the dividend, as if it were a remainder. We then say, 6 in 42, 7 times, and place the 7 under the 2.
36. Divide 36060 by 6. 37. Divide 49000 by 7. 38. Divide 43900 by 9. 39. Divide 568000 by 8.
40. Allowing 5 yards of cloth for a suit of clothes, how many suits can be made from 1525 yards ?
Ans. 305 suits. 41. A company of 3 men agree to pay a bill of 321 dollars : how many dollars must each man pay?
42. Divide 14350 by 7. 43. Divide 30420 by 6. 44. Divide 25105 by 5. 45. Divide 643240 by 8.
46. A merchant wished to divide 49 oranges equally among 4 boys : how many must he give to each ? Operation.
After giving them 12 apiece, it
will be seen that there is one re4)49 Ans. 12–1 remainder. is not divided.
mainder, or 1 orange left, which
Now it is plain that the whole dividend must be divided, in order to render the division complete. But 4 is not contained in 1; hence the division must be represented by writing the 4 under the 1, thus ), (Art. 67,) and in order to complete the quotient, the į must be annexed to the 12. The true quotient, therefore, is 12 and 1 divided by 4, and should be written thus, 124. Hence,
71. When there is a remainder, after dividing the last figure of the dividend, it should always be written over the divisor and annexed to the quotient.
47. A shoemaker has 375 pair of boots, which he wishes to pack in 6 boxes : how many pair can be put in a box?
Ans. 622 48. A baker wishes to lay out 756 dollars in flour : how much can he buy when the price is 5 dollars a barrel?
Quest.–71. When there is a remainder, after dividing the last figure of the dividend, what must be done with it?
49. How many yearlings, at 9 dollars a head, can be bought for 468 dollars ?
50. How many acres of land, at 6 dollars an acre, I buy for 973 dollars ?
72. The preceding method of dividing is called Short Division. From the illustrations and principles now explained, we derive the following
RULE FOR SHORT DIVISION.
1. Write the divisor on the left hand of the dividend with a curve line between them.
Then beginning at the left hand, divide successively each figure of the dividend by the divisor, and place each quotient figure directly under the figure divided. (Art. 68.)
II. If there is a remainder after dividing any figure, prefix 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. 69, 70.)
III. When a remainder occurs after dividing the last figure, write it over the divisor and annex it to the quotient. (Art. 71.)
73. 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. Take the last example:
Proof. 162 quotient. Solution.
6 divisor. 6)973
972 Quot. 162 and 1 rem.
Add the rem.
1 The result is 973, the div.
Quest.–72. What is the rule for short division ? 73. How is division proved? Obs. How does it appear that the product of the divisor and quotient should be equal to the dividend? What other way of proving division is mentioned ?
Obs. 1. Since the quotient shows how many times the divisor is contained in the dividend, (Art. 64,) it follows, that if the divisor is repeated as many times as there are units in the quotient, it must produce the dividend.
2. Division may also be proved by subtracting the remainder, if any, from the dividend, then dividing the result by the quotient.
PROOF OF MULTIPLICATION BY DIVISION.
74. Divide the product by one of the factors, and if the quotient thus arising is equal to the other factor, the work is right.
Obs. This method of proof depends on this obvious principle, viz: if the divisor and quotient, multiplied together, produce the dividend, the product of the two numbers, divided by one of those numbers, must give the other number.
Divisor. Divid. Quot.
Ex. 1. A father bought 741 acres of land, which he divided equally among his 3 sons : how many acres did each receive ?
Note. This example has been solved by short division. (Art. 70. Ex. 28.) We have introduced it here for the purpose of illustrating a different mode of dividing. Having written the divisor and divi.
Operation. dend as before, we find 3 is contained in 7, 2 times, and place the 2 on the
3) 741 (247 right of the dividend, with a curve line
6 between them. We next multiply the divisor by this quotient figure-2 times
14 3 are 6-and, placing the product under 12 7, the figure divided, subtract it there
21 from. We now bring down the next
21 figure of the dividend, and placing it on the right of the remainder 1, we have 14. Now 3 is in 14, 4 times. Set the 4 on the right of the last quotient
QUEST.-74. How is multiplication proved by division ? Obs. Upon what principle does this proof depend? How are the numbers written for long division? Where begin to divide ? Where is the quotient placed ?