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. Now, as there are 4 sons to share equally this dollar, how much ought each son to have? A. †, or one quarter of a dollar apiece. In this expression, 4, we use the remainder, 1, and the divisor, 4: how, then, may division be carried out more exactly? By writing the divisor under the remainder with a line be tween. A. From these remarks and illustrations we derive the following RULE. I. How do you begin to divide? A. As in Short Division II. How many steps are there? A. Four. III. What are they? A. 1st. Find how many times; 2d. Multiply; 3d. Subtract; IV. Where do you write the quotient? A. At the right hand of the dividend. V. In performing the operation, whenever you have subtracted, what must the remainder be less than? A. Than the divisor. VI. 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.-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 and 20 cents. 3. There are 7 days in one week; there in 877 days? A. 1254 weeks. dollars, or 125 dollary how many weeks are 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; in divid ing them equally, how many will each have? A. 28 bushels 6. If a man is to travel 1201 miles in 12 months, how many is that a month? A. 100 miles. 7. If 1600 bushels of corn are to be divided equally among 40 men, how inany 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. 6880 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. A. 4000 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? beans. 12. For 36 boys? For 54 boys? A. 1250 beans. 13. Divide 29876543 by 13. 14. Divide 6283459 by 29. 15. Divide 37895429 by 112. 16. Divide 29070 by 15'; by 18. 17. Divide 29070 by 19; by 17. 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. 22981951 A. 768. A. 25203. ¶ XVIII. When the divisor is a composite number. 1. Bought 20 yards of cloth for 80 dollars; how much was that a yard? Now, as 2 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. 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. In ¶ 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. What will be the effect, if we cut off two ciphers? A. It will be the same as dividing by 100. Why does it have this effect? 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 I. What, then, 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. II. What are the figures cut off?. A. The remainder Exercises for the Slate. 1. A prize, valued at 25526 dollars, is to be equally divided among 100 men; what will be each man's part? OPERATION. 255 26 2552 dollars. 2. Divide 1786582 by 10000. A. 178-6582. 3. Divide 87653428 by 10; by 100; by 1000; by 10000; by 100000; 28 3428 by 1000000. A. Remainder to each, fo' foo 1000, 00001 53428 653428 100000 10000 Quotients, total, 9739257. ¶ XX. When there are ciphers at the right hand of the Divisor. 1. Divide 4960 OPERATION. dollars among 80 8 times 10 are 80)496'0 men. 62 dollars. In this example we have a divisor, 80, which is a composite number; (thus, 8 times 10 are 80 ;) how, then, may we proceed to divide by 10, one of the component parts? A. By cutting off one place at the right hand of the dividend, as in ¶ XIX. How do you obtain the 62? A. By dividing the 496 by 8, as usual. RULE I. As any number, which has a cipher or ciphers at the right, can be produced by two other numbers, one of which may be either 10, 100, 1000, &c., how, then, would you proceed to d vide when there are ciphers at the right of the divisor? A. Cut them off, and the same number of figures from the right of the dividend. II. How do you divide the remaining figures of the dividend? A. As usual. III. What is to be done with the figures of the dividend which are cut off? A. Bring them down to the right hand of the remainder. Exercises for the Slate. 2. How many oxen, at 30 dollars a head, may be bought for 38040 dollars? A. 1268. 3. Divide 783567 by 2100. A. rem. 95876 4. Divide 2082784895876 by 1200000. A. 1200000 rem. 5. Divide 7942851265321 by 12500000. A. 1265321 12500000 rem. 875 Miscellaneous Questions on the foregoing. Q. What is the subject which you have now been attending to called? 4. Arithmetic. Q. From what you have seen of it, how would you define it? A. It teaches the various methods of computing by numbers. Q. What rules have you now been through? A. Notation or Numeration, Addition, Subtraction, Multiplication, and Division. Q. How many rules do these make? Q. What are these rules sometimes called? A. The funda mental rules of arithmetic. Q. Why? A. Because they are the foundation of all the other rules. Q. To denote the operation of these different rules, we have certain characters; what is the name of these characters? A. Signs. Q. What do two horizontal straight lines signify; t 100 cents 1 dollar? A. Equal to: as, 100 cents = Ala, read, 100 cents are equal to 1 dollar. Q. What does a horizontal line crossing a perpendic you to do; thus, 6+10=16? A. To add: thus, 6+, read, 6 and 10 are 16. Q. What else does this sign denote? A. A remainder te dividing. 3? Q. What does one_horizontal straight line tell you to do, thus, 8-6-2: A. To subtract: thus, 8-6-2, read, 6 from 8 leaves 2. Q What do two lines, crossing each other in the form of the Roman letter X, tell you to do; thus, 6×8-48? A. Tomuitiply: thus, 6×8=48, read, 6 times 8 are 48. Q. What does a horizontal line, with a dot above and below it, tell you to do; thus, 8÷2=4? A. To divide; thus, 8÷2=4, read, 2 in 8, 4 times. Q. By consulting ¶ XVII. you will perceive that division may be represented in a different manner; how is this done? A. By writing the divisor under the dividend, with a line be tween them; thus, =2, read, 4 in 8, 2 times. Q. What does signify, then? 20 signify? 36? 42? 108? 144? 35? Let me see you write down on the slate the signs of Addition, Subtraction, Multiplication, and Division. Perform the following examples on the slate, as the signs indicate. 1. 87834+284+65+32+100-88315, Ans. 5. 2600-600-2000+1828-3828, Ans. 8. 198+36—18, Ansi 9.18836114+15+20=28, Ans. 10. What is the whole number of inhabitants in the world, there being, according to Hassel, in each grand division as follows ;-in Europe, one hundred and eighty millions; America, twenty-one millions; A. 682000000 What was the number of inhabitants in the following England towns in 1820, there being in dr one ortland, usuortsmouth, Salem, 8,581; Boston, 43,298, 7,327 Providence, 11,767; 12,731; New Haven, |