PROOF-Which terms do you multiply together to prove the operation? A. The divisor and quotient. What is to be done with the remainder, if there be any? A Add it to this product. What must the amount be like? A. The dividend. More Exercises for the Slate. 3. Rufis 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 le pay a head? A. 53 dollars. 7. How much flour, at 7 dollars a barrel, can be bought for 1512 dollars? 4. 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 jol 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. 13. Divide 256587 by 2. 14. Divide 378567 by 2; by 3. A. 128293-1 rem. A. 94155-3 rem. 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, there fore, 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 ? Long Division. OPERATION. Dividend. Quotient. Divisor, 4) 957 (239 8 15 12 37 36 Short Division. OPERATION. 4) 957 239+ Quotient. As Long and Short Division are exactly alike, except in Short Division the whole operation is not written down, to begin, then, in Short Division we should say, 4 in 9, 2 times, and 1 over. This we discover by saying in 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. We are next to say, 4 times 2 are 8: this 8, you know, must be subtracted from 3: where would it be convenient to place the 8? A. Under the 9. By taking 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. How do you get the 3 in the quotient? A. I say, 4 in 15, 3 times. How do you proceed next? A. I say, 3 times 4 are 12; and 12 from 15 leaves 3. What do you do with the 3? A. I bring down 7 of the dividend to the right hand of the 3, making 37. 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. 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? A. By writing the divisor under the remainder with a line be tween. 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; IV. Where do you write the quotient? A. At the right hand of the dividend. V. In performing the operation, whenever you have subtract ed, 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 (26 dollars equally among 5 men how many will that be apiece? A. 125 and 20 cents. dollars, or 125 dollar 3. There are 7 days in one week; how many weeks are there in 877 days? A. 125 weeks. 4. A man, having 5520 Lushels or 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. 23 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 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. 6848 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. A. 3240. I 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 yard will be but one half (3) 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 annering 1 cipher to any num ber multiplied ut 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. 11. What are the figures cut off?. 1. The remainder 1 III. What are the other figures? A. The quotient. 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|26 255 dollars. 2. Divide 1786582 by 10000. A 178-6582. 3. Divide 87653428 by 10; by 100; by 1000; by 10000; by 100000; 428 3428 10000. by 1000000. A. Remainder to each, fo' foo, Tooo, Tobo, BRARE, 85342 Quotients, total, 9739257. 53428 ¶ XX. When there are ciphers at the right hand c 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 di |