From the illustrations now given, we derive the following RULE. Q. At which hand of the dividend do you place the divisor? A. At the left. Q. How many figures do you take first? A. Enough to contain the divisor once, or more. Q. What do you set down underneath? A. The quotient. Q. If there should be a remainder, how would you proceed? A. I join or carry it to the next figure of the dividend, as so many tens. Q. For example, suppose 3 remain, and the next figure be 8, how would you say? A. I would say, 3 (to carry) to 8, makes 38. Q. How do you proceed if the divisor be not contained in the next figure of the dividend? A. Write a cipher in the quotient, and join this figure to the figure next to it, as so many tens. PROOF. Q. Which terms do you multiply together to prove the operation? A. The divisor and quotient. Q. What is to be done with the remainder, if there be any? Q. What must the amount be like? A. The dividend. More Exercises for the Slate. 2. Rufus divided 42 oranges equally between his two little brothers; how many had they apiece? A. 21. 3. If 3 bushels of apples cost 360 cents, how much is that a bushel? A. 120 cents. 4. How many months are there in 452 weeks, there being 4 weeks in each month? A. 113 months. 5. A man, having 416 dollars, laid them all out in cider, at 4 dollars a barrel; how much cider did he buy? A. 104 barrels. 6. A man bought 6 oxen for 318 dollars; how much did he pay a head? A. 53 dollars. 7. How much flour, at 7 dollars a barrel, can be bought for 1512 dollars? A. 216 barrels. 8. At 8 cents apiece, how many oranges will 8896 cents buy? A. 1112 oranges. 9. At 10 dollars a barrel, how many barrels of flour may be bought for 1720 dollars? A. 172 barrels. 10. 12 men by contract are to receive 1500 dollars for a job of work; how many dollars will be each man's part, if they be divided equally among them? A. 125 dollars. 11. 2 men, trading in company, gained 2450 dollars; how much was each man's part? A. 1225 dollars. 12. At 3 dollars a barrel, how many barrels of pork can be bought for 5463 dollars? A. 1821 bbls. Note. The total remainder is found by adding together what remains after each operation. A. 128293, 1 rem. A. 315472, 1 rem. 13. Divide 256587 by 2. 14. Divide 378567 by 2; by 3. 15. Divide 278934 by 2; by 3. 16. Divide 256788 by 3; by 4. 17. Divide 256788 by 5; by 6. 18. Divide 65342167 by 4; by 5. 19. Divide 65342167 by 6; by 7. 20. Divide 523467898 by 4; by 6. 21. Divide 523467898 by 7; by 8. 22. Divide 2653286 by 7; by 8. 23. Divide 2653286 by 9; by 10. 24. Divide 52345 by 9; by 10. 25. Divide 52345 by 11; by 12. Q. The operation, thus far, has been carried on partly in the mind, and partly by writing the numbers down; but oftentimes the divisor will be too large to be thus performed. When, therefore, we write the operation out at length, what is the process called? A. Long Division. LONG DIVISION. XVII. 1. A man, dying, left 957 dollars to be divided equally among his 4 sons; what was each son's part? the mind, 4 times 2 are 8, and 8 from 9 leaves 1. Now, to express in figures this operation, we may write the numbers where we please: where, then, for the sake of convenience, may the 2 (times the quotient figure) be placed? A. At the right hand of the dividend? Q. We are next to say, 4 times 2 are 8: this 8, you know, must be subtracted from 9: where would it be convenient to place the 8? A. Under the 9. Q. By taking 8 from 9, we have 1 remaining, which we should, in Short Division, carry or join to 5, the next figure of the dividend; how can we do this now? A. By joining or bringing down the 5 to the right hand of the 1, making 15. Q. How do you get the 3 in the quotient? A. I say, 4 in 15, 3 times. Q. How do you proceed next? A. I say, 3 times 4 are 12; and 12 from 15 leaves 3. Q. What do you do with the 3? A. I bring down 7 of the dividend to the right hand of the 3, making 37. Q. How do you get the 9 in the quotient? A. I say, 4 times 9 are 36, and subtracting 36 from 37 leaves 1, remainder. Q. It now appears that each son has 239 dollars. and there is 1 dollar still remaining undivided: to explain the division of this, tell me how many quarters there are in a dollar. A. Four. Q. Now, as there are 4 sons to share equally this dollar, how much ought each son to have? A. 4, or one quarter of a dollar apiece. Q. In this expression, 4, we use the remainder, 1, and the divisor, 4: how, then, may Division be carried out more exactly? A. By writing the divisor under the remainder, with a line between. From these remarks and illustrations we derive the following RULE. Q. How do you begin to divide ? Q. What are they? A. 1st. Find how many times; 2d. Multiply; 3d. Subtract; 4th. Bring down. Q. Where do you write the quotient ? A. At the right hand of the dividend. Q. In performing the operation, whenever you have subtracted, what must the remainder be less than A. Than the divisor. Q. When you have brought down a figure, and the divisor is not contained in the new dividend thus formed, what is to be done? A. Place a cipher in the quotient, and bring down another figure; after which divide as before. PROOF. Q. How do you prove the operations? A. As in Short Division. More Exercises for the Slate. 2. A man wishes to divide 626 dollars equally among 5 men; how many will that be apiece? A. 125 dollars, or 125 dollars and 20 cents. 3. There are 7 days in one week; how many weeks are there in 877 days? A. 125 weeks. 4. A man, having 5520 bushels of corn, wishes to put it into bins, each holding 16 bushels; how many bins will it take? A. 345 bins. 5. Four boys had gathered 113 bushels of walnuts; în dividing them equally, how many will each have? 6. If a man is to travel 1201 miles in 12 months, how many A. 28 bushels. is that a month? A. 100 miles. 7. If 1600 bushels of corn are to be divided equally among 40 men, how many is that apiece? A. 40 bushels. 8. 27000 dollars are to be divided equally among 30 soldiers; how many will each have? A. 900 dollars. 9. The salary of the president of the United States is 25000 dollars a year; how much is that a day, reckoning 365 days to the year? A. 68 dollars. 10. A regiment of soldiers, consisting of 500 men, are allowed 1000 pounds of pork per day; how much is each man's part? A. 2 pounds. 11. James says that he has a half bushel that holds 27000 beans; how many will that be apiece for 9 boys, if they be divided equally? How many apiece for 27 boys? A. 4000 beans. 12. For 36 boys? For 54 boys? A. 1250 beans. 13. Divide 29876543 by 13. A. 2298195 A. 216671. A. 338352 A. 3553. 4. 3240. 8 18. Divide 10368 by 27; by 36. 19. Divide 10368 by 54; by 18. 20. Divide 2688 by 112; by 224. 21. Divide 101442075 by 4025. A. 672. A. 25203. ¶ XVIII. WHEN THE DIVISOR IS A COMPOSITE NUMBER. Q. 1. Bought 20 yards of cloth for 80 dollars; how much was that a yard? Now, as times 10 are 20 (a composite number), it is plain that, if there had been but 10 yards, the cost of 1 yard would be 8 dollars, for 10 in 80, 8 times; but as there are 2 times 10 yards, it is evident that the cost of 1 yard will be but one half (1) as much: how much, then, will it be? RULE. Q. What, then, appears to be the rule for dividing by a composite number? A. Divide by one of its component parts first, and this quotient by the other. ¶ XIX. TO DIVIDE BY 10, 100, 1000, &c. Q. In T XII. it was observed, that annexing 1 cipher to any num ber multiplied it by 10, 2 ciphers by 100, &c. Now, Division being the reverse of Multiplication, what will be the effect, if we cut off a cipher at the right of any number? A. It must decrease or divide it by 10. Q. What will be the effect, if we cut off two ciphers? A. By cutting off one cipher or figure at the right, the tens take the units' place, and hundreds the tens' place, and so on. RULE. Q. What, tnen, is the rule for dividing by 10, 100, &c.? A. Cut off as many places or figures at the right hand of the dividend, as there are ciphers in the divisor. |